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Philosophic Foundations of Quantum Mechanics
Philosophic Foundations
of Quantum Mechanics
By HANS REICHENBACH
PROFESSOR OF PHILOSOPHY IN THE UNIVERSITY OF CALIFORNIA
UNIVERSITY OF CALIFORNIA PRESS
BERKELEY AND LOS ANGELES 1944
UNIVERSITY OF CALIFORNIA PRESS
BERKELEY AND LOS ANGELES
CALIFORNIA
CAVJRIDGE UNIVERSITY PRESS
nf'onDON, ENGLAND
COPYRIGHT, 1Q44, BY
THE REGENTS Ofr THE UNIVERSITY OF CALIFORNIA
PRINTED IN THE UNITED STATES OF AMERICA
BY THE UNIVERSITY OP CALIFORNIA PRESS
PREFACE
Two GREAT theoretical constructions have shaped the face of modern
physics: the theory of relativity and the theory of quanta. The first has
been, on the whole, the discovery of one man, since the work of Albert
Einstein has remained unparalleled by the contributions of others who, like
Hendrik Anton Lorentz, came very close to the foundations of special rela-
tivity, or, like Hermann Minkowski, determined the geometrical form of the
theory. It is different with the theory of quanta. This theory has been de-
veloped by the collaboration of a number of men each of whom has contributed
an essential part, and each of whom, in his work, has made use of the results
of others.
^'The necessity of such teamwork seems to be deeply rooted in the subject
matter of quantum theory. In the first place, the development of this theory
has been greatly dependent on the production of observational results and on
the exactness of the numerical valuta nhtfl.inftfi. Without the help of the army
of experimenters who photographed spectral lines or watched the behavior of
elementary particles by means of ingenious devices, it would have been im-
possible ever to carry through the theory of the quanta even after its founda-
tions had been laid. In the second place, these foundations are very different in
logical form from those of the theory of relativity. They have never had the
form of one unifying principle, not even after the theory has been completed.
They consist of a set of principles which, despite their mathematical elegance,
do not possess the suggestive character of a principle which convinces us at first
sight, as does the principle of relativity. And, finally, they depart much further
from the principles of classical physics than the theory of relativity ever did in
its criticism of space and time; their implications include, in addition to a
transition from causal laws to probability laws, a revision of philosophical
ideas about the existence of unobserved objects, even of the principles of logic,
and reach down to the deepest fundamentals of the theory of knowledge.
In the development of the theoretical form of quantum physics, we can dis-
tinguish four phases. The first phase is associated with the names of Max
Planck, Albert Einstein, and Nils Bohr. Planfik'p intrndnnf.jop nf f.h^ qnonfn in
1900 was followed by Einstein's extension of the quantum conr^pt toward that
of a needle radiation (1905). The decisive step, however, was made IP Rnhr'a
application (1913^ of thelmanfom idea .to the analvsHof the structure of the
atgin^which led to a new world of physical discoveries.
The second phase, which began in 1925, represents the work of a younger
generation which had been trained in the physics of Planck, Einstein, and
Bohr, and started where the older ones had stopped. It, is a most astonishing
fact that this phase, which led up to quantum nr>eftlmrnfg T bepan without a clear
insight into what was actually bein^ done. Louis de Broglie introduced waves
a!T"companions of particles; Erwin Schrodinger, guided by mathematical
VI PREFACE
analogies with wave optics, discovered the^two fundamental differential equa-
tions of quantum mechanicsTMax Born, Werner Heisenberg, Pascual Jordan,
and, independently nf thfo grnnp, f^ul A. M. Dirac constructed the matrix
mechanics which seemed to defy any wave interpretation. Tj^is period repre-
sents an amazing triumph of mathematical technique which, masterly applied
and guided by a physical instinct more than, by IngipaJ principles l determined
the path to the discovery o f q ^ifif^iV w *"fh wgg Q Hft t-fl embrace all observable
3afta. All this was done in a very short time; by 1926 the mathematical shape
of the new theory had become clear.
The third phase followed immediately: it consisted in the physical interpre-
tation of the results obtained. Schrodinger showed the identity of wq.ve me-
chanics and matrix mechanics. Born recognized the probability interpretation
of the waves. Heisenberp; saw that the m^P^niliipflil mnfr TFiisni of the theory
involves an unsurmountable uncertainty of predictions and a disturbance of
the object by the measurement. And here once more Bohr^ntervenc^M.n tlie
\flork of theyoungergeneration anJT showe J that the"description of nature
given by the theory was to leave open a specific ambiguity which Jiejormu-
lated in his principle of complementarity .
The fourth phase continues up to the present day; it is filled with constant
extensions of the results obtained toward further and further applications,
including the application to new experimental results. These extensions are
combined with mathematical refinements ; in particular, the adaptation of the
mathematical method to the postulates of relativity is in the foreground of the
investigations. We shall not speak of these problems here, since our inquiry is
concerned with the logical foundations of the theory.
It was with the phase of the physical interpretations that the novelty of the
logical form of quantum mechanics was realized. Something had Been achieved
in this new theory which was Contrary to traditional coTipppts nf fc n 9wledge
and reality. It was not easy, however, to say what had happened, i.e., to
proceed to the philosophical interpretation of the theory. Based on the physical
interpretations given, a philosophy for common use was developed by the
physicists which spoke of the relation of subject and object, of pictures of
reality which must remain vague and unsatisfactory, of operationalism which
is satisfied when observational predictions are correctly made, and renounces
interpretations as unnecessary ballast. Such concepts may appear useful for
the purpose of carrying on the merely technical work of the physicist. But it
seems to us that the physicist, whenever he tried to be conscious of what he
did, could not help feeling a little uneasy with this philosophy. He then became
aware that he was walking, so to speak, on the thin ice of a superficially frozen
lake, and he realized that he might slip and break through at any moment.
It was this feeling of uneasiness which led the author to attempt a philo-
sophical analysis of the foundations of quantum mechanics. Fully aware that
philosophy should not try to construct physical results, nor try to prevent
PREFACE Vll
physicists from finding such results, he nonetheless believed that a logical
analysis of physics which did not use vague concepts and unfair excuses was
possible. The philosophy of physics should be as neat and clear as physics
itself; it should not take refuge in conceptions of speculative philosophy which
must appear outmoded in the age of empiricism, nor use the operational form
of empiricism as a way to evade problems of the logic of interpretations. Di-
rected by this principle the author has tried in the present book to develop a
philosophical interpretation of quantum physics which is free from meta-
physics, and yet allows us to consider quantum mechanical results as state-
ments about an atomic world as real as the ordinary physical world.
It scarcely will appear necessary to emphasize that this philosophical analysis
is carried through in deepest admiration of the work of the physicists, and that
it does not pretend to interfere with the method of physical inquiry. All that
isjntended in this book iff fllflj-ifWfinn nf pnrmfipf.fi ; nowhere in this presenta-
tion, therefore, is any contribution toward the solution of physical problems
to be expected. Whereas physics consists in the analysis of the physical world,
philosophy consists in the analysis of our knowledge of the physical world.
The present book is meant to be philosophical in this sense.
The division of the book is so planned that the first part presents the jgeneral
ideas on which quantum mechanics is based; this part, therefore, outlines our
and siiT^ni^lJ^^s^tg.zesviltfi. The presentation is
such that it does not presuppose mathematical knowledge, nor an acquaintance
with the methods of quantum physics. In the second part we present the out-
lines of the mathematical [ methods_of quantum mechanics: this is so written
that a knowledge of the calculus should enable the reader to understand the
exposition. Since we possess today a number of excellent textbooks on quantum
mechanics, such an exposition may appear unnecessary; we give it, however, in
order to open a short cut toward the mathematical foundations of quantum
mechanics for all those who do not have the time for thorough studies of the
subject, or who would like to see in a short review the methods which they
have applied in many individual problems. Our presentation, of course, makes
no claim to be complete. The third part deals with the various interpretations
of quantum mechanics; it is here that we make use of both the philosophical
ideas ot the tirst part and the mathematical formulations of the second. The
properties of the different interpretations are discussed, and an interpretation
in terms of a three-valued logic is constructed which appears as a satisfactory
logical form of quantum mechanics.
I am greatly indebted to Dr. Valentin Bargmann of the Institute of Ad-
vanced Studies in Princeton for his advice in mathematical and physical
questions; numerous improvements in the presentation, in Part II in particular,
are due to his suggestions. I wish to thank Dr. Norman C. Dalkey of the Uni-
versity of California, Los Angeles, and Dr. Ernest H. Hutten, formerly at Los
Angeles, now in the University of Chicago, for the opportunity of discussing
Vlll PREFACE
with them questions of a logical nature, and for their assistance in matters of
style and terminology. Finally I wish to thank the staff of the University of
California Press for the care and consideration with which they have edited
my book and for their liberality in following my wishes concerning some
deviations from established usage in punctuation.
A presentation of the views developed in this book, including an exposition
of the system of three-valued logic introduced in 32, was given by the author
at the Unity of Science Meeting in the University of Chicago on September
5, 1941.
HANS REICHENBACH
Department of Philosophy,
University of California,
Los Angeles
June, 1942
CONTENTS
PART I: GENERAL CONSIDERATIONS
PAGE
1. Causal laws and probability la*ws 1
2. The probability distributions 5
3. The principle of indeterminacy 9
4. The disturbance of the object by the observation 14
5. The determination of unobserved objects 17
6. Waves and corpuscles 20
7. Analysis of an interference experiment 24
8. Exhaustive and restrictive interpretations 32
PART II: OUTLINES OF THE MATHEMATICS OF
QUANTUM MECHANICS
9. Expansion of a function in terms of an orthogonal set 45
10. Geometrical interpretation in the function space 52
11. Reversion and iteration of transformations 58
12. Functions of several variables and the configuration space 64
13. Derivation of Schrodinger's equation from de Broglie's principle 66
14. Operators, eigen-functions, and eigen-values of physical entities ... 72
15. The commutation rule 76
16. Operator matrices 78
17. Determination of the probability distributions 81
18. Time dependence of the ^-function 85
19. Transformation to other state functions 90
20. Observational determination of the ^-function 91
21. Mathematical theory of measurement 95
22. The rules of probability and the disturbance by the measurement . 100
23. The nature of probabilities and of statistical assemblages in
quantum mechanics 105
x CONTENTS
PART III : INTERPRETATIONS
24. Comparison of classical and quantum mechanical statistics Ill
25. The corpuscle interpretation 1 18
26. The impossibility of a chain structure 122
27. The wave interpretation . . 129
28. Observational language and quantum mechanical language '. . 136
29. Interpretation by a restricted meaning 139
30. Interpretation through a three-valued logic 144
31. The rules of two-valued logic 148
32. The rules of three-valued logic 150
33. Suppression of causal anomalies through a three- valued logic 160
34. Indeterminacy in the object language 166
35. The limitation of measurability 169
36. Correlated systems 170
37. Conclusion 176
Index 179
Parti
GENERAL CONSIDERATIONS
1. Causal Laws and Probability Laws
The philosophical problems of quantum mechanics are centered around two
main issues. The first concerns the transition from causal laws to probability
laws^Jhe second concerns thejnterpretation of unobserved objects. We begin
with the discussion of the first issue, and shall enter into the analysis of the
second in later sections.
The question of replacing causal laws by statistical laws made its appearance
in the history of physics long before the times of the theory of quanta. Since
the time of Boltzmann's great discovery which revealed the second principle
of thermodynamics jbo be a statistical instead of a causal law, the opinion has
be"enTFepeatedly uttered that a similar late may meet all other physical laws.
The idea of determinism, i.e., ofjtrifit^causal laws governing the elementary
phflfinnnftim nf nfl/hirp, was recognized as an extrapolation inferred from the
causal regularities of the macrocosm. The validity of this extrapolation was
questioned as soon as it turned out tha/Lmacrocosmic regularity is equally
compatible with irregularity in the microcosmic domain, sii^ce the law of great
numbers will transform the probability character of the elementary phenomena
into the_practical certainty of stafcfttJcallawft. Observations in the macrocosmic
domain will never furnish any evidence for causality of atomic occurrences so
long as only effects of great numbers of atomic particle^" are considered. This
was the result of unprejudiced philosophical analysis of the physics of Boltz-
mann. 1
With this result a decision of the question was postponed until it was possible
to observe macrocosmic effects of individual atomic phenomena. Even with
the use of observations of this kind, however, the question is not easily
answered, but requires the development of a more profound logical analysis.
Whenever we speak of strictly causal laws we assume them to hold between
idealized physical states; and we know that actual physical states never cor-
1 It is scarcely possible to say who was the first to formulate this philosophical idea.
\Ve have no published utterances of Boltzmann indicating that he thought of the possi-
bility of abandoning the principle of causality. In the decade preceding the formulation
of quantum mechanics the idea was often discussed. F. Exner, in his book, Vorlesungen
uber die physikalischen Grundlagen der Naturwissenschaften (Vienna, 1919), is perhaps the
first to have clearly stated the criticism which we gave above: "Let us not forget that
the principle of causality and the need for causality has been suggested to us exclusively
by experiences with macrocosmic phenomena and that a transference of the principle to
, microcosmic phenomena, viz. the assumption that every individual occurrence be strictly
i, has no longer any justification based on experience." p. 691. Witn
; causally determined, has no longer any justification based on experience,
reference to Exner, E. Schrodinger has expressed similar ide ' ~ ' '
in Zurich, 1922, published in Naturwissenschaften, 17:9 (1929).
CO
2 PART I. GENERAL CONSIDERATIONS
respond exactly to the conditions assumed for the laws. This discrepancy has
often been disregarded as irrelevant, as befog due to the imperfection of the
experimenter and therefore negligible in a statement about causality as a propi
erty of nature. With such an attitude, however, the way to a solution of the
problem of causality is barred. Statements about the physical world have
meaning only so far a. they arfi CQiffleqtefl wfoh verifiable results ; and a state-
ment about strict causality must be translatable into statements about observ-
able relations if it is to have a utilizable meaning. Following this principle we
can interpret the statement of causality in the following way.
If we characterize physical states in observational terms, i.e., in terms of
observations as they are actually made, we know that we can construct prob-
ability relations between these states. For instance, if we know the inclination
of the barrel of a gun, the powder charge, and the weight of the shell, we can
predict the point of impact with a certain probability. Let A be the so-defined
initial conditions and B a description of the point of impact; then we have a
probability implication A -=>- B (V\
which states that if A is given, B will happen with a determinatejprnbability p.
From this empirically verifiable relation we pass to an ideal relation by con-
sidering ideal states A f and B r and stating a logical implication
A' D B f (2)
between them, which represents a law of mechanics. Since we know, however,
that from the observational state A we can infer only with some probability
the existence of the ideal state A', and that similarly we have only a probability
relation between B and B r , the logical implication (2) cannot be utilized. It
derives its physical meaning only from the fact that in all cases of applications
to observable phenomena it can be replaced by the probability implication (1).
What then is the meaning of a statement saying that if we kne^exactly. jiha
initial conditions we could predict with certainty the Juto^^tat^-PE^auJ^^
from them? Such a statement can be meaningfully said only in the s$nse olji
* Instead of characterizing the initial conditions of shooting
only by the mentioned three parameters, the inclination of the barrel, the
powder charge, and the weight of the shell, we can consider further parameters,
such as the resistance of the air, the rotation of the earth, etc. As a consequence,
the predicted value will change; but we know that with such a more precise
characterization also the probability of the prediction increases. From expe-
riences of this kind we have inferred that the probability p caq be madeto
tihft Ygjue 1 as closely as we want bv the introductionjoL^ctEer
' nfrk +-hfi analysis of physical states. It is in this form that we must
state the principle of causality if it is to have physical meaning. The statement
that nature is governed by strict causal laws means that we can predict the
future with a determinate probability and that we can push this probability as
1. CAUSALITY AND PROBABILITY 3
close to certainty as we want by using a sufficiently elaborate analysis of the
phenomena under consideration.
With this formulation the principle of causality is stripped of its disguise as
a principle a priori, in which it has been presented within many a philosophical
system. If causality is stated as a limit of probability implications, it .is clear
that this principle can be maintained only in the sense of an empirical hypoth-
esig. There is, logically, no need for saying that the probability of predictions
can be made to approach certainty by the introduction of more and more
parameters. In this form the possibility of a limit of predictability was recog-
nized even before quantum mechanics led to the assertion of such a limit. 2
The objection has been raised that we can know only a finite number of
parameters, and that therefore we must leave open the possibility of discover-
ing, at a later time, new parameters which lead to better predictions. Although,
of course, we have no means of excluding with certainty such a possibility, we
must answer that there may be strong inductive evidence against such an
assumption, and that such evidence will be regarded as given if continued
attempts at finding new para/mp^m frave failed. Physical laws, like the law of
conservation of energy, have been based on evidence derived from repeated
failures of attempts to prove the contrary. If the existence of causal laws is
denied, this assertion will always be grounded only in inductive evidence. The
critics of the belief in causality will not commit the mistake of their adversaries,
and will not try to adduce a supposed evidence a priori for their contentions.
The quantum mechanical criticism of causality must therefore be considered
as the logical continuation of a line of development which began with the intro-
duction of statistical laws into physics within the kinetic theory of gases, and
was continued in the empiricist analysis of the concept of causality. The
specific form, however, in which this criticism finally was presented through
Heisenberg's principle of indeterminacy was different from the form of the
criticism so far explained.
In the preceding analysis we have assumed that it is possible to measure the
independent parameters of physical occurrences as exactly as we wish; or more
precisely, to measure the simultaneous values of these parameters as exactly
as we wish. The breakdown of causality then consists in the fact that these
values do not strictly determine the values of dependent entities, including the
values of the same parameters at later times. Our analysis therefore contains
an assumption of the measurement of simultaneous values of independent
parameters. It is this assumption which Heisenberg has shown to be wrong.
The laws of classical physics are throughout temporally directed laws, i.e.,
laws stating dependences of entities at different times and which thus establish
causal lines extending in the direction of time. If simultaneous values of differ-
2 Cf. the author's "Die Kausalstruktur der Welt/' Ber. d. Bayer. Akad., Math. Kl.
(Munich, 1925), p. 138: and his paper, "Die Kausalbehauptung und die Mpglichkeit ihrer
empirischen Nachprinung," which was written in 1923 and published in Erkenntnis 3
(1932), p. 32.
4 PART I. GENERAL CONSIDERATIONS
ent entities are regarded as dependent on one another, this dependence is
always construed as derivable from temporally^directed laws. Thus the cor-
respondence of various indicators of a physical state is reduced to the influence
of the same physical cause acting on the instruments. If, for instance, barom-
eters in different rooms of a house always show the same indication, we explain
this correspondence as due to the effect of the same mass of air on the instru-
ments, i.e., as due to the effect of a common cause. It is possible, however, to
assume the existence of cross-section laws, i.e., laws which directly connect
simultaneous values of physical entities without being reducible to the -effects
of common causes. It is such a cross-section law which Heisenberg has stated
in his relation of indeterminacy.
This cross-section law has the form of a limitation of measurability. It states
that the simultaneous values of the independent parameters cannot be meas-
ured as exactly as we wish. We can measure only one half of all the parameters
to a desired degree of exactness; the other half then must remain inexactly
known. There exists a coupling of simultaneously measurable values such that
greater exactness in the determination of one half of the totality involves less
exactness in the determination of the other half, and vice versa. This law does
not make half of the parameters functions of the others; if one half is known,
the other half remains entirely unknown unless it is measured. We know,
however, that this measurement is restricted to a certain exactness.
This cross-section law leads to a specific version of the criticism of causality.
If the values of the independent parameters are inexactly known, we cannot
expect to be able to make strict predictions of future observations. We then
can establish' only statistical laws for these observations. The idea that there
are causal laws "behind" these statistical laws, which determine exactly the
results of future observations, is then destined to remain an unverifiable state-
ment; its verification is excluded by a physical law, the cross-section law
mentioned. According to the verifiability theory of meaning, which has been
generally accepted for the interpretation of physics, the statement that there
are causal laws therefore must be considered as physically meaningless. It is an
empty assertion which cannot be converted into relations between observa-
tional data.
There is only one way left in which a physically meaningful statement about
causality can be made. If statements of causal relations between the exact
values of certain entities cannot be verified, we can try to introduce them at
least in the form of conventions or definitions; that is, we may try to establish
arbitrarily causal relations between the strict values. This means that we can
attempt to assign definite values to the unmeasured, or not exactly measured,
entities in such a way that the observed results appear as the causal conse-
quences of the values introduced by our assumption. If this were possible, the
causal relations introduced could not be used for an improvement of predic-
tions; they could be used only after observations had been made in the sense
2. PROBABILITY DISTRIBUTIONS 5
of a causal construction post hoc. Even if we wish to follow such a procedure,
however, we must answer the question of whether such a causal supplementa-
tion of observable data by interpolation of unobserved values can be consistently
done. Although the interpolation is based on conventions, the answer to the
latter question is not a matter of convention, but depends on the structure of
the physical world. Heisenberg's principle of indeterminacy, therefore, leads
to a revision of the statement of causality; if this statement is to be physically
meaningful, it must be made as an assertion about a possible causal supple-
mentation of the observational world.
With these considerations the plan of the following inquiry is made clear.
We shall first explain Hcisenberg's principle, showing its nature as a cross-
section law, and discuss the reasons why it must be regarded as being well
founded on empirical evidence. We then shall turn to the question of the inter-
polation of unobserved values by definitions. We shall show that the question
stated above is to be answered negatively; that the relations of quantum
mechanics are so constructed that they do not admit of a causal supplementa-
tion by interpolation. With these results the principle of causality is shown to
be in no sense compatible with quantum physics; causal determinism holds
neither in the form of a verifiable statement, nor in the form of a convention
directing a possible interpolation of unobserved values between verifiable data.
2. The Probability Distributions
Let us analyze more closely the structure of causal laws by means of an
example taken from classical mechanics and then turn to the modification of
this structure produced by the introduction of probability considerations.
In classical physics the physical state of a free mass particle which has no
rotation, or whose rotation can be neglected, is determined if we know the
position q, the velocity v, and the mass m of the particle. The values q and v, of
course, must be corresponding values, i.e., they must be observed at the same
time. Instead of the velocity v, the momentum p = m v can be used. The
future states of the mass particle, if it is not submitted to any forces, is then
determined; the velocity, and with it, the momentum, will remain constant,
and the position q can be calculated for every time t. If external forces inter-
vene, we can also determine the future states of the particle if these forces are
mathematically known.
If we consider the fact that p and q cannot be exactly determined, we must
replace strict statements about p and q by probability statements. We then
introduce probability distributions
d(q) and d(p) (1)
which coordinate to every value q and to every value p a probability that this
value will occur. The symbol d( ) is used here in the general meaning of distri-
PART I. GENERAL CONSIDERATIONS
button of; the ｅxpressions d(q) and d(p) denote, therefore, different mathemati-
cal functions. As usual, the probability given by the function is coordinated,
not to a sharp value q or p, but to a small interval dq or dp such that only the
ｅxpressions (2)
represent probabilities, whereas the functions (1) are probability densities.
This can also be stated in the form that the integrals
f q 'd(q)dq and / ** d(p)dp ' (3)
J qi J pi
represent the probabilities of finding a value of q between qi and g 2 , or a value
of p between pi and p 2 .
Probability distributions can be determined only for sets of measurements,
not for an individual measurement. When we speak of the exactness of a
measurement we therefore mean, more precisely, the exactness of a type of
measurement made in a certain type of physical system. In this sense we can
say that every measurement ends with the determination of probability func-
tions d. Usually d is a Gauss function, i.e., a bell-shaped curve following an
exponential law (cf. figure 1) ; the steeper this curve, the more precise is the
measurement. In classical physics we make the assumption that each of these
curves can be made as steep as we want, if only we take sufficient care in the
elaboration of the measurement. In quantum mechanics this assumption is
discarded for the following reasons.
Whereas, in classical physics, we consider the two curves d(q) and d(p) as
independent of each other, quantum mechanics introduces the rule that they
are not. This is the cross-section law mentioned in 1 . The idea is expressed
through a mathematical principle which determines both curves d(q) and d(p),
at a given time t, as derivable from a mathematical function \f/(q) ; the deriva-
tion is so given that a certain logical connection between the shapes of the
curves d(q) and d(p) follows. This contraction of the two probability distri-
butions into one function ^ is one of the basic principles of quantum mechanics.
It turns out that the connection between the distributions established by the
principle has such a structure that if one of the curves is very steep, the other
must be rather flat. Physically speaking, this means that measurements of p
and q cannot be made independently and that an arrangement which permits
a precise determination of q must make any determination of p unprecise,
and vice versa.
The function \[/(q) has the character of a wave; it is even a complex wave,
i.e., a wave determined by complex numbers ^. Historically speaking, the intro-
duction of this wave by L. de Broglie and Schrodinger goes back to the strug-
gle between the wave interpretation and the corpuscle interpretation in the
theory of light. The ^-function is the last offspring of generations of wave
concepts stemming from Huygens's wave theory of light; but Huygens would
2. PROBABILITY DISTRIBUTIONS 7
scarcely recognize his ideas in the form which they have assumed today in
Born's probability interpretation of the ^-function. Let us put aside for the
present the discussion of the physical nature of this wave; we shall be con-
cerned with this important question in later sections of our inquiry. In the
present section we shall consider the ^-waves merely as a mathematical instru-
ment used to determine probability distributions; i.e., we shall restrict our
presentation to show the way in which the probability distributions d(q) and
d(p) can be derived from \fs(q).
Thfc derivation which we are going to explain coordinates to a curve $(q) at
a given time the curves d(q) and d(p) ; this is the reason that t does not enter
into the following equations. If, at a later time,
\l/(q) should have a different shape, different func-
tions d(q) and d(p) would ensue. Thus, in gen-
eral, we have functions ^(g,)> d(q,f), and d(p,t).
We omit the t for the sake of convenience.
The derivation will be formulated in two rules,
the first determining d(q), and the second de-
termining d(p). We shall state these rules here
only for the simple case of free particles. The
extension to more complicated mechanical sys- d(q)
terns will be given later (17). We present first t
the rule for the determination of d(q).
Rule of the squared \f/-f unction: The probability
of observing a value q is determined by the square C. 1 ^ - ., .
J * * . Fig. 1. Curve 1-1-1 represents
of the "({/-function according to the relation a precise measurement, curve
2-2-2 a less precise measure-
d(q) = | $(q) | 2 (4) ment of q. Both curves are Gauss
distributions, or normal curves.
The explanation of the rule for the determina-
tion of d(p) requires some introductory mathematical remarks. According to
Fourier a wave of any shape can be considered as the superposition of many
individual waves having the form of sine curves. This is well known from
sound waves, where the individual waves are called fundamental tone and
overtoneSj or harmonics. In optics the individual waves are called monochro-
matic waves, and their totality is called the spectrum. The individual wave
is characterized by its frequency v, or its wave length X, these two charac-
teristics being connected by the relation v X = w, where w is the velocity
of the waves. In addition, every individual wave has an amplitude a- which
does not depend on q, but is a constant for the whole individual wave. The
general mathematical form of the Fourier expansion is explained in 9; for
the purposes of the present part it is not necessary to introduce the mathe-
matical way of writing.
The Fourier superposition can be applied to the wave ^, although this wave
is considered by us, at present, not as a physical entity, but merely as a mathe-
8 PART I. GENERAL CONSIDERATIONS
matical instrument. In case the wave \l/ consists of periodic oscillations ex-
tended over a certain time, such as in the case of sound waves produced by
musical instruments, the spectrum furnished by the Fourier expansion is
discrete. Thus the individual waves of musical instruments have the wave
lengths A, -,-,-,- where A is the wave length of the fundamental tone
and the other values represent the harmonics. In case the wave ^ consists of
only one simple impact moving along the g-axis, i.e., in case the function \f/ is
not periodic, the Fourier expansion furnishes a continuous spectrum, i.e., the
frequencies of the individual waves constitute, not a discrete, but a continuous
set. As before, each of these individual waves possesses an amplitude cr, which
can be written <r(X), since it depends on the wave length X but is independent
of q.
It is the amplitudes o-(X) which are connected with the momentum. We
shall not try to explain here the trend of thought which led to this connection
and which is associated with the names of Planck, Einstein, and L. de Broglie.
Such an exposition may be postponed to a later section (13). Let us suppress
therefore any question of why this connection holds true, and let us rely,
instead, upon the authority of the physicist who says that this is the case.
Suffice it to say, therefore, that every wave of the length X is coordinated to a
momentum of the amount ,
h /e , x
P = - (5)
A
where h is Planck's constant. The probability of finding a momentum p then
is connected with the amplitude cr belonging to the coordinated wave X. This is
expressed in the following rule. 1
Rule of spectral decomposition: The probability of observing a value p is deter-
mined by the square of the amplitude o-(X) occurring within the spectral decompo-
sition of\l/(q), in the form ^
(6)
The factor results from the relation between p and X expressed in (5). 2
A 3
The two rules show clearly the connection which the ^-function establishes
between the two distributions d(q) and d(p), so far as it reduces these two
distributions to one root. We shall later show that this kind of connection is not
1 The name "principle of spectral decomposition" has been introduced by L. de Broglie,
Introduction a I' Etude de la Mecanique ondulatoire (Paris, 1930), p. 151. In his later book,
La Mecanique ondulatoire (Paris, 1939), p. 47, he uses also the name "principle of Born,"
since this principle was introduced by Born. For the rule of the squared ^-function he
uses the name "principle of interference" and in his later book the name "principle of
localization".
2 Mathematically speaking, this factor corresponds to a density function r as introduced
in (22), 9. The third power in h originates from the fact that we assume the waves
to be three-dimensional.
3- PRINCIPLE OF INDETERMINACY 9
restricted to the simple case of one mass particle, and that the same logical
pattern is established by quantum mechanics for the analysis of all physical
situations. For every physical situation there exists a ^-function, and the prob-
ability distributions of the entities involved are determined by two rules of the
kind described. This is one of the basic principles of quantum mechanics. We
shall now construct the implications of this principle, returning once more to
the simple case of the mass particle.
3. The Principle of Indeterminacy
It can be shown that the derivation of the two distributions d(q) and d(p)
from a function \l/ leads immediately to the principle of indeterminacy. Let us
consider a particle moving in a straight line, and let us assume that the func-
Fig. 2
respectively^ v or p
Fig. 3
Fig. 2. Distribution of the position q, in the form of a Gauss curve.
Fig. 3. The dotted line indicates the direct Fourier expansion of the
curve of fig. 2. The solid line is constructed through the Fourier expan-
sion of a ^-function from which the curve d(q) of fig. 2 is derivable, and
represents the distribution d(p) of the momentum, coordinated to d(q).
tion \f/ is practically equal to zero except for a certain interval along the line.
The function \$(q) | 2 , i.e., the function d(q), then will have the same property;
let us assume that it is a Gauss curve such as is shown in figure 2. The shape
of the curve means that we do not know the location of the particle exactly;
with practical certainty it is within the interval where the curve is noticeably
different from zero, but for a given place within this interval we know only
with a determinate probability that the particle is there. Our diagram, of
course, represents the situation only for a given time t; for a later time, when
the particle has moved to the right, we shall have a similar curve, but it will be
shifted to the right. 1
1 The curve will also gradually change its[ form. This, however, is irrelevant for the
present discussion.
1O PART I. GENERAL CONSIDERATIONS
Now let us apply the principle of spectral decomposition. This decomposi-
tion, it is true, is to be applied to the complex function ^(g), and not to the real
function \\l/(q) | 2 of our diagram. For the study of the mathematical relations
in the decomposition, however, we shall first apply it to the real function of
the diagram; the results can then be transferred to the complex case.
The Fourier expansion constructs the curve of figure 2 out of an infinite set
of individual waves. Each of these individual waves is a pure harmonic wave
of infinite length; i.e., its oscillations have a sine form and extend alofig the
whole infinite line. Their amplitudes differ, however. The maximal amplitude
will be associated with a certain mean frequency j> ; for frequencies greater
and smaller than VQ the amplitude will be smaller, and outside a certain range
on each side of VQ the amplitude of the individual waves will be practically
zero. Let us call the range within which the amplitudes have a noticeable size,
the practical range. A bunch of harmonic waves of the kind described is also
called a wave packet, since the superposition of all these harmonic waves results
in one packet of the form given in figure 2.
Now it is one of the theorems of Fourier analysis that the practical range of
a packet of harmonic waves is great when the curve of figure 2 is rather steep,
but is small when this curve is flat. We can illustrate this by a diagram when
we draw as abscissae the frequencies v, and as ordinates the corresponding
amplitudes s(v) of the harmonic analysis, such as is done for the dotted line
in figure 3. We choose the notation s(v) for these amplitudes of the Fourier
expansion of the real function d(q), or \^(q) | 2 , in order to distinguish them
from the amplitudes a(v) of the Fourier expansion of the complex function
\fs(q). In our case, -since we had assumed the curve d(q) to be a Gauss curve,
the curve s(v) is also a Gauss curve, but of a much flatter shape, 2 as shown by
the dotted -line curve of figure 3. This illustrates the theorem mentioned ; it can
be proved generally that the steeper the curve of figure 2, the flatter will be the
dotted-line curve of figure 3, and vice versa. We therefore have an inverse
correlation between the shape of the original curve and the shape of the curve
expressing its harmonic analysis. We call this the law of inverse correlation of
harmonic analysis.
An instructive illustration of this law is found in some problems of radio
transmission. If the wave of a radio transmitter does not carry any sound, it is
a pure sine wave of a sharply defined frequency. If it is modulated, however,
i.e., if its amplitude varies according to the intensity of impressed sound
waves, it no longer represents one sharp frequency, but a spectrum of fre-
quencies varying continuously within a certain range. This range is given by
the highest pitch of the sound frequencies. Consequently, a receiver with a
2 The maximum of the s(v) is at the place v = 0, and the curve is symmetrical for positive
and negative frequencies v. This results from the fact that we have assumed the curve
d(q) in the form of a Gauss curve. For the curve d(p) there exists no such restriction,
because the function \fs(q) is not determined by d(q), but left widely arbitrary; the maxi-
mum of d(p) may therefore be situated at any value v = j> , or correspondingly, at any
value p.
3- PRINCIPLE OF INDETERMINACY H
sharp resonance system will pick out only a narrow domain of the transmitted
waves; it will therefore drop the higher sound frequencies and reproduce trans-
mitted music in a distorted form. on the other hand, if the resonance curve of
the receiver is sufficiently flat for a high fidelity reproduction of music, the
receiver will not sufficiently separate two radio stations transmitting adjacent
wave lengths. The principle of inverse correlation is here expressed in the fact
that it is impossible to unite high fidelity and high selectivity in the same
adjustment of the receiver.
Thp application of these considerations to the determination of the proba-
bility distribution of the momentum of the particle involves some complica-
tions, which, however, in principle, do not change the result. As explained
above, the spectral decomposition which furnishes the momenta must be
applied, not to the probability curve d(q) of figure 2, but to a complex ^-curve
from which this curve is derivable by the rule of the squared ^-function. The
complex amplitudes a(v) of the resulting harmonic waves then must be squared
according to the rule of spectral decomposition. It is only by means of this
detour through the complex domain that we arrive at the probability distribu-
tion d(p) of the momentum as shown in the solid line of figure 3. Like the
dotted line, this curve is also a rather flat Gauss distribution, although not
quite so flat. But it can be shown that the law of inverse correlation holds as
well for the two curves d(q) and d(p). We therefore shall speak of the law of
inverse correlation of the probability distributions of momentum and position.
More precisely, this inverse correlation is to be understood as follows. If only
the curve d(q) is given, the curve d(p) is not determined; it can have various
forms depending on the shape of the function \l/(q) from which d(q) is derived.
But there is a limit to the steepness of d(p), represented by the solid line of
figure 3. Should d(q) be derived from another ^-function than that assumed
for the diagram, the resulting curve d(p) can only be flatter. This general
theorem can be stated without a further discussion of the question concerning
the practical determination of the function \l/(q) in a given physical situation.
The answer to the latter question, which requires the mathematical apparatus
of quantum mechanics, must be postponed to a later section ( 20).
We said that the results obtained for the mass particle can be extended to
all physical situations. Now the difference between q and p is transferred to
situations in general as a difference between kinematic and dynamic parameters.
We therefore shall speak generally of the law of inverse correlation of kinematic
and dynamic parameters. It is in the form of this law of inverse correlation that
we express the principle of indeterminacy. The universality of this principle
follows from the fact that, whatever be the physical situation, our observa-
tional knowledge of it is summarized in a ^-function.
The law of inverse correlation can be extended to the two parameters lime
and energy. This extension can be made clear as follows. A measurement of
time is analogous to a determination of position. When we speak of the position
1 2 PART I. GENERAL CONSIDERATIONS
q of a particle we mean the position at a given time t; inversely we can ask for
the time t at which the particle will be at a given space point q. This value t
will be determinable only with a certain probability, and we can therefore
introduce a probability distribution d(f) analogous to d(q). Similarly we can
introduce a probability function d(H) stating the probability that the particle
will have a certain energy H . H is connected with the frequency of the har-
monics by the Planck relation IT _ /, n \
corresponding to (5), 2, and the probability d(H) is therefore determined
by the principle of spectral decomposition. Hence the two curves d(f) and d(H)
are subject to the principle of inverse correlation in the same way as the curves
d(q) and d(p). We shall therefore include time in the category of kinematic
parameters, and energy in that of the dynamic parameters. The general law of
inverse correlation of kinematic and dynamic parameters then includes the
inverse correlation of time and energy.
This general law can be formulated somewhat differently when we use the
concept of standard deviation. Let g represent the mean value of g, that is, the
value of the abscissa for which the curve d(q) reaches its maximum; and let Ag
be the standard deviation. Then the area between the curve and the axis of
abscissae is divided by the ordinates g Ag and g + Ag (drawn in figure 2)
2
in such a way that approximately - of the whole area is situated between these
o
two ordinates. It is shown in the theory of probability that this ratio is inde-
pendent of the shape of the Gauss curve. The probability of finding a value q
2
within the interval' g Ag is therefore approximately = -. Because of these
o
properties, the quantity Ag represents a measure of the steepness of the Gauss
curve, and is therefore used as an ｅxpression characterizing the exactness of
the distribution of measurements. If the standard deviation is small, the meas-
urements are exact; if it is great, the measurements are inexact. In figure 2 and
figure 3 the standard deviations Ag and Ap belonging, respectively, to the
curves d(q) and d(p) are indicated on the axis. Now it can be shown that for
the case of curves of this kind, which are derived from the same ^-function,
the relation ,
Ag - Ap ^ - (2)
47T
holds where h is the Planck constant. For time and energy we have the cor-
responding relation : ,
At- AH 2:- (3)
4?r
The inequalities (2) and (3) represent the form in which the relation of inde-
terminacy has been established by Heisenberg. (2) expresses the inverse cor-
g. PRINCIPLE OF INDETERMINACY ig
relation of measurements of position and momentum by stating that a small
standard deviation in q implies a great standard deviation in p, and vice versa.
(3) states the corresponding relation between At and A#. These relations show
at the same time the significance of the constant h. Since h has a very small
value, the indeterminacy will be visible only in observations within the micro-
cosmic domain; there, however, the indeterminacy cannot be neglected. The
case of classical physics corresponds to the assumption that h = 0.
Relation (2) can be interpreted in the form: When the position of a particle
is well determined, the momentum is not sharply determined, and vice versa.
(3) can be interpreted in a similar way. This form makes it clear that the cross-
section law of inverse correlation between kinematic and dynamic parameters
states a limitation of measurability .
We now are in a position to answer the question raised above about the
legitimacy of this cross-section law. If the basic principles of quantum me-
chanics are correct, the principle of indeterminacy must hold because it is a
logical consequence of these basic principles. Furthermore, it must hold for all
physical situations, because it is derivable directly from the rules of the squared
^-function and of spectral decomposition, without reference to any special
form of the ^-function. The issue of legitimacy is reduced with this to the valid-
ity of the basic principles of quantum mechanics. Now these principles are, of
course, empirical principles, and no physicist claims absolute truth for them.
But what can be claimed for them is the truth of a well-established theory.
Since it is a consequence of the limitation of measurability that all relations
between observational data are restricted to statistical relations, we can there-
fore say: With the same right with which the physicist maintains any one of his
fundamental theorems, he is entitled to assert a limitation of predictability. We
may add that the same limitation follows for the determination of past data in
terms of given observations, and that we therefore must also speak of a
limitation ofpostdictability.
It has sometimes been said that quantum mechanics possesses a mathe-
matical proof of the limitation of predictability. Such a statement can reason-
ably be meant only in the sense that there is a mathematical proof deriving
the statement of the limitation from the basic principles of quantum me-
chanics. The principle of indeterminacy is an empirical statement; all that can
be said mathematically in its favor is that it is supported by the very evidence
on which the basic principles of quantum mechanics are founded. This is,
however, very strong evidence.
We occasionally meet with the objection that the laws of quantum me-
chanics, perhaps, hold only for a certain kind of parameter; that at a later
stage of science other parameters may be found for which the relation of uncer-
tainty does not hold; and that the new parameters may enable us to make
strict predictions. Logically speaking, such a possibility cannot be denied. It
then might be possible, for instance, to combine a measurement of the new
14 PARTI, GENERAL CONSIDERATIONS
parameters with a measurement of the kinematic parameters in such a way
that the results of measurements of the dynamic parameters could be pre-
dicted. The law of inverse correlation between kinematic and dynamic param-
eters then still would hold for the old parameters so long as the new ones were
not used; but when the new observables were to be applied for the selection of
types of physical systems, the law of inverse correlation would no longer hold
within the assemblages so constructed, even for the old parameters, and there-
fore their values could be strictly predicted. This would mean, in other words,
that we could empirically define types of physical systems for which the sta-
tistical relations controlling their parameters were not expressible in terms of
^-functions. 3
In such a case quantum mechanics would be considered as a statistical part
of science imbedded in a universal science of causal character. Although, as we
said, we cannot adduce logical reasons excluding such a further development
of physics, and, although some eminent physicists believe in such a possibility,
we cannot find much empirical evidence for such an assumption. If a physical
principle embracing all known entities has been established, it seems plausible
to assume that it holds universally, and that there is no unknown class of
physical entities which do not conform to this principle. Such an inductive
inference from all known entities to all entities has always been considered
legitimate. The principle of describing all physical situations in terms of
^-functions is a well-established principle, and though, certainly, quantum
mechanics is still confronted by many unsolved problems and may experience
important improvements, nothing indicates that the principle of the ^-func-
tion will be abandoned. Since the relation of uncertainty and the limitation of
predictability follow directly from the principle of the ^-function, these theo-
rems must be regarded as being as well founded in their universal claims as all
other general theorems of physics.
4. The Disturbance of the Object by the Observation
We now turn to considerations involving the second main issue confronting
the philosophy of quantum mechanics the issue of the interpretation of un-
observed objects. This question finds a first answer in the statement that the
object is disturbed by the means of observation. Heisenberg, who recognized
this feature in combination with his discovery of the principle of indetermi-
8 We use the term "expressible in terms of ^-functions" in order to include both the pure
case and the mixture; cf. 23. In his book, Mathematische Grundlagen der Quantenmechanik
(Berlin, 1932), p. 160-173, J. v. Neumann has given a proof that no "hidden parameters' '
can exist. But this proof is based on the assumption that for all kinds of statistical assem-
blages the laws of quantum mechanics, expressed in terms of ^-functions, are valid. If
the indeterminism of quantum mechanics is criticized, this assumption will be equally
questioned. J. v. Neumann's proof therefore cannot exclude the case to which we refer
in the text. It shows only that the assumption of hidden parameters is not compatible
with a universal validity of quantum mechanics.
4- DISTURBANCE OF THE OBJECT 15
nacy, used it as an explanation of the latter principle; he maintained that the
indeterminacy of all measurements is a consequence of the disturbance by the
means of observation.
This statement has aroused a wave of philosophical speculation. Some
philosophers, and some physicists as well, have interpreted Heisenberg's
statement as the confirmation, in terms of physics, of traditional philosophical *
ideas concerning the influence of the perceiving subject on its percepts. They
have iterated this idea by seeing in Heisenberg's principle a statement that the
subject cannot be strictly separated from the external world and that the line
of demarcation between subject and object can only be arbitrarily set up; or
that the subject creates the object in the act of perception; or that the object
seen is only a thing of appearance, whereas the thing in itself forever escapes
human knowledge; or that the things of nature must be transformed according
to certain conditions before they can enter into human consciousness, etc. We
cannot admit that any version of such a philosophical mysticism has a basis
in quantum mechanics. Like all other parts of physics, quantum mechanics
deals with nothing but relations between physical things; all its statements
can be made without reference to an observer. The disturbance by the means
of observation which is certainly one of the basic facts asserted in quantum
mechanics is an entirely physical affair which does not include any reference
to effects emanating from human beings as observers.
This is made clear by the following consideration. We can replace the observ-
ing person by physical devices, such as photoelectric cells, etc., which register
the observations and present them as data written on a strip of paper. The act
of observation then consists in reading the numbers and signs written on the
paper. Since the interaction between the reading eye and the paper is a macro-
cosmic occurrence, the disturbance by the observation can be neglected for this
process. It follows that all that can be said about the disturbance by the means
of observation must be inferable from the linguistic ｅxpressions on the paper
strip, and must therefore be statable in terms of physical devices and their
interrelations. Quantum mechanics should not be misused for attempts to
revive philosophical speculations which are not on a level with the clarity and
precision of the language of physics. The solution of its philosophical problems
can only be given within a scientific philosophy such as has been developed in
the analysis of science and in symbolic logic.
There was a similar period in the discussion of Einstein's theory of relativity
in which the relativity of time and motion was ascribed to the subjectivity of
the observer. Later analysis has shown that the dependence of statements about
space and time on the system of reference is in no way connected with the
privacy of every person's sense data, but represents the ｅxpression of arbitrary
definitions involved in every description of the physical world. We shall see
that a similar solution can be given to the problems of quantum mechanics,
although the situation there is even more complicated than in the case of the
l6 PART I. GENERAL CONSIDERATIONS
theory of relativity. The difference is that on the arbitrariness of definitions
is superimposed, in quantum mechanics, an uncertainty in the prediction of
observable results, a feature which has no analogue in the theory of relativity.
We must begin our analysis with a revision of Heisenberg's statement that
the uncertainty of predictions is a consequence of the disturbance by the means
of observation. We do not think that the statement is correct in this form,
although it is true that there is a disturbance by the observation and that there
is a logical connection between this principle and the principle of indetermi-
nacy. This connection should rather be stated inversely, namely, in the form
that the principle of indeterminacy implies the statement of a disturbance of
objects by the means of observation.
To say that the indeterminacy of predictions originates from the disturbance
by the instruments of observation means that whenever there is a non-
negligible disturbance by observation there will always be a limitation of pre-
dictability. A consideration of classical physics shows that this is not true.
There are many cases in classical physics where the influence of the instrument
of measurement cannot be neglected, and where, nevertheless, exact predic-
tions are possible. Such cases are dealt with by the establishment of a physical
theory which includes a theory of the instrument of measurement. When we
put a thermometer into a glass of water we know that the temperature of the
water will be changed by the introduction of the thermometer; therefore we
cannot interpret the reading taken from the thermometer as giving the water
temperature before the measurement, but must consider this reading as an
observation from which we can determine the original temperature of the
water only by means of inferences. These inferences can be made when we
include in them a theory of the thermometer.
Why is it not possible to apply this logical procedure to the case of quantum
mechanics? Heisenberg has shown that for a precise determination of the
position of a particle we need light waves of a very short wave length, that is,
waves which carry rather large quanta of energy and which change the velocity
of the particle by their impacts, with the consequence that this velocity cannot
be measured by the same experiment. If, on the other hand, we wish to deter-
mine the velocity of a particle, we must use rather long wave lengths in order
not to change the velocity to be measured; but then we shall not be able to
ascertain precisely the position of the particle. If, however, the observation of
a particle by illuminating it with a light ray produces an impact which throws
the particle off its path, why can we not construct a theory which tells us by
means of inferences starting from the result of the observation what the
original velocity of the particle was? It is here that the cross-section law of
Heisenberg intervenes. This principle states that whatever be the observational
results, the corresponding distributions of position and momentum must be
derivable from a ^-function, and therefore must be inversely correlated. Thus,
a measurement of position involves physical processes of such a kind that, rela-
5. UNOBSERVED OBJECTS 17
tive to the observational effects of these processes, the velocity distribution is
a rather flat curve. This is the reason that we cannot determine exactly the
velocity of the particle in the experiment mentioned. The relation between
disturbance through observation and indeterminacy must therefore be stated
as follows: The disturbance by the observation is the reason that the deter-
mination of the physical entity considered is not immediately given with the
measurement, but requires inferences using physical laws; since these infer-
ences are bound to the use of a ^-function, they are limited by the principle of
indeterminacy, and therefore it is impossible to come to an exact determina-
tion. This formulation makes clear that the disturbance by the observation, in
itself, does not lead to the indeterminacy of the observation. It does so only in
combination with the principle of indeterminacy. 1
In view of such objections Heisenberg's principle has sometimes been formu-
lated as meaning : We have no exact knowledge of physical states because the
observation disturbs in an unpredictable way. In this form the statement is
correct; but then it can no longer be interpreted as substantiating the principle
of indeterminacy. It states this principle, but does not give a reason for it. And
we recognize that the "disturbance in an unpredictable way" is but a special
case of a general cross-section law of nature stating the inverse correlation of
all physical data available. The instrument of measurement disturbs, not
because it is an instrument used by human observers, but because it is a phys-
ical thing like all other physical things. Instruments of measurement do not
represent exceptions to physical laws; the general limitation of inferences
leading to simultaneous values of parameters includes the case of inferences
referring to the effects of instruments of measurement. It is in this form that
we must state the principle of the disturbance by the means of observation.
This is, however, only the first step in our analysis of the disturbance by the
observation. We have so far taken it for granted that we know what we mean
by saying that the observation disturbs the object. In order to come to a deeper
understanding of the relations involved here, we must first construct a precise
formulation of this statement.
5. The Determination of Unobserved Objects
When we say that the object is disturbed by the observation, or that the un-
observed object is different from the observed object, we must have some
knowledge of the unobserved object; otherwise our statement would be un-
justifiable. Before we can enter into an analysis of the particular situation in
1 A mathematical proof that the disturbance of the object by the observation does not
entail the principle of indeterminacy will be given later (p. 104). The idea that it is not
the disturbance in itself which leaas to the indeterminacy was first expressed by the
author in "Ziele und Wege der physikalischen Erkenntnis," Handbuch der Physik, Vol. IV
(ed. by Geiger-Scheel, Berlin, 1929), p. 78. The same idea has also been expressed by E.
Zilsel, Erkenntnis 5 (1935), p. 59. The precise formulation of the principle of uncertainty
requires a qualification which will be explained in 30.
1 8 PART I. GENERAL CONSIDERATIONS
quantum mechanics we must therefore discuss in general the problem of our
knowledge of unobserved things. How do things look when we do not look at
them? This is the question to which we must find an answer.
It has sometimes been said that this problem is specific for quantum me-
chanics, whereas for classical physics there is no such problem. This is, how-
ever, a misunderstanding of the nature of the problem. Even in classical physics
we meet with the problem of the nature of unobserved things; and only after
giving a correct treatment of this problem on classical grounds shall we be able
to answer the corresponding question for quantum mechanics. The logical
methods by which the answer is to be formulated are the same in both cases.
To begin our analysis with an example, let us assume we look at a tree, and
then turn our head away. How do we know that the tree remains in its place
when we do not look at it? It would not help us to answer that we can easily
turn our head forward and thus "verify" that the tree did not disappear.
What we thus verify is only that the tree is always there when we look at it;
but this does not exclude the possibility that it always disappears when we do
not look at it, if only it reappears when we turn our head toward it. We could
make an assumption of the latter kind. According to this assumption the obser-
vation produces a certain change of the object in such a way that there
appears to be no change. We have no means to prove that this assumption is
false. If it is suggested that another person may observe the tree when we do
not see it and thus confirm the statement that the tree does not disappear, we
may restrict our assumption to cases in which no person looks at the tree, thus
ascribing the power of reproducing the tree to the observation of any human
being. If it is suggested that we may derive the existence of the tree from cer-
tain effects remaining observable even when we do not see the tree, such as the
shadow of the tree, we may answer that we can assume a change in the laws of
optics such that there is a shadow although there is no tree. The argument
therefore proves only that an assumption concerning the existence, or dis-
appearance, of the unobserved object is to be connected with an assumption
concerning the laws of nature in both cases.
It would be a mistake to say that there is inductive evidence for the assump-
tion that the tree does not disappear when we do not see it, and that this
assumption is at least highly probable. There is no such inductive evidence.
We cannot say: "We have so often found the unobserved tree to be unchanged
that we assume this to hold always". The premise of this inductive inference
is not true, since, in fact, we never have seen an unobserved tree. What we
have often seen is that when we turned our head to the tree it was there; from
this set of facts we can inductively infer that the tree will always be there when
we look at it, but there is no inductive inference leading from these facts to
statements about the unobserved tree. We therefore cannot even say that the
unchanged existence of the unobserved object is at least probable.
We are inclined to discard considerations of the given kind as "nonsense",
5- UNOBSERVED OBJECTS 1Q
because it seems so obvious that the tree is not created by the observation.
Such an answer, however, does not meet the problem. The correct answer
requires deeper analysis.
We must say that there is more than one true description of unobserved
objects, that there is a class of equivalent descriptions, and that all these de-
scriptions can be used equally well. The number of these descriptions is not
limited. Thus we can easily introduce an assumption according to which the
tree splits into two trees every time we do not look at it; this is permissible if
only We change the optics of unobserved things in a corresponding way, such
that the two trees produce only one shadow. on the other hand, we see that
not all descriptions are true. Thus it is false to say that there are two unob-
served trees, and that the laws of ordinary optics hold for them. It follows that
the statements about unobserved things are to be made in a rather complicated
way. Descriptions of unobserved things must be divided into admissible and
inadmissible descriptions; each admissible description can be called true, and
each inadmissible description must be called false. Looking for general features
of unobserved things, we must not try to find the true description, but must
consider the whole class of admissible descriptions; it is in properties of this
class as a whole that the nature of unobserved things is expressed.
In the case of classical physics this class contains one description which
satisfies the following two principles :
1) The laws of nature are the same whether or not the objects are observed.
2) The state of the objects is the same whether or not the objects are observed.
Let us call this descriptional system the normal system. It is this system which
we usually consider as the "true" system. We see that this interpretation is
incorrect. We may, however, make the following statement. In case a class of
descriptions contains a normal system, each of the descriptions is equivalent
to the normal system. If we now consider one of the unreasonable descriptions
of the class, such as the statement that a tree splits into two trees whenever it
is not observed, we see that these anomalies are harmless. They result from the
use of a different language, whereas the description as a whole says the same
as the normal system. This is the reason that we can select the normal system
as the only description to be used.
The convention that the normal system be used is always tacitly assumed in
the language of daily life when we speak of inductive evidence for or against
changes of unobserved objects. This convention is understood when we say
that our house remains in its place as long as we are absent; and the same con-
vention is understood when we say that the girl is not in the box of the magi-
cian while he is sawing the box into two pieces, although we have seen the girl
in it before. It is because of the use of this convention that ordinary statements
about unobserved things are testable. The same convention is used in scientific
language; it simplifies the language considerably. We must, however, realize
SO PART I. GENERAL CONSIDERATIONS
that this choice of language has the character of a definition and that the sim-
plicity of the normal system does not make this system "more true" than the
others. We are concerned here only with what has been called a difference in
descriptive simplicity, 1 such as we find in the case of the metrical system as
compared with the yard-inch system.
In stating that a class of descriptions includes a normal system, we make a
statement about the whole class. This way of stating a property of the class by
means of a statement about the existence of a normal system may be illustrated
by an example from differential geometry. Properties of curvature are statable
in terms of systems of coordinates and their properties. Thus the surface of the
sphere can be characterized by the statement that it is not possible to intro-
duce on it a system of orthogonal straight-line coordinates which covers large
areas. only for an infinitesimal area is this possible; i.e., for small areas it is
possible to introduce approximately orthogonal straight-line coordinates, and
the degree of approximation increases for smaller areas. For the plane, how-
ever, such a system covering the whole plane can be introduced. It is not
necessary, though, to use this "normal system" of coordinates for the plane,
since any kind of curved coordinates can be used equally well; but the fact that
there is such a normal system distinguishes the class of possible systems of
coordinates holding for the plane from the corresponding class holding for a
curved surface.
Similar considerations have been developed for Einstein's theory of rela-
tivity, which is the classical domain of application for the theory of classes of
equivalent descriptions. Every system of reference, including systems in
different states of motion, furnishes a complete description, and we have there-
fore in the class of systems of reference a class of equivalent descriptions. If the
class of such systems includes one for which the laws of special relativity hold,
we say that the considered space does not possess a "real" gravitational field.
This is true, although we can introduce in such a world unreasonable systems
which contain pseudogravitational fields; they are pseudogravitational be-
cause they can be "transformed away". 2
6. Waves and Corpuscle^
Turning from these general considerations to quantum mechanics, we first
must clarify what is to be meant by observable and by unobservable occur-
rences. Using the word "observable" in the strict epistemological sense, we
must say that none of the quantum mechanical occurrences is observable; they
are all inferred from macrocosmic data which constitute the only basis acces-
sible to observation by human sense organs. There is, however, a class of
1 Cf. the author's Experience and Prediction (Chicago, 1938), 42. Descriptive simplicity
is distinguished from inductive simplicity; the latter involves predictional differences.
2 Cf. the author's Philosophic der Raum-Zeit-Lehre (Berlin, 1928), p. 271.
6. WAVES AND CORPUSCLES 1
occurrences which are so easily inferable from macrocosmic data that they may
be considered as observable in a wider sense. We mean all those occurrences
which consist in coincidences, such as coincidences between electrons, or elec-
trons and protons, etc. We shall call occurrences of this kind phenomena. The
phenomena are connected with macrocosmic occurrences by rather short
causal chains; we therefore say that they can be "directly" verified by such
devices as the Geiger counter, a photographic film, a Wilson cloud chamber,
etc.
We 'then shall consider as unobservable all those occurrences which happen
between the coincidences, such as the movement of an electron, or of a light
ray from its source to a collision with matter. We call this class of occurrences
the interphenomena. Occurrences of this kind are introduced by inferential
chains of a much more complicated sort; they are constructed in the form of an
interpolation within the world of phenomena, and we can therefore consider
the distinction between phenomena and interphenomena as the quantum me-
chanical analogue of the distinction between observed and unobserved things.
The determination of phenomena is practically unambiguous. Speaking more
precisely, this means that in the inferences leading from macrocosmic data to
phenomena we use only the laws of classical physics; the phenomena are
therefore determinate in the same sense as the unobserved objects of classical
physics. Putting aside as irrelevant for our purposes the problem of the un-
observed things of classical physics, we therefore can consider the phenomena
as verifiable occurrences. It is different with the interphenomena. The intro-
duction of the interphenomena can only be given within the frame of quantum
mechanical laws; it is in this connection that the principle of indeterminacy
leads to some ambiguities which find their ｅxpression in the duality of waves
and corpuscles.
The history of the theories of light and matter since the time of Newton and
Huygens shows a continuous struggle between the interpretation by cor-
puscles and the interpretation by waves. Toward the end of the nineteenth
century this struggle had reached a phase in which it seemed practically set-
tled; light and other kinds of electromagnetic radiation were regarded as
consisting of waves, whereas matter was assumed to consist of corpuscles. It
was Planck's theory of quanta which, in its further development, conferred a
serious shock to this conception. In his theory of needle radiation Einstein
showed that light rays behave in many respects like particles; later L. de
Broglie and Schrodinger developed ideas according to which material particles
inversely are accompanied by waves. The wave nature of electrons then was
demonstrated by Davisson and Germer in an experiment of a type which, a
dozen or so years before, had been made by M. v. Laue with respect to X-rays,
and which had been considered at that time as the definitive proof that
X-rays do not consist of particles. With these results the struggle between the
conceptions of waves and corpuscles seemed to be revived, and once more
22 PART I. GENERAL CONSIDERATIONS
physics seemed to be confronted by the dilemma of two contradictory concep-
tions each of which seemed to be equally demonstrable. one sort of experiment
seemed to require the wave interpretation, another the corpuscle interpreta-
tion; and in spite of the apparent inconsistency of the two interpretations,
physicists displayed a certain skill in applying sometimes the one, sometimes
the other, with the fortunate result that there was never any disagreement with
facts so far as verifiable data were concerned.
An attempt to reconcile the two interpretations was made by Born who
introduced the assumption that the waves do not represent fields of a kind of
matter spread through space, but that they constitute only a mathematical
instrument of expressing the statistical behavior of particles; in this conception
the waves formulate the probabilities for observations of particles. It is this
interpretation which we have used in 2. It has turned out, however, that
even this ingenious combination of the two interpretations cannot be carried
through consistently. We shall describe in 7 experiments which do not con-
form with the Born conception. on the other hand, the latter conception has
been incorporated into quantum physics so far as it has been made the defini-
tive form of the corpuscle interpretation. Whenever we speak of corpuscles we
assume them to be controlled by probability waves, i.e., by laws of probability
formulated in terms of waves. The duality of interpretations, therefore, is given
by a wave interpretation according to which matter consists of waves; and a
corpuscle interpretation, according to which matter consists of particles con-
trolled by probability waves. As to the waves the struggle between the two
interpretations, therefore, amounts to the question whether the waves have
thing-character or behavior-character, i.e., whether they constitute the ultimate
objects of the physical world or only express the statistical behavior of such
objects, the latter being represented by atomic particles.
The decisive turn in the evaluation of this state of affairs was made by Bohr
in his principle of complementarity. This principle states that both the wave
conception and the corpuscle conception can be used, and that it is impossible
ever to verify the one and to falsify the other. This indiscernability was shown
to be a consequence of the principle of indeterminacy, which with this result
appeared to be the key unlocking the door through which an escape from the
dilemma of two equally demonstrable and contradictory conceptions was
possible. The contradictions disappear, since it can be shown that they are
restricted to occurrences situated inside the range of indeterminacy; they are
therefore excluded from verification.
Although we should like to consider this Bohr-Heisenberg interpretation as
ultimately correct, it seems to us that this interpretation has not been stated
in a form which makes sufficiently clear its grounds and its implications. In the
form so far presented it leaves a feeling of uneasiness to everyone who wants
to consider physical theories as complete descriptions of nature; the path
towards this aim seems either to be barred by rigorous rules forbidding us to
6. WAVES AND CORPUSCLES 2g
ask questions of a certain kind, or open only to vague pictures which have no
claim to be regarded as an adequate ｅxpression of reality. It seems to us that
this state of affairs is due, not so much to mistakes in the quantum mechanical
part of the interpretation, as to an erroneous interpretation of the correspond-
ing problem of classical physics which has not been seen in the full extent of its
logical complications. In the following considerations we shall try to give a
solution of these problems which follows the outlines of the ideas of Bohr and
Heiseriberg, but which seems to us to avoid the unsatisfactory parts of this
conception.
We shall use for our analysis a formulation of the ideas of Bohr and Heisen-
berg which has been developed by Lande. 1 Land6 states the duality of the two
interpretations, by waves and by corpuscles, in the following form. It is incor-
rect to say that some experiments require the wave interpretation and others
require the corpuscle interpretation; such a conception, which represents the
state of affairs before the Bohr-Heisenberg theory, is not permissible, because
it would make physical theory inconsistent. Instead, we must say that all
experiments can be explained through both interpretations. It will never be
possible to construct an experiment which is incompatible with one of the
interpretations.
Combining this formulation with our theory of equivalent descriptions, and
using the terminology explained above, we can state Landd's conception as
follows. Given the world of phenomena, we can introduce the world of inter-
phenomena in different ways; we then shall obtain a class of equivalent descrip-
tions of interphenomena, each of which is equally true, and all of which belong
to the same world of phenomena. In other words: In the class of equivalent
descriptions of the world the interphenomena vary with the descriptions,
whereas the phenomena constitute the invariants of the class. With this, the
arbitrariness of descriptions is eliminated from the world of phenomena and
restricted to the world of interphenomena; but there it is harmless, as we know
that a similar arbitrariness of descriptions holds for the unobserved things of
classical physics. Nowhere do we find an unambiguous supplementation of
observations; interpolation of unobserved values can be given only by a class
of equivalent descriptions.
From this result we turn to the question whether the class of equivalent
descriptions of quantum mechanics contains a normal system, i.e., a descrip-
tion which satisfies the two principles established on page 19. Now it is obvious
that the second principle will be violated by all descriptions, as there is always
a disturbance of the object by the observation. We must therefore modify our
definition of the normal system and restrict it to the requirement that at least
the first principle be satisfied. 2 The question must therefore be asked in the
1 A. Land6, Principles of Quantum Mechanics (Cambridge, England, 1937).
2 Professor W. Pauli drew my attention to the fact that even in classical physics the
second principle is mostly to be suspended for the introduction of normal systems. When
we see a physical object, the fact producing the observation is the entering of light rays
24 PART I. GENERAL CONSIDERATIONS
form: Is there a normal system in the wider sense, i.e., a system at least
satisfying the first principle?
We must not believe that the existence of a normal system can be postulated
by philosophical considerations. We cannot admit that there is any synthetic
a priori principle, i.e., a principle which is not logically empty and which
physical theory is bound to satisfy. The question of whether there is a normal
system can only be answered by experience. If there is such a system, the world
of interphenomena is revealed to be of a rather simple structure; if there is no
such system, this world turns out to be more complicated than we would
perhaps like it to be. But by no means is it permissible to refuse an answer to
the question as meaningless, or to evade it by drawing the attention to other
sides of the problem. The general properties of the world of interphenomena,
as it is constructable on the basis of quantum mechanics, are expressed in the
answer we give to this question.
7. Analysis of an Interference Experiment
To find an answer to the question of whether there is a normal system, let us
analyze some experiments which can be considered as typical for various kinds
of logical situations. First, let us consider (figure 4) the case of a diaphragm
containing one slit B through which radiation of light, or electrons, or other
particles of matter passes towards a screen. We then shall obtain an inter-
ference pattern on the screen. We know, however, that if we use very low in-
tensities of the radiation, we obtain on the screen, not the whole pattern at
once, but individual flashes in strictly localized areas, for instance, in C. These
flashes could be verified, say, by a Geiger counter in C. If we let the experi-
ment go on for a certain time, the distribution of the flashes, occurring one after
the other, will follow the interference pattern mentioned above; it is this sum
of individual flashes which a photographic film placed on the screen would
show us.
The phenomena of this experiment are given by the individual flashes on the
screen; besides, we have the macrocosmic objects consisting in the source of
radiation, the diaphragm, and the screen. Let us ask which kind of inter-
phenomena we can introduce here by the method of interpolation. First, we
into the retina of the human eye; but in this action the light ray is absorbed and thus
changed by the interaction of the means of observation. The statement that there is a
physical object which is undisturbed by the observation is therefore, physically speaking,
obtained by an inference based on an observation which does not satisfy the second prin-
ciple. only psychologically speaking is this not true, since the inference is performed
automatically by the sensory mechanism; the eye is gauged in stimulus language. This is
the reason why it appears advisable in classical physics to interpret the term "observa-
tion" in such a way that the second principle can be maintained. It would be equally
possible to cancel the second principle even for classical physics. This corresponds to our
view that the abandonment of the second principle in quantum mechanics is rather irrele-
vant and that a normal system in a wider sense can be defined including the case that the
second principle is violated. It is the first principle which expresses the conditio sine qua
non of the normal system.
7- AN INTERFERENCE EXPERIMENT 5} 5
can use the corpuscle interpretation. 1 We then say that individual particles are
emitted from the source of radiation, and move on straight lines such as indi-
cated in figure 4; at the place B these particles are subject to impacts or other
forms of interaction imparted to them by the particles of which the substance
of the diaphragm is composed. They thus deviate from their paths. These
impacts follow statistical laws in such a way that some parts of the screen are
frequently hit, others less frequently. The interference pattern of the photo-
graphic film indicates, therefore, the probability distribution of the impacts
given in B to the passing particles. Of course, there are also other particles
A .-
O"
source
of
radiation
diaphragm screen
Fig. 4. Diffraction of radiation at a slit B.
leaving the source A; but when they reach the diaphragm in places other than
the slit B they are absorbed or reflected and will not appear on the screen.
In this interpretation we have a certain probability
P(A,B) (1)
that a particle leaving the source A will arrive at 5, and a probability
P(A.B,C) (2)
that a particle leaving A and passing through B will arrive at C. Both values
can be statistically determined by counting on a surrounding screen all par-
ticles leaving A t then all those particles arriving on the screen (the number
arrived at meaning the number of particles passing through the slit B) y and
then all particles arriving at C.
We see that we have here an interpretation of interphenomena which satisfies
the first principle required for a normal system. The only deviation from clas-
sical physics consists in the fact that we have merely a probability law govern-
ing the transition from -B to C; but this extension of the concept of causality
holds equally for the phenomena of quantum mechanics. Therefore, in this
interpretation both phenomena and interphenomena are governed by the same
laws.
1 The way of describing this experiment in corpuscle interpretation is represented in
A. Land6, Principles of Quantum Mechanics (Cambridge, England, 1937), 9.
2 6 PART I. GENERAL CONSIDERATIONS
Now let us use the wave interpretation. We then say that spherical waves
leave A t that only a small part of these waves passes through the slit B and
then spreads toward the screen. This part of the waves consists of different
trains of waves, each of which has a different center; all these centers lie on
points within the slit B (principle of Huygens). The superposition of these
different trains furnishes the interference pattern on the screen.
So long as we consider only the results of the process obtained during a long
time, for instance, in the pattern obtained on a photographic film, this explana-
tion leads to no difficulties. It has even the advantage over the corpuscle inter-
pretation in that it does not use statistical laws, but strictly causal laws. The
numerical values of the probabilities (1) and (2) appear here as the intensities
of waves which determine directly the amount of blackening in the various
points of the screen. It is different as soon as we consider the individual flashes
which we know can be verified on the screen. Let us assume, for instance, that
the screen is replaced by a set of Geiger counters; the statistics of this system of
counters then will be equivalent to the interference pattern on the film, with
the addition, however, that it reveals the process as being composed of indi-
vidual impacts. In face of these facts the assumption of waves leads into diffi-
culties which were first pointed out by Einstein. So long as the wave has not yet
reached the screen, it covers an extended surface, namely, a hemisphere, with
its center in B; but when it reaches the screen it will produce a flash only at one
point, say, at C, and will then automatically disappear at all other points. The
wave is swallowed, so to speak, by the flash at C. This process of the disappear-
ance of the wave constitutes a causal anomaly so far as it contradicts the laws
established for observable occurrences. We see that in this description the
laws of interphenomena are different from the laws of phenomena ; the given
description therefore does not represent a normal system.
With the intention of escaping disagreeable consequences of the wave inter-
pretation, the suggestion has been advanced that it should be forbidden to ask
questions about what becomes of the wave after a flash on the screen has been
observed. We shall later discuss an interpretation in which such an interdiction
of questions is carried through; with such an interpretation, however, we have
abandoned the wave theory. Within the wave interpretation the legitimacy of
such questions cannot be denied. The different reasons which have been
offered for ruling out such questions do not stand a logical test. Thus it has
been said that within the wave description we cannot speak of effects localized
in space. This, however, is incorrect; the wave itself is conceived as a function
of space, and if a certain effect is observed in one place on the screen, it is per-
fectly correct to ask what effects are caused by the wave in other places. It has
been said, furthermore, that the flash on the screen belongs to the corpuscle
interpretation, and that therefore we cannot incorporate it into the wave
interpretation. This is incorrect because the flash on the screen is a verifiable
phenomenon, and therefore is not included in the duality of interpretations
7- AN INTERFERENCE EXPERIMENT 2 7
which concerns only the interphenomena. The flash belongs neither to the one
nor to the other of the interpretations, but is one of the verifiable data on which
both interpretations are based. Now we must demand that each interpretation
of interphenomena be compatible with the given set of phenomena; and if the
wave interpretation is used, it must be extended to include a statement con-
cerning the transformation of a wave into a localized flash.
It appears understandable that in the case of such an experiment as the one
considered we prefer the corpuscle interpretation, since it does not include
causal anomalies, and therefore represents a normal system. But the wave
interpretation is as true as the other; it is true in the same sense as is the inter-
pretation in the example given above, according to which the tree always splits
B,
source
of
radiation
diaphragm screen
Fig. 5. Diffraction of radiation at two slits, B A and B 2 .
into two trees at the moment it is not observed. This anomaly need not bother
us because we know that it can be "transformed away" by the use of another
description. In the same sense the anomaly of the wave description should not
bother us because we know that it can be transformed away by the use of the
corpuscle description. But if the wave interpretation is used, it includes the
consequence of the disappearance of the whole wave with the appearance of
the flash in one point, however distant this point may be from other points of
the wave. We must have the courage to face this consequence which is neces-
sarily combined with this interpretation of interphenomena.
Let us now turn to a second experiment, which we indicate in figure 5. We
use the same arrangement as before, with the difference that the diaphragm
has two slits, at BI and J5 2 . We know that in this case we shall also obtain an
interference pattern on the screen, which is, however, different from the pat-
tern obtained in the first experiment. Let us consider this experiment in
different interpretations.
We first use the corpuscle interpretation. Assuming, as before, low intensities
of the radiation, we know that there will be individual flashes on the screen.
We can explain this by the assumption that individual particles leave the
2 8 PART I. GENERAL CONSIDERATIONS
source intermittently; sometimes a particle passes through the slit J5i, some-
times through the slit B 2 , sometimes it is absorbed by the diaphragm all this
being determined by the direction in which the particle leaves A. If there
occurs a flash in C, we shall say that the particle has passed either through
The probability that a particle reaches C can be stated by the rule of elimina-
tion established in the calculus of probability :
P(A,C) = P(A,BJ - P(A.B,,C) + P(A,Bi) P(A.B 2 ,C) ' (3)
The meaning of these terms follows from the explanation given with respect
to (1) and (2). We should be inclined to assume that the numerical values of
the probabilities on the right hand side of (3) are the same as obtained in
experiments of the first type; it turns out, however, that this assumption is
wrong.
This can be proved in the following way. We first close the slit at B 2 , and let
the process of radiation go on for a certain time; then we close the slit at B\,
and let the process go on for an equal time. Using a film as a screen we then
obtain a superposition of both interference patterns. The question arises: Is
this pattern of interference the same as that which will result when both slits
are opened simultaneously? If it is, we can assume that the values of the
occurring probabilities are the same as in the first experiment. If it is not, the
probabilities P(A.B ly C) and P(A.B 2) C) must have changed.
It is well known that the experiment decides in favor of the second alterna-
tive. We therefore must assume that the probability with which a particle
passing through BI reaches C depends on whether the slit B 2 is open. 2 This is a
causal anomaly; it states that there is an effect originating in B 2 and being
spread to B\ such that it influences the impacts given in B\ to passing particles.
We see that it is in this case the corpuscle interpretation which leads to causal
anomalies. 3
It would be erroneous to say that because of these anomalies the corpuscle
interpretation is false. Some physicists who have uttered opinions of this kind
have based their judgments on the fact that we have no means of knowing
through which of the two slits, BI or B 2 , the particle has passed after the flash
in C has been observed. The latter statement, of course, is true, since an obser-
vation in BI, or in J3 2 , would disturb the experiment. We are not even able to
determine the probability P(A.C,Bi) that the particle, observed in the flash C,
2 It can be easily seen that this consideration is a special case of the consideration
given in 22, according to which an intermediate measurement of an entity v influences
the probability leading from u to w, even if the result of the measurement is not included
in the statement of the latter probability. Closing the slit Z? 2 is equivalent to making a
measurement of position in BI.
3 Further anomalies arise if we bring the screen closer to the diaphragm. If we choose
the point C on the screen always in such a way that the direction B\C remains the same,
we find that the probability P(A.B\,C) cannot be considered as remaining constant; i.e.,
this probability is not a function depending only on the direction in which the particle
leaves BI. This follows from considerations using waves.
7. AN INTERFERENCE EXPERIMENT 2Q
has passed through BI. This inverse probability could be determined by means
of the rule of Bayes 4 in the form
if we knew the value of the forward probability P(A.#i,C); but since this
probability is different from the value P(A.B,C) obtained with the use of only
one slit, it cannot be determined. Every such determination would require
observations of the particle in BI, and therefore would lead to a disturbance of
the experiment. Applying the verificability theory of meaning, even in the
modified form of probability meaning, 5 we therefore must say that the sen-
tence "the particle has passed through BI" is meaningless if we consider it as a
statement about physical facts. It is permissible, however, to use this sentence
if we consider it as a definition. In order to make the corpuscular description
complete we coordinate by definition to every flash in C a path of a particle,
passing either through BI or B z ; the choice of these paths is arbitrary. Even if
we follow the requirement (which in itself has only the character of a definition)
that the probability P(A,Bi) and P(A,B 2 ) be the same as obtained in experi-
ments with one slit, the choice of the paths will remain arbitrary within wide
limits.
The situation presented reminds us of a similar situation in the problem of
simultaneity. If a light signal leaving a point Q at the time k is reflected at a
point R, and then returns to Q at the time fa, we may coordinate to the time of its
arrival in R every numerical value between ti and fa. With this choice we make
a definition of simultaneity at the points Q and R. The sentence: one of the
events in Q between t\ and fa is simultaneous with the arrival of the light
signal in R" is meaningless if it is considered as an empirical statement, since
it is not verifiable ; but it is meaningful if we introduce it as a definition. In the
same sense we can use the sentence: "the particle passed through B\ J as a
definition. In both cases such a definition is necessary in order to make our
description complete.
It follows that the corpuscular interpretation can be carried through con-
sistently, and that there is nothing incorrect in it. Its only disadvantage is that
it leads to causal anomalies as explained. The influence of the slit B% on the
occurrences in BI is of such a kind as to violate the principle of action by con-
tact. There is no spreading of the effect from 7? 2 to /^; this can be seen from
the fact that changes in the material of the diaphragm between BI and 5 2 , or
changes of its shape such as would result from corrugating the diaphragm,
would not influence the effect.
If we want to construct for the experiment described an interpretation free
4 Cf. any textbook on probability, or the author's Wahrscheinlichkeitslehre (Leiden,
1935), formula (6), 21.
6 Cf. the author's Experience and Prediction (Chicago, 1938), 7.
PART I. GENERAL CONSIDERATIONS
from anomalies, we must use a wave interpretation. It would not be sufficient,
however, to use an interpretation in which the wave spreads through all the
open space; this would lead to the same anomalies as explained for the wave
interpretation of the first experiment. We must use a wave interpretation
according to which the wave is limited to two narrow canals such as indicated
in figure 6. It can be shown that any changes made at a point inside the
"two-canal element" would produce a change in the probability P(A,C) that a
flash in C occurs, whereas changes outside do not influence this probability. We
could, for instance, fill the space outside the strips included in the dotted lines
with absorbing matter, without changing P(A,C). 6 This can be proved through
ft
source "
of
radiation
,--.:--
-*
diaphragm screen
Fig. 6. The "two-canal clement" representing the radiation.
the principle of superposition which we used above; the interference pattern on
the screen can be considered as the superposition of the blackcnings, produced
one after the other, by two-canal elements having their top ends in different
points C on the screen. We therefore obtain a normal system for the considered
experiment if we use a description in which the interphenomena are regarded
as two-canal waves spreading from A to various points C on the screen, and
following each other intermittently.
We may add here the remark that for the case of only one slit, i.e., for the
experiment of figure 4, we have also a wave interpretation which is free from
causal anomalies; we then speak of a one-canal wave, such as Einstein origi-
nally assumed in his needle radiation. We therefore have here two normal
systems. But the difference between these two descriptions is not very great,
and we therefore usually speak only of the corpuscle interpretation. Every
statement about corpuscles can therefore be replaced by a statement about
needle radiation.
6 Strictly speaking, this is only approximately true. The degree of approximation in-
creases when the canals chosen are wider in their middle parts, while the widths at BI,
B%, and C remain unchanged. The mathematical theorem to which we refer here consists
in the statement that it is possible to introduce two sets of waves starting from BI and B 2
interfering on the screen in such a way that only in a small area at C an intensity is left.
Our ' 'two-canal element" may be considered as a simplified illustration of these two sets
of waves.
7- AN INTERFERENCE EXPERIMENT gl
It has sometimes been said that the differences of wave and corpuscle inter-
pretation are combined with an alternative concerning the use of space-time
concepts and causal concepts. The wave interpretation, according to this con-
ception, satisfies the principle of causality so far as the waves are controlled by
a differential equation, the Schrodinger equation (cf. 13), but does not allow
us to give a spatio-temporal description of physical objects. The corpuscle
interpretation, on the other hand, is said to satisfy the requirements of a
spatio-temporal description but to violate the principle of causality. Apart
from the latter statement we cannot consider this conception as correct. It is
not true that the wave description satisfies throughout the conditions of
causality. It does so only so far as the wave field, conceived as a physical
reality, spreads through space in a form expressible by a differential equation;
in this respect it represents an action by contact, at least when free particles
are concerned. 7 But, as we have shown, there are other points in which the
latter principle is violated. Thus, the disappearance of the wave after the
occurrence of a flash on the screen is a process not following the Schrodinger
equation, and therefore not conforming to the principle of action by contact.
In addition, we must say that a wave interpretation which does not satisfy
the conditions of a space-time description eo ipso cannot satisfy the postulates
of a normal causality either. Spatio-temporal order is closely connected with
causal order, as has been brought to light in the analysis of the theory of
relativity. 8 If the wave cannot be imagined as imbedded in a space-time
manifold in which every part of the wave process satisfies the principle of action
by contact, it cannot be said to conform to the requirements of a normal cau-
sality.
Our exposition has shown that the problems involved in quantum mechanics
cannot be reduced to the alternative space-time versus causality. The space-
time order to be assumed is always the ordinary one, and can be ascertained in
many cases by the usual macrocosmic methods; thus the distance between the
two slits Bi and B 2 may be large enough to be measured by macrocosmic
appliances. In both interpretations it is violations of the requirements of
normal causality which are concerned. The kind of these violations only varies
with the interpretation.
Such violations will occur also within a third interpretation which we must
mention now and which is given by a combination of both wave and corpuscle
interpretation. According to this interpretation we have a wave field spreading
through the whole space and being diffracted at the slits BI and B z in the usual
way; in addition to this field we have corpuscles whose movement is controlled
7 For systems composed of several particles the waves spread, not in a three-dimen-
sional space, but in the n-dimensional configuration space. But even if we were to consider
the configuration space as "real" space, the waves would not satisfy the requirements of
normal causality. In such a space the disappearance of the wave after the flash would
lead to the same difficulties as in ordinary space.
8 Cf . the author's Philosophic der Raum-Zeit-Lehre (Berlin, 1928), 27, 42.
3* PART I. GENERAL CONSIDERATIONS
by the field in such a way that the intensity of the field determines the proba-
bility of finding a corpuscle. Using a term introduced by L. de Broglie, we speak
here of pilot waves. This interpretation has its anomalies in the existence of a
field which follows laws different from those holding for other kinds of waves;
in particular, this wave field possesses no energy, since the energy is supposed
to be concentrated in the particles. Furthermore, the influence of the field on
the particles is governed by unusual laws. If the particles are assumed to move
in straight lines, these laws would violate the principle of action by contact,
since, then, the probability that a particle passing through BI will turn- in the
direction of C would depend, not on the intensity of the field near B ly but
on its intensity at <7. 9 Assuming that the particles move in oscillating lines we
encounter other anomalies. 10
This description is therefore not a normal system, but it has some advan-
tages which make its application advisable in many cases. It is not necessary
to assume that the pilot waves appear intermittently with the particles, and
that these waves disappear with the flash in <7. The waves may be assumed to
go on continuously; their existence, however, will be verifiable only at the
moments when there are particles traveling through them.
8. Exhaustive and Restrictive Interpretations
The above analysis shows that neither the corpuscle interpretation nor the
wave interpretation can be carried through without causal anomalies. Using
the particle interpretation we can explain some experiments in such a way that
the laws of phenomena and interphenomena are the same; but then we en-
counter anomalies in other cases. Using the wave interpretation we can explain
these other cases in such a way that the laws of phenomena and interphe-
nomena are the same; but then anomalies appear in the explanation of experi-
ments of the first kind. Finally, a combination of the two into an interpretation
of pilot waves shows other anomalies.
The question arises whether there is another interpretation, perhaps un-
known to us, which is free from causal anomalies. The preceding investigations
cannot be considered as a proof that there is no such interpretation. Such a
proof cannot be given by trying out one interpretation after another; we then
are never sure whether a better interpretation which escaped our attention
remains. The proof must be based on a general theory of the relations between
quantum mechanical entities. We shall give this proof in 26; the conception
of causality assumed for it, which takes account of possible modifications of
this notion, will be explained at the end of 24. Our results can be formulated
as follows: It is impossible to give a definition of interphenomena in such a
9 This follows from considerations similar to those indicated in fn. 3, p. 28.
10 The anomalies of this interpretation are very clearly presented in L. de Broglie's book,
Introduction d V Etude de la Mecanique ondulatoire (Paris, 1930), chap. 9.
8. KINDS OF INTERPRETATIONS 33
way that the postulates of causality are satisfied. The class of descriptions of
interphenomena contains no normal system. This can be proved to be a conse-
quence of the basic principles of quantum mechanics. We shall call this result
the principle of anomaly.
In view of this negative result two different conceptions can be carried
through. The first calls for a duality of interpretations. Among the class of
equivalent descriptions we have two, the corpuscle interpretation and the
wave interpretation, which are more expedient than the others; since we have
no normal system, we can use, instead, either of these two interpretations as a
minimum system, i.e., a system for which the deviations from a normal system
constitute a minimum. In this conception causal anomalies cannot be avoided;
but they can at least be reduced to a minimum.
The second conception represents a more radical remedy. Since no normal
description of interphenomena exists, it has been suggested we should renounce
any description of interphenomena; we should restrict quantum mechanics to
statements about phenomena then no difficulties of causality will arise. The
impossibility of a normal system is construed, in this conception, as a reason
for abandoning all descriptions of interphenomena. We shall call conceptions of
this kind restrictive interpretations of quantum mechanics, since they restrict
the assertions of quantum mechanics to statements about phenomena. The
rule expressing this restriction can assume various forms, and we shall there-
fore have several restrictive interpretations. Interpretations which do not use
restrictions, like the corpuscle and the wave interpretation, will be called
exhaustive interpretations, since they include a complete description of inter-
phenomena.
The adherents of restrictive interpretations have maintained that a descrip-
tion of interphenomena is unnecessary; for the purpose of observational
predictions, they say, it is sufficient to have an interpretation which refers only
to phenomena. The latter statement is true; but it cannot be considered as
proof that exhaustive descriptions should be abandoned. We should clearly
keep in mind that neither of the two conceptions can be proved to be true.
These conceptions represent volitional decisions concerning the form of physics ;
either of them is as justifiable as the other.
Speaking in terms of the class of equivalent descriptions, the situation can
be characterized as follows. The system of phenomena is the same for each
description of this class; it is therefore the invariant of this class. Now the class
is dependent on its invariant; so far, any restrictive interpretation deter-
mines the whole class of exhaustive descriptions. The latter descriptions, how-
ever, reveal a feature which we would not know if we knew only a restrictive
description: this is the fact that no interpretation free from causal anomalies
can be given. Since this is a property of the class of exhaustive descriptions, it
represents an inherent property of every restrictive interpretation. This prop-
erty is expressed in the restrictive interpretations through the fact that they
34 PART I. GENERAL CONSIDERATIONS
rule out certain statements; but the reason for this rule can only be formulated
in terms of a statement about the properties of the class of exhaustive descrip-
tions.
We therefore shall turn now to a further analysis of the class of exhaustive
interpretations, while we postpone the discussion of restrictive interpretations.
Within the first class, we said, the two interpretations in terms of corpuscles
and waves hold a special position so far as they represent minimum systems.
To this we now must add a second statement which secures a unique position
to these two interpretations, and which at the same time attenuates the* conse-
quences resulting from the absence of a normal system.
Although we have no exhaustive description free from anomalies holding for
all interphenomena, we can construct such a description for every interphe-
nomenon by using either the wave or the corpuscle interpretation. It is this
fact which we express in speaking of the duality of wave and corpuscle inter-
pretation. We mean by this that for a given experiment at least one of the two
will be a normal description and will thus define interphenomena in such a way
that they follow the same laws as the phenomena; it is only in other experi-
ments that the interpretation so chosen will lead to causal anomalies. Let us
call this statement the principle of eliminability of causal anomalies. The differ-
ence between all and every , which we used to formulate this principle, is well
known to symbolic logic. Using this grammatical distinction in another form
we may also say: It is false to say that all interphenomena follow the laws
holding for phenomena; but it is correct to say that every interphenomenon
does so. We do not have one normal system for all inter -phenomena , but we do
have a normal system for every interphenomenon.
As before, an analogy from differential geometry may illustrate these formu-
lations. When we use a system of orthogonal coordinates on the sphere, such as
that given in the circles of longitude and latitude (such a system is possible
because it does not consist throughout of straightest lines, the circles of latitude
not being greatest circles), this system has singularities at the North Pole and
the South Pole; i.e., these points do not have a definite longitude. These
singularities, however, are due only to the system of coordinates; the poles
themselves are not distinguished geometrically from any other point of the
sphere. The singularities can therefore be "transformed away" by the intro-
duction of another system of coordinates; thus, the sailor will use, near the
poles, a notation of points which determines positions relative to a chosen
initial point and two chosen directions rectangular to each other. These co-
ordinates could even be produced and used to cover the whole sphere, at least
if one set of lines is not assumed to consist of straightest lines; but then singu-
larities will appear in other points of the system. We may call a system of
orthogonal coordinates without singularities a normal system. Then we may
say that we can introduce a normal system for every extended area on the
sphere, but we cannot introduce one normal system for all areas, i.e., for the
8. KINDS OF INTERPRETATIONS 35
whole sphere. We thus express a statement about the sphere in terms of a
statement concerning the class of possible systems of coordinates.
The difference between this case and the case considered above, which con-
cerns orthogonal straight-line coordinates, is as follows. A system of coordi-
nates being both orthogonal and straight-lined is possible only for infinitesimal
areas; for extended areas of some size it cannot even be carried through
approximatively. If we renounce the requirement of straight lines we can con-
struct a system which covers great areas of the sphere, and which is strictly
orthogonal; but such a system will lead to singularities in two points. The
advantage of this case over the first is that, with this definition of the normal
system, we obtain a normal system for extended areas, not for infinitesimal
areas only.
Returning to quantum mechanics, we must say that the situation there cor-
responds to the second case. The causal anomalies can be transformed away
strictly for "extended areas", i.e., for a whole experiment, by a suitable
description. They will reappear only for other experiments or for questions in
which experiments of different kinds are compared; for the answer to such
questions we then can introduce a new description such that once more the
anomalies disappear. The reason that this is always possible is given in the re-
lation of indeterminacy. If we could observe a particle passing through the
slit Bi in the experiment of figure 5, we could not introduce a wave description,
and therefore would have no normal description of the experiment, i.e., no de-
scription free from anomalies. on the other hand, if we could prove that a wave
arrived simultaneously at different points C of the screen in the experiment of
figure 4, we could not introduce a corpuscle description, and therefore would
have no normal description of this experiment. We see that the principle of
elimindbility of anomalies is made possible through the principle of indeterminacy,
since the latter principle makes it impossible ever to construct a crucial experi-
ment between wave interpretation and corpuscle interpretation. 1
The ultimate root of the duality of wave interpretation and corpuscle inter-
pretation is therefore given in the principle of indeterminacy; but this principle
also points the way out of the dilemma of causal anomalies, a result stated by
us in the principle of eliminability. We spoke above of the skill displayed by
physicists in applying sometimes the wave interpretation and sometimes the
corpuscle interpretation; we now see that we can give a justification of this
change of interpretations which proves that the switching over to a normal
interpretation is a legitimate means of physical analysis. When the physicist, in
face of a particular experiment, introduces a suitable description which elimi-
nates causal anomalies within the frame of his question, he may be compared
1 It may be questioned whether it is actually possible in all cases to eliminate causal
anomalies by a suitable description. What can be shown is that the principle of elimina-
bility holds, at least for single particles or for swarms of particles which do not interact
with each other such as electron swarms or light rays. Difficulties arise for complicated
structures composed of several particles. Cf. 27.
36 PART I. GENERAL CONSIDERATIONS
to the sailor who, at the North Pole, discontinues determining his position in
terms of longitude, and prefers to use another system of coordinates free from
singularities. Such a procedure is permissible because nature has not deter-
mined one normal system for all interphenomena, but only a separate normal
system for each interphenomenon.
Let us consider some examples. Using the wave interpretation, we arrive at
the question why the whole wave disappears after a flash has been observed on
one point of the screen. We eliminate the causal anomaly presented in this
description of the interphenomenon by introducing the particle interpretation.
The wave then is transformed into a probability, and, instead of the disap-
pearance of a wave, we have the simple statement that although the proba-
bility P(A,C 2 ) of finding a flash on the screen in a point C 2 has a certain positive
value, the probability P(A.Ci,Cz) of finding a flash in C 2 after a flash has been
observed in Ci, is zero. Instead of the contraction of a wave into a point, we
have here the trivial logical fact that probabilities are relative. It was by con-
siderations of this kind that Born was led to the introduction of the statistical
interpretation of the waves which originally had been conceived by Schrodinger
as waves of electrical density.
Another example where the normal description is given by the particle inter-
pretation is represented by the following consideration. The probability that a
particle leaving the source A will pass through either of the slits is the sum of
the probabilities that a particle will pass through one of the slits; we may
indicate this by the symbolic ｅxpression :
P(A, Bi V &) = P(A,BO + P(A,B 8 ) (1)
(The logistic sign " V " means "or".) This relation follows from the rules of
probability because the particles can go through only one of the holes at a time.
It can be tested by observations, as follows: To ascertain the value of the left
hand side we count all flashes occurring on the screen when both slits are open;
to determine the two values of the right hand side we count all flashes on the
screen occurring, respectively, when one of the slits is closed. Since statistics so
compiled show that (1) holds, this relation must hold also for the wave inter-
pretation. Here, however, the explanation leads to causal anomalies. We then
must assume that whenever a wave leaves A in the direction of Bi there is
another wave leaving simultaneously in the direction of B 2 , but that the wave
going toward the open slit BI will sometimes disappear due to an influence of
the closed slit B 2 (namely, in all those cases when, in the corpuscle interpreta-
tion, a particle is emitted only in the direction of J3 2 ), and is thus controlled by
an influence which represents an action at a distance. The physicist who wishes
to explain the well-confirmed relation (1) will therefore prefer the corpuscle
interpretation by the use of which he can derive the relation without the
assumption of anomalies.
On the other hand, there are questions which only by the use of the wave
8. KINDS OF INTERPRETATIONS 37
nterpretation can be answered without reference to anomalies. We saw that
he corpuscle interpretation of the experiment indicated in figure 5, 7,
nvolves an action at a distance 'between the two slits BI and JB 2 . In the wave
nterpretation this action at a distance is eliminated and replaced by a state-
nent about a phase relation between the waves arriving in BI and 2, which is
lue to their common origin from the source A. In answering questions of this
dnd the physicist will therefore prefer the wave interpretation.
Our examples show that it is even preferable to speak, not of the normal
jysteni for every interphenomenon, but, of the normal system for every question
joncerning interphenomena. It is the question which determines the normal
;ystem, and, relative to the same experimental arrangement, different ques-
ions can be asked which require different normal systems. Thus the question
joncerning the probability relation (1) is asked with respect to an experimental
irrangement which, for other questions, necessitates a wave interpretation.
SVhen we say "for every question", we mean, of course, that the question is
sufficiently limited, and not constructed as an "and"-combination of different
questions. With this qualification we can formulate the principle of elimina-
aility as stating: We have no normal system for all questions concerning
interphenomena, but we do have a normal system for every such question.
The switching over from one interpretation to another is justifiable, we said,
as a means of eliminating causal anomalies. This is, however, its only justifica-
tion, and it would be incorrect to adduce reasons of another kind. We some-
times read that questions like "what becomes of the wave after a flash on the
screen has been observed" or "why does a particle going through the slit BI
move differently according as the slit B z is closed or open" must be forbidden
because they are not adequate to the respective interpretation. But this word
"adequate" means only that the answer to such questions leads to causal
anomalies. The occurrence of such anomalies does not make the question, or
the answer, unreasonable. If we have decided to use one of the exhaustive
interpretations, such questions are not meaningless. We then must become
accustomed to the fact that for a given interpretation there are always
questions which can only be answered by the assumption of causal anomalies.
If we prefer to use in the answer to such questions an interpretation which is
free from anomalies, we have good reasons to do so; but we should not believe
that the answer so constructed is the only meaningful answer, or the only ad-
missible answer, or the only true answer. All the merits of such an interpretation
consist in the fact that it is free from causal anomalies for the interphenomenon
considered; but neither does this fact make it more true than others, nor is it a
necessary condition of meaning within an exhaustive interpretation. Judg-
ments of this kind are based on a confusion of exhaustive and restrictive inter-
pretations. only for the latter will the mentioned questions be meaningless;
but for such an interpretation the complete description of the experiment
which is free from anomalies is meaningless as well. Within an exhaustive
38 PART I. GENERAL CONSIDERATIONS
description, however, we shall be entitled to state causal anomalies as well as
cases of normal causality.
On the other hand, the opposite mistake has been frequently made, i.e., the
mistake of saying that we have no true description at all of the interphenom-
ena; and that the duality of interpretations proves that we can construct only
pictures of the interphenomena, correct in some features and incorrect in
others. Whereas the previously criticized attitude appears dogmatic, the latter
attitude must be called too modest. All admissible descriptions are equally
true in all their details, including the anomalies. If somebody wishes to call
descriptions of interphenomena pictures he may do so in order to stress the
possibility of choice; but he must not forget that then it is also a use of pictures
to say that the tree on the street remains in its place when nobody looks at it.
We saw that in this case also we have a choice of descriptions, and that the
question of the unobserved tree can be unambiguously answered by inductive
evidence only after the postulate of unchanged laws of nature has been intro-
duced. It is only the combination of this postulate with a description of the
unobserved object which can be empirically verified. For the interphenomena
of quantum mechanics the situation is the same; an unambiguous statement
concerning interphenomena can be made only after the postulate of unchanged
laws of nature has been introduced, and the combination of a description of
interphenomena with this postulate can be empirically verified. It will deter-
mine one interpretation for every experiment, but not one for all.
If once the problem of the description of interphenomena has been formu-
lated in this way, it is clear that even in the macrocosm there is no logical need
for the existence of a normal system. Imagine that whenever we turn our eyes
away from a tree and observe only its shadow we see two equally shaped
shadows, whereas when we see both the tree and its shadow there is only one
tree and one shadow. We then have the choice of saying either that there are
always two trees when we do not look at a tree, or that there is only one un-
observed tree for which, however, the known laws of optics do not hold. There-
fore one of the two principles defining the normal system in the classical sense
must be abandoned in such a case. Imagine, furthermore, observations showing
that all changes happening to one of the shadows take place in the other shadow
as well; for instance, if we feel a blast of wind and see a certain branch of one of
the tree shadows being moved, we see an equal movement in the corresponding
branch of the other tree shadow; if the shadow of a bird appears on one of the
tree shadows, an equal shadow of a bird appears on the other tree shadow, etc.
If, in this case, we want to use a description in which the laws of optics are
unchanged, we must assume a duplication of occurrences, which would repre-
sent a precstablished harmony, or an action at a distance, which produces
duplicates of occurrences in different places. Since this assumption signifies a
causal anomaly, we have, in the case considered, no normal description in the
sense of a description satisfying the first of our principles.
8. KINDS OF INTERPRETATIONS gQ
A macrocosmic analogy resembling more closely the experiment of figure 5,
7, can be constructed as follows. Imagine in A a machine gun which is
turned irregularly by a machine so that it shoots bullets in all directions in an
irregular sequence. Let the diaphragm have a certain thickness so that on the
walls of the short slit canals the bullets can be reflected. We then shall observe
an irregular distribution of bullets on the screen which may consist of lead so
that the bullets are caught in it. Imagine, furthermore, that the bullets move
so fast 'that we cannot see through which hole they pass, and that we have no
other means to verify this. Let us now assume the following observations to be
made. If both slits are open, the number of bullets hitting the screen is twice
as large as in the case when only one slit is open. on certain spots of the screen,
however, we find no hits whenever both slits are open, whereas we do find hits
on these spots when only one of the slits is open. In such a case we have the
same choice of interpretations as in the case of the radiation experiment de-
scribed above. We may assume that the bullets on their path through the air
remain individual particles, but that there is an action at a distance between
the two slits; or that the interphenomena consist in waves spreading through
both holes and uniting later to form the bullets found in the lead of the screen.
Such analogies may make it clear that there is nothing unimaginable in the
state of affairs ascertained by quantum mechanics, and that it is possible to
construct macrocosmic models of it. The physics of these models, of course,
will be different from that of our actual macrocosm. But should our macrocosm
follow a similar pattern, we should after some time get accustomed to it; we
should consider it as a matter of course that we could not give one normal
description for all interphenomena, and should learn to use, for the purpose of
answering a certain question, the description which at least for that question
does not involve anomalies. Fortunately our daily world does not show this
kind of structure. It is different with the atomic world; quantum mechanics has
shown that its structure is of the kind depicted in these analogies.
This means that in the world of atomic dimensions the postulate of un-
changed laws of nature cannot be carried through for the totality of interphe-
nomena, and therefore does not determine one interpretation as the normal
interpretation of all interphenomena, although it determines one normal inter-
pretation for every interphenomenon. This result must be considered as the
most general statement which physics, in its present status, can make about the
structure of the physical world. It may seem strange that the physical world
cannot be caught in the network of one normal description; that the idea of
uniformity of nature, so often claimed to be the ultimate result of science, can-
not be extended to include the interphenomena of the world of quanta. We
are dealing here, however, with a question which must be answered, not by
wishful thinking, but by experimental inquiry. Since physics has come to the
result as stated we must now take it seriously, and not palliate it by calling it a
breakdown of human capacities for inventing pictures.
40 PART I. GENERAL CONSIDERATIONS
In order to be complete in our statement of general properties of the physi-
cal world, we must add to this negative result the positive statement formu-
lated in the principle of eliminability. Nature allows us to construct, at least
partly, the world of interphenomena in agreement with the laws of phenomena.
This fact has consequences of great bearing. one is that we can answer all
questions by constructing suitable interphenomena which follow normal laws.
Another is that the anomalies incurred with other descriptions can never be
used to produce anomalous effects in the world of phenomena. Thus we 'cannot
use the action at a distance existing between the two slits B\ and B 2 in the cor-
puscle description in order to send signals from one slit to the other. No such
anomalous effect is possible because otherwise it would also occur in the normal
description of the experiment; there, however, it is excluded through the
normal behavior of the interphenomena. The causal anomalies which we en-
counter in anomalous descriptions may therefore be considered as pseudo
anomalies; they are due to the form of the chosen description and can be
eliminated. We see here the far-reaching significance of the principle of inde-
terminacy: It reveals the discrepancy between the laws of phenomena and the
laws of interphenomena as not being of a malignant nature. The causal
anomalies have a ghostlike existence; they can always be banished from the
part of the world in which we happen to be interested, although they cannot
be banished from the world as a whole.
It is this specious character of the causal anomalies which suggests the use of
restrictive interpretations. Every exhaustive interpretation states too much
so far as it speaks of causal anomalies which have no bearing upon the world
of observable phenomena. It may therefore seem advisable to renounce
exhaustiveness, and to prefer a restrictive interpretation which is free from
statements involving such causal anomalies. We are thus led back to the prob-
lem of restrictive interpretations to which we must now turn for closer con-
sideration.
A restrictive interpretation has been introduced by Bohr and Heisenberg.
The rule of restriction states that only statements about measured entities, i.e.,
about phenomena, are admissible; statements about unmeasured entities, or
interphenomena, are called meaningless. This has the immediate consequence
that statements about the simultaneous values of complementary entities
cannot be made. The interpretation so introduced is neither a corpuscle nor a
wave interpretation; since it leaves the status of interphenomena widely inde-
terminate, we cannot say whether these interphenomena consist of particles
or of waves. It resembles a corpuscle interpretation when the measured entity
is the position, since then one rather sharply defined space point is attributed
to the entity. But since a statement about the momentum is left open, we do
not know whether the entity so determined is a particle. If we consider the
whole spectrum of possible momenta as simultaneously realized, the entity
might as well be a wave packet. on the other hand, if a measurement of mo-
8. KINDS OF INTERPRETATIONS 41
tientum is made, we can consider this value either as the momentum of a par-
icle or as the frequency of a wave; the restrictive interpretation leaves this
luestion open.
Let us consider an example in order to show the rule of restriction at work.
The interference experiment of figure 5, 7, is an arrangement which allows
is to measure a frequency, and therefore the momentum of a particle ; so, all
tatements about the position of the particle are ruled out. This means that
tot only a statement of the kind "the particle went through slit BI" is inad-
nissibie, but that even a statement of the form "the particle went either
hrough slit BI or through slit B%" is forbidden. It is clear that this rule works.
Ye then cannot say that if the particle went through slit B\ it must have
>een influenced by the existence of slit # 2 ; the clause with "if" belongs to the
orbidden domain. The causal anomaly has therefore disappeared from the
lomain of admissible statements. The rule of restriction, like a surgical opera-
ion, cuts off all unhealthy parts of quantum mechanical language. Unfor-
unately, like all such operations, it also cuts off some sound parts. Thus it is
tard to abandon a statement like the one concerning the particle's going
hrough one slit or the other. All that can be said against this statement is
hat it leads to undesired consequences.
It should be realized that the elimination of causal anomalies is the only
ustification of the restrictive interpretation of Bohr and Heisenberg. If it were
>ossible to construct an exhaustive interpretation free from causal anomalies,
10 one would question the legitimacy of the definitions used in such an in-
erpretation, even if the relation of indeterminacy should hold and make it
mpossible to replace these definitions by verifiable statements. Thus, if the ex-
>eriment should show that the interference pattern on the screen in figure 5, 7,
srere equal to the superposition of the two interference patterns resulting when
irst one slit is open and then the other, no one would doubt that the particles
vent either through one slit or the other, although, because of the disturbance
>y the observation, it could not be known through which slit an individual
>article went. 2 It then would be considered a reasonable supplementation of
observable data to say that a percentage of the particles given by P(A,Bi)
vent through slit BI, and a percentage given by P(A,B^) went through slit #2.
Vhen the restrictive interpretation rules this statement out, it does so only
2 The disturbance by the observation, in this case, may be of the same kind as is
,ctually assumed in quantum mechanics: If we observe a particle passing through a slit,
t will be pushed off its path by the light ray. It is true that if in this case the particle is
,ot observed at the slit where the observation is made, we know that it passed through
he other slit; therefore we have here a knowledge of the particle's position without hav-
ng disturbed its path. But this knowledge is acquired by inference, not by observation,
ince what is observed is the absence of the particle at one slit, not its passage at the
ither. Our fictitious experiment therefore represents a case where, although the obser-
'ation disturbs, a normal supplementation of observable data by interpolation can be
;iven. This is possible because, in this case, the interference pattern on the screen is of
uch a kind that it permits us to consider the motion of the particle independent of what
lappens at a slit through which the particle did not pass.
42 PART I. GENERAL CONSIDERATIONS
because the experiment shows that the interference pattern on the screen is not
a superposition of the two individual patterns and thus makes it necessary to
assume that the path of a particle passing through one slit will depend on
whether the other slit is open or not; i.e., it does so because of the causal
anomalies derivable from the statement. We often read that the principles of
empiricism laid down in the verifiability theory of meaning lead to the rule
of restriction formulated by Bohr and Heisenberg. This argumentation is
incorrect. Meaning is a matter of definition, and various definitions of mean-
ing can be given; all that can be asked by the philosopher is: Which are the
consequences to which a given definition of meaning leads? 3 The restricted
meaning of Bohr and Heisenberg's interpretation has the advantage that it
eliminates causal anomalies; this is a strong argument in its favor, but it is
the only argument. The exhaustive interpretations given in the corpuscle and
the wave interpretation are equally compatible with the principles of empiri-
cism if they are conceived as being based on definitions.
It must be added that even a restrictive interpretation is not free from defi-
nitions. What we call phenomena are certainly not immediate objects of obser-
vations; they are inferred from observations by indirect methods (cf. p. 21).
These inferences contain a definition, and we shall give in 29 the exact form
of this definition. The logical difference between the physics of phenomena and
the physics of interphenomena is therefore a matter of degree; the latter
contains more definitions than the former. It is a question of volitional decision
which of these two systems we prefer; none can be said to be completely
restricted to observational data.
Whereas the Bohr-Heisenberg interpretation uses a restricted meaning, we
can construct a second form of restrictive interpretation which uses a three-
valued logic. Ordinary logic is written in terms of the two truth values true and
false. To these we shall add, for the purposes of quantum mechanics, a third
truth value which we call indeterminate. Statements about unobserved entities
then are considered as meaningful; but they are neither true nor false, they
are indeterminate. This means that it is impossible to verify or falsify such
statements.
The interpretation so constructed is superior to the interpretation by a re-
stricted meaning because it possesses a system of rules which makes it possible
to connect statements about unobserved entities with statements about ob-
served entities and thus to manipulate all these statements by means of strictly
logical operations. It can be shown that owing to these rules statements ex-
pressing causal anomalies always will obtain the truth value of indeterminacy,
and therefore can never be asserted as true. on the other hand, part of the
statements about unobserved entities are even retained as true, or considered
as true in a somewhat wider meaning; they cannot be used, however, for the
derivation of causal anomalies because the rules of three-valued logic, differing
8 Cf. the author's Experience and Prediction (Chicago, 1938), chap. I.
8. KINDS OF INTERPRETATIONS 43
from those of two-valued logic, make such derivations impossible. We shall
show, for instance, that the statement: "The particle passes either through slit
Bi or 5 2 " need not be completely abandoned, but can be retained in a some-
what wider sense, whereas a statement about an action at a distance between
the slits is not derivable (cf. 33). Three- valued logic is therefore the adequate
form of quantum mechanics once the- decision for the use of a restrictive inter-
pretation has been made.
We have, therefore, good reasons to say that the language of quantum me-
chanics is written in terms of a three-valued logic. We must not forget, how-
ever, that the subject matter of a science, without the addition of further
qualifications, does not determine a particular form of logic. Quantum me-
chanics can be constructed in the form of a two- valued logic; this is demon-
strated by the existence of exhaustive interpretations. only when we introduce
the postulate that causal anomalies be not derivable are we obliged to turn to
a three-valued logic. It is in this form that the structure of the subject matter
expresses itself in the structure of its language. When we apply the same postu-
late to classical physics we arrive at a two-valued logic. The nature of quantum
mechanical occurrences is of such a kind that statements of causal anomalies
can be eliminated from the domain of true statements only if a three-valued
logic is used; this is the form in which we must express the causal structure of
the microcosm.
The three-valued language appears adequate to quantum mechanics because
the causal anomalies, as formulated in exhaustive interpretations, appear to
be superfluous complications; they need not be taken into account so far as
predictions of observable phenomena are concerned. In consideration of this
fact a restrictive interpretation will appear natural to one who works in
quantum mechanical problems. To this logical insight, habit will add its
leveling influence; the desire to ask questions transcending the limits of the re-
strictive interpretation will disappear ; and the restrictive form of quantum me-
chanics may finally seem to answer everything that can be reasonably asked.
This attitude will be as much help to the physicist as the above-mentioned
switching over from one interpretation to the other to which he resorts in
exhaustive interpretations. Similarly, a man in a world in which bullets be-
haved as unreasonably as electrons, might learn to restrict his questions in such
a way as to obtain only reasonable answers. We should not forget, however,
that such an attitude, advisable as it may be, means making a virtue of neces-
sity. When we exclude some kinds of statements about interphenomena and
admit others, we have no other reason to do so than that the first statements
lead to causal anomalies, whereas the others do not.
Our final judgment concerning the logical significance of restrictive interpre-
tations can therefore be stated as follows. The peculiar form of the causal
structure of the microcosm, visible in the causal anomalies of the exhaustive
interpretations, finds a corresponding ｅxpression in the rule of restriction, or
44 PART I. GENERAL CONSIDERATIONS
the existence of indeterminate statements, in the restrictive interpretations.
In other words: The physical status of the quantum mechanical world, ex-
pressed through a restrictive interpretation, is the same as the status expressed
through exhaustive interpretations with causal anomalies which can be trans-
formed away locally. The restrictive interpretations do not say the causal
anomalies, but they do not remove them.
The causal anomalies cannot be removed because they are inherent in the
nature of the physical world. The principle of indeterminacy formulates only
one part of this nature; it states that it is impossible to verify certain,, state-
ments about interphenomena. To this is added, by the system of quantum
mechanics, another principle which we have called the principle of anomaly. It
states that no definition of interphenomena can be given which satisfies the re-
quirement of a normal causality; it therefore maintains the impossibility of a
normal supplementation of the world of phenomena by interpolation. This
includes the restrictive interpretations, since they do not establish a normal
causality either.
The limitations of scientific interpretations of the world of quanta, expressed
in these two principles, must not be considered as limitations of the power of
the human intellect. It is not human ignorance, nor lack of knowledge, which
leads to the conditions imposed upon descriptions of the physical world ex-
pressed in the laws of quantum mechanics. It is positive knowledge, deep
insight into the nature of the atomic world, which constitutes the basis of this
strange network of rules, formulated as rules limiting descriptions, but express-
ing implicitly rules holding for all physical occurrences. Beneath the disguise
of a theory of physical knowledge we discern the outlines of a physical world
different from what centuries of scientific research had dreamed it to be, but
nevertheless demanding recognition as the world of reality.
Part II
OUTLINES OF THE MATHEMATICS OF
QUANTUM MECHANICS
9. 'Expansion of a Function in Terms of an Orthogonal Set
The mathematical formalism of quantum mechanics is based on the general
mathematical method of expanding a function in terms of a set of other func-
tions, which are called the basic functions of the expansion.
The functions considered are complex functions of real variables, i.e., whereas
the special values of the argument variables are real numbers, the functions
have complex numbers as their special values. We denote complex entities
by small Greek letters, real entities by small Latin letters. The complex
conjugate of an entity \l/ is indicated by ^*; the square | \l/ \ 2 of a complex value
is the same as \f/ ^*. The relations to be presented hold, of course, also for
real functions; such functions result from the general case by the condition
^ = \l/*. Let us add the remark that the complex functions used in quantum
mechanics are assumed to satisfy certain conditions of regularity, and that the
infinite series used in the expansions must fulfill some requirement of conver-
gence. We shall not state these conditions here explicitly, leaving this task to
the more technical expositions of the subject. It will be the aim of our presen-
tation to point out only the general relations holding for expansions.
A function \l/(x) is expanded in terms of the set <pi(x) within the domain D of x,
if within D the relation holds
(1)
The summation runs from i = 1 to i = ; since this is the same in all the
following formulae we do not indicate it in the notation. The expansion
coefficients a are constants, i.e., they do not depend on x; for every function
<?i(x) there is such a particular constant <r*.
The set <pi(x) is called complete if the expansion (1) can be given, by means
of suitable coefficients o-;, for every function \l/(x) within D. In addition, the
set <f>*(%) is usually required to satisfy the further condition that the two
relations hold:
(2)
46 PART II. MATHEMATICAL OUTLINES
The integration in (2) extends over the domain D; this is the same in all fol-
lowing formulae and therefore does not need a special ｅxpression in the nota-
tion. Relation (2) determines the expansion coefficients a in terms of the given
function \l/(x) and the basic functions <f>i(x). This relation shows a structure
similar to that of (1), with the difference that here an integration replaces the
summation. The symmetry between \f/(x) and <n finds a further ｅxpression
in (3). A set of functions <?i(x) which, for every pair of functions $(x) and <r t -
connected by (1), satisfies the relations (2) and (3) is called orthogonal and
normalized.
It is understood, in this definition, that the ｅxpression (2) is unique, i.e.,
that there is only one set of coefficients <r; coordinated to a given function
$(x). This includes the consequence, that, if \l/(x) = throughout the domain
D, all <Tt must be = 0. The normalization finds its ｅxpression in (3). For the
case that the domain D is infinite, (3) includes the condition that the integral
on the left hand side is finite; this condition of convergence is one of the
conditions restricting the choice of functions $. We say that \l/(x) must be
quadratically integrable. If, in particular, the value of the quadratical integral
in (3) is = 1, we say also that \l/(x) is normalized. Similarly, (3) imposes upon
the <?i a condition of convergence. Sometimes functions \f/ must be considered
for which the integral on the left hand side of (3) is not finite, and which there-
fore cannot be normalized. It is not necessary to deal with such functions in
this book.
In (2) and (3) we have given, for the orthogonal and normalized character
of the set <pi(x) y an indirect formulation which makes use of other functions
\f/(x) and <n. The question arises whether the conditions of orthogonality and
normalization of the <pt(x) can be expressed by a direct formulation, i.e., in
terms of these functions alone, without reference to other functions. This aim
can be easily reached when the expansions occurring admit a commutation of
summations and integrations. In this case, the orthogonality and normaliza-
tion of the set ^(x) can be defined by the conditions :
<Pi(x)<pk*(x)dx = Bit (4)
The Weierstrass symbol dik used here has the meaning:
(5)
It is easy to show that, if (1) and (4) hold, the relations (2) and (3) are
Q. EXPANSION OF A FUNCTION 47
derivable. We prove (2) by multiplying both sides of (1) by <ph*(x) and inte-
grating with respect to x:
Jt(x) Vk *(x)dx =J* V k*(x)
-*/"*(*) w*(i)fo (6)
t J
Similarly, (3) is proved as follows:
t*(x}dx _ _
J i k
"V ^ I I n /7\
= >, kr ( 7 )
Relation (4) is therefore a sufficient condition 1 of orthogonality and normali-
zation in case summations and integrations are commutative. We may ask in
which case it is a necessary condition; i.e., we may ask what requirement must
be made with respect to (l)-(3) in order that (4) be derivable. The answer is
that it must be required that the functions ^i(z) are themselves functions to
which the expansion can be applied, i.e., functions which can be put in the
place of $(x). This is proved as follows. If we choose, for instance, a particular
function &(x) as the function $(x), we shall have a solution by putting c\- = 1,
ff k = for k 7* i. Since we assumed the expansion to be unique, this must
be the only solution. Using (2) and substituting there &(x) for \l/(x), we have
1 = I w(x)<f>i*(x)dx
= / w(x)<pk*(x)dx k
(8)
We thus obtain the relations (4) if the substitution <pi(x) for \l/(x) is made, suc-
cessively, for every subscript i.
A set of basic functions of this kind may be called reflexive, since it includes
the basic functions among the functions to be expanded in terms of the set.
As it can be shown that the basic functions of (l)-(3) must also be quad-
ratically integrable, it follows that if a set of this type is complete it must also
1 (4) is sufficient to derive (2) and (3) from (1), as shown. If, inversely, (1) and (3) are
to be derived from (2), relation (4) cannot be used; instead, a condition corresponding to
(16) would haye to be introduced, which is combined with the difficulties existing for
continuous variables.
48 PART II. MATHEMATICAL OUTLINES
be reflexive. Therefore (4) is a consequence of completeness. on the other hand,
even if summations and integrations are not commutative, it is possible to
derive (2) and (3) from (1) and (4) by more complicated methods, provided
that the convergence of the summation in (1) is suitably defined. In an expan-
sion of the type (1), therefore, the indirect characterization of orthogonality
can be satisfactorily replaced by a direct characterization.
The definition of completeness given above is also an indirect characteriza-
tion. Its replacement by a direct characterization represents a rather difficult
mathematical problem into which we shall not enter here. Condition (4), in
any case, is not sufficient to guarantee completeness. This can be shown as
follows. When we omit one of the functions <pi(x\ say the function <pi(x), the re-
maining set will satisfy (4) ; but it will be impossible to expand into this set a
function which, expanded into the original set, has a <TI ^ 0. For, if this were
the case, we should have two series, with different coefficients <n, for the expan-
sion of the function into the original set, with 0-1 = in the second series. This
shows that completeness represents a specific condition not included in ortho-
gonality and normalization.
The determination whether a certain given set of functions is complete,
orthogonal, and normalized, requires special analysis. It can be shown, for
instance, that the trigonometrical functions <p n (#) = r e inx satisfy these con-
V27T
ditions for whole numbers n within the domain D extending from to 2?r.
These functions are known from the Fourier expansion. Other sets vi(x) obtain
when the <pi(x) are defined as the solutions of certain differential equations ; they
then are called the eigen-functions of these equations. We shall deal with such
sets later.
The method of expansion can be extended to functions of a different type.
Thus, we can consider discrete functions fa (k = 1, 2, 3 . . .) which consist of
series of discrete numbers; they are called functions because we can consider
the value fa as a function of the subscript. Instead of the functions <pi(x) we then
use a set <pik consisting of a discrete matrix of numbers, i.e., a totality of num-
bers arranged in rows and columns. The matrix will in general be infinite in
both directions. The expansion then assumes the form
(9)
9- EXPANSION OF A FUNCTION 49
The conditions of orthogonality and normalization can be expressed by
analogy with (4) in the form :
2 .
Y") <pik<pmk* = &m
k
We have here complete symmetry between fa and <ri because both are discrete
functions. This is the reason that we have here two forms of the condition of
orthogonality and normalization. If the summations are commutative, the first
form is used to derive (10) and (11) from (9) ; the second form is used, together
with the same assumption, in order to derive (9) and (10) from (11). These
proofs are easily given by analogy with (6) and (7) . In this discrete case, how-
ever, it is possible to eliminate the assumption of commutative summations.
The two forms given in (12) represent independent conditions, the combina-
tion of which constitutes a very strong condition: It can be generally shown
that if both forms (12) hold, the system <pik is orthogonal, normalized, and even
complete, with respect to all functions fa possessing a finite amount of/^\
This includes a proof that the &* are reflexive, since the finite amount of
2^ \<pik\ 2 follows from the second form (12) for k = n. In the discrete case
i
we therefore have a rather simple form for a direct definition of the three
fundamental properties, a definition which is not limited to commutative sum-
mations. The proof of this general theorem cannot be given in this book; we
refer the reader to the mathematical textbooks on the subject.
Another type of expansion obtains when the functions <?i(x) are replaced by
functions of two variables, <p(y,x), i.e., when the subscript i is replaced by the
variable y. The function <p(y,x) may be considered as a continuous matrix, and
we speak here of the continuous case of an expansion. The expansion assumes
the form:
(13)
(14)
(15)
An expansion of this kind is used, for instance, for Fourier expansions
when the domain D is infinite; the Fourier functions then have the form
The continuous and the discrete case are homogeneous cases of expansions;
the first case may be called the heterogeneous case. It turns out, however, that
50 PART II. MATHEMATICAL OUTLINES
the three cases are not completely analogous to each other; the continuous
case shows some peculiarities. They concern, in particular, the conditions of
orthogonality and normalization.
A set of continuous basic functions is never reflexive; therefore relation (4)
is here not derivable. Furthermore, the integrations are not commutative,
since commutation leads here to indeterminate values. When we wish, in spite
of that, to express orthogonality and normalization directly, by a relation
analogous to (4), it can be done in a fictitious form. We then write
I <p(y,x)<p*(z,x)dx = 8(y,z) I v(y,x)v*(y,z)dy = d(x,z) (16)
The symbol 5(#,z), which is the continuous analogue of the symbol d ik) is defined
by the conditions
d(x,z) = for x ^ z n(x,z}dx = 1 n(x,z)dz = 1 (17)
But the symbol has only a fictitious meaning, since no function can be con-
structed which has these properties. The use of such a fictitious symbol is
permissible if a set of rules is given which translate formulae containing the
symbol into ordinary formulae. If, in addition, we were given a set of rules for
the manipulation of the symbol such that the validity of results derived by
means of the symbol were guaranteed, the symbol would have a legitimate
place in mathematics. It has so far not been possible, however, to give such
rules in general; the symbol, therefore, must be ' 'handled with care" in
mathematical derivations.
That the symbol &(x,z) must have the properties (17) results from the fact
that relations (16) can be derived only by means of an impermissible commuta-
tion of integrations. Substituting the value (14) for a(y) in (13) we obtain,
when we write z for x in (14) :
(18)
Changing the order of integrations, we arrive at :
(19)
(20)
Let us first consider the symbol 8(x,z) used here as an abbreviation whose
meaning is given by (20). From (19) we see that the symbol has the property
that, multiplied by a function $(z) and integrated over z, it reproduces the
function ^. This, as can be easily seen, is precisely the property defined by the
conditions (17). Notwithstanding the incorrect way of the derivation of (19),
Q. EXPANSION OF A FUNCTION 5 1
the symbol d(x,z) in (20) can therefore be identified with the d(x,z) symbol of
(17), if used for the purpose of certain integrations.
The 8(x,z) -function is called the Dirac function, having been introduced by
Dirac. It is used in quantum mechanics not only to formulate the condition of
orthogonality, but also for various other purposes. It is a fictitious symbol
because the conditions (17) cannot be directly realized; they have a meaning
only in the sense of approximations. We can define functions d n (x f z) which
satisfy (17) approximately; for a given value z they are = except when z is
within a small interval A#, within which d n represents a steep and high peak.
As n increases, this interval becomes narrower, and the height of the peak
grows towards infinite values; the integral over x, however, is always = 1. The
limiting case Ax = is a degenerate case, and it appears understandable that
the use of the Dirac function is criticized by mathematicians. The use of the
5(z,z)-symbol has only a psychological justification: The symbol suggests cor-
rect solutions which later can be derived strictly without the use of the symbol.
That for continuous matrices commutation of integrations is not permissible
is shown by the continuous Fourier functions <p(y,x) = ^e*. These func-
tions do not satisfy (16) ; the integral occurring in these relations is here an
indefinite ｅxpression and does not vanish for x 7* z. For continuous basic
functions we therefore prefer to define orthogonality and normalization indi-
rectly by the use of relations (13)-(15). Thus it can be shown that the con-
tinuous Fourier functions are normalized and orthogonal in this sense. We
may add the remark that in certain cases, for instance for the latter functions,
a direct formulation of orthogonality and normalization also can be given in
a rigorous way, without the use of the 5-symbol; but the presentation of such
methods goes beyond the frame of this book. Let it be said in general that all
results of quantum mechanics can be derived by rigorous methods. It may
suffice for us to avoid the 5-symbol whenever this can be easily done, and we
may be excused when we shall use this symbol occasionally in order to simplify
our presentation.
A further peculiarity of the continuous case concerns the changing of
variables. Such a change will introduce a density function r into the expansion.
If, for instance, we replace y by a variable u such that
du
and put
r(u) = ^ (22)
we arrive at the expansion
r
i)<p(u,x)r(u)du (23)
52 PART II. MATHEMATICAL OUTLINES
The density function r(u) must also be introduced into the relations of the
S-symbol. Thus we have, instead of the first integral in (17),
ti)dtt- 1 (24)
where 5 (r) (w,s) is to be conceived as the function resulting from &(y,z) by the
transformation (21). Relations (16) and (14) then remain unchanged, t}ut (15)
assumes the form
u) | *r(u)du (25)
Similar changes result if we replace x by another variable.
10. Geometrical Interpretation in the Function Space
The discrete case lends itself to a geometrical interpretation. Let us assume for
a moment that the subscripts run only from 1 to 3, and that fa and a are real
numbers. Then both ^ and <r can be considered as vectors in a three-dimen-
sional space, having the three components, respectively, ^i, ^ 2 , ^s, and 01, o- 2 , 03.
The relation (6), 9, then represents a transformation of vectors, given by
the matrix ^.
The structure of a space is determined by its metric. The space which we
consider here is Euclidean, i.e., its metric is given, for orthogonal coordinates,
by the theorem of Pythagoras, which determines the square of the length of a
vector a as the sum of the squares of its components:
a 2 = ar* + 02 2 + a 3 2 (1)
A generalization of this ｅxpression, constructed for two vectors, is the inner
product (also called scalar product) :
(a,b) = ai&i + 0262 + a 3 & 3 (2)
This mathematical ｅxpression is used to determine the relation of orthogonality.
Two non-zero vectors are perpendicular to each other if, and only if, their
inner product vanishes. The length can be regarded as a special case of an
inner product, namely, as the inner product of a vector with itself.
The orthogonal coordinates which scaffold the space can be conceived as
being determined by a set of unit vectors w a) , t*<*), w<s), which are orthogonal
to each other. The vector UM has the components 1, 0, 0; the vector u& has
the components 0, 1,0; and similarly the vector uw has the components 0, 0,
1. The orthogonality and the unit character of this set is expressed by the
condition
* (3)
10. GEOMETRICAL INTERPRETATION 53
A vector a can then be expressed as a linear function of the unit vectors :
a = ai u ( D + 02 i*c2> + 03 w<3) (4)
Taking the inner product, on both sides, with W ( D, and using (3), we obtain
ai (5)
Since a similar relation can be derived for the other components, we can write
down the general ｅxpression determining the components of a vector in terms
of the unit vectors :
ai (6)
The inner product is a linear function of the vectors; i.e.] it satisfies the
two relations
(a,fc-6) = k (a,6) (7)
(a,6 + c) = (a,6) + (a,c) (8)
where k represents any real number, and a, 6, and c any real vectors.
When we wish to introduce, instead of the set UM, another set v^) of unit vec-
tors, this is done by the transformation
V ( k) = ]j Cwtt<o (9)
i
The subscribt of summation, here and in the following formulae, runs from
1 to 3. In order to make the set VM orthogonal, the coefficients CM must satisfy
the condition of orthogonality
2 CtiChm = Sim 2 CkiC m i = Bkm (10)
k i
The orthogonality of the v^) then follows by the relations
(0<*), 0(m>) = ( 2 CwW(f), ^ C mn W(n) ) = 2 S Ck < Cmn ( U M> W <">)
\ i n /in
= ]T) ^ CkHOmniin = ^ CwCm* = 5 Am (11)
in i
The reverse transformation to (9) is given by
This follows with the use of (10) when we multiply (9) on both sides by c* m ,
sum up on k, and finally replace the subscript m by i :
(13)
54 PART II. MATHEMATICAL OUTLINES
The components of the vector a in the new system of reference are given by
(
(14)
This shows that the components of a vector follow the same transformation
as the unit vectors. The same holds for the inverse transformation :
Oi = ]T) Ckia >k (15)
k
which results from (14) by a proof analogous to (13).
Both length and inner product are invariants of orthogonal transformations,
i.e., the relations hold:
(a,a) = ai 2 + a 2 2 + a 3 2 = a/ 2 + a/ 2 + a 3 /2 }
\ (16)
It is sufficient to show this for the inner product, since the length is a special
case of such a product. We have
/
' = E *' 6 *'
km k
All these relations can be extended to an infinite number of dimensions, if
only precautions are taken concerning the convergence of ｅxpressions like the
length and the inner product; the components a; of a vector then are assumed
to converge toward zero with growing subscript i in such a way that these
ｅxpressions remain finite. Similar precautions permit us to consider summa-
tions as commutative. With these qualifications the formulae presented hold
equally when the subscript of summation runs from 1 to .
Furthermore, the relations can be generalized for the case of complex
vectors, i.e., vectors whose components are complex numbers. We then define
the inner product by the ｅxpression
GM = ]><** (18)
i
and similarly the square of the length by
(19)
10. GEOMETRICAL INTERPRETATION 55
It follows that the length is a positive real number, which vanishes only if \p
vanishes. The inner product is, in general, a complex number, which satisfies
the relation (*,*) = (*,*)* (20)
and, in addition, has the character of linearity as expressed in (7)-(8) :
= K* - (fcer) (21)
Here K is a complex constant, not a vector. For real numbers all these relations
are identical with the above given. The extension to complex numbers must
be considered as a convention enabling us to handle complex vectors by
analogy with real vectors. Thus we shall say that two complex vectors are
orthogonal to each other when the inner product (18) vanishes.
Combining the two forms of generalization, we shall consider the formulae
(18)-(21) as holding also for the case of an infinite number of dimensions, i.e.,
for subscripts i running from 1 to . The existence of the quantities (18) and
(19) is guaranteed by the definition of the vector; i.e., entities ^ and <r for
which length and inner product do not possess determinate finite values, are
not called vectors.
The space defined by the metric (18) and (19) is called unitary space, or
Hilbert space. This term denotes, for an unrestricted number of dimensions, the
complex analogue of Euclidean space, namely, a space in which a system of
orthogonal straight-line coordinates is possible. Introducing complex unit
vectors <) satisfying the relations of orthogonality
5ik (22)
we can express every vector ^ as a linear function of the unit vectors :
(23)
As before, the components fa are given by the relation
(24)
A transformation to another set of orthogonal unit vectors 17 > is called a unitary
transformation and is given by
where the coefficients satisfy the relation of orthogonality
<f>ik<f>in* =
56 PART II. MATHEMATICAL OUTLINES
By a derivation analogous to (13) we can show that the second of the relations
(25) follows from the first, when the relation (26) are assumed. The proof,
that owing to these conditions the vectors r?<o have the length 1 and are orthog-
onal to each other, is easily given by analogy with (11). The same coefficients
<?ik determine the transformation of the components fa of the vector \l/ into
new components <n :
These relations are proved by analogy with (14). Length and inner product
are invariants of such a transformation; i.e., the relations hold:
** (28)
r* (29)
k i
if X is a second vector in the -system and the n represent its components in
the ij-system.
Relations (26) and (27) correspond to the relations (12), 9, and (9)-(10),
9; (28) corresponds to (11), 9. We see that the expansion into a set of
orthogonal functions corresponds to a unitary transformation in a unitary
space of an infinite number of dimensions.
According to a remark by F. Klein 1 any such transformation can be inter-
preted in two ways, which may be distinguished as the passive and the active
interpretation. In the passive interpretation, which we have used so far, we
consider ^ and cr as the same vector, but represented in different systems of
coordinates. The matrix <p ik then represents a transformation of coordinates, and
(9), 9, determines the components of the vector in the ^-system as functions
of the components of the same vector in the <r-system. Since the transforma-
tion is unitary, it consists in a mere turning of the system of coordinates,
combined with mirror reflections.
In the active interpretation we consider ^ and <r as different vectors, but
represented in the same system of coordinates. The matrix ^a then represents
an operator. An operator, in the general sense of the term, is a rule which
coordinates to a given entity, or function, another entity, or function. Such a
coordination in terms of an operator can always be interpreted as a deforma-
tion of the space, which transforms the one entity, or function, into the other.
In the case considered here the operator ^ coordinates to every given vector
another vector. Owing to the unitary character of the transformation, the
deformation of the space produced by it is of a rather simple 'kind: it consists
merely in a turning of the space, combined with mirror reflections. In another
1 F. Klein, Elementarmathematik vom hoheren Standpunkte aus, Vol. II (Berlin, 1925),
p. 74.
10. GEOMETRICAL INTERPRETATION 57
version of the active interpretation we speak, instead of a deformation of the
space, of a coordination of two different spaces to each other performed by the
matrix <&&
The geometrical language can be extended to the continuous case (13)-(15),
9. The functions \l/ and <r then are considered as vectors in a space F whose
number of dimensions has the magnitude of the continuum. We must realize
that, in the space F, x and y are not coordinates; they rather perform the
numeration of dimensions, such that a special value of x determines a di-
mension, as does the subscript i for discrete functions, whereas the value of
the coordinate in this dimension is the value $(x). The totality of all values
\l/(x) is the vector ^. The space F is therefore called the function space, since
each of its points corresponds to a whole function. As before, the metric of the
space is determined by the ｅxpressions introduced for length and inner product,
the existence of which represents a condition defining vectors. These expres-
sions are given here by the relations :
(30)
(31)
Relation (30) corresponds to (15), 9. Unitary transformations can be defined
either by the fictitious way of writing used in (16), 9, or by the requirement
that the ｅxpressions (30) and (31) be invariants of the transformation. In the
passive interpretation of the transformation, the axes determined by the
various values of x are different from the axes determined by the various values
of y. In the active interpretation these axes can be identified.
The geometrical language can also be used for the heterogeneous case. Using
the passive interpretation we then consider \l/(x) and cr< as representations of the
same vector in different systems of coordinates, the transformation between
these systems being given by the heterogeneous matrix <f>i(x). The invariance of
the length is expressed by (3), 9. It follows that in this interpretation the
same space can be built up either by a denumerable infinity or by a continuous
infinity of dimensions. 2 In the active interpretation of this case we use the
version of different spaces. The mixed matrix w(x) then coordinates the vectors
of a space of a denumerable infinity of dimensions to those of a space of a con-
tinuous infinity of dimensions.
2 By the term 'Dumber of dimensions" we understand here the number of parameters
determining a point in the space. Another meaning of the term is used when it is defined
by the number of linearily independent vectors, which is denumerable also for a space
having, in the first terminology, a continuous number of dimensions. only for a finite
number of dimensions do both meanings coincide.
58 PART II. MATHEMATICAL OUTLINES
11. Reversion and Iteration of Transformations
The geometrical language enables us to discuss mathematical operations with
orthogonal expansions in terms of unitary transformations. In order to analyze
the nature of transformations it is necessary to answer two questions: first,
which is the converse of a given transformation, and second, which is the
transformation resulting from the iteration of two transformations. In answer-
ing these questions for unitary transformations, we shall develop further prop-
erties of orthogonal expansions. We begin with the discrete case.
The question of the reversed transformation is easily answered by (10), 9.
Let (fr 1 be the converse of 40; by analogy with (9), 9, we then define <p~ l by
the relation
l (1)
This definition is chosen in such a way that, as in (9), 9, the summation
runs over the first subscript of pw~ l . The choice of the letters i and k is of course
irrelevant. Now a comparison with (10), 9, shows that
<Pki~ l = <pik* = <pki (2)
This means that the reversed transformation obtains by reversing the sub-
scripts and taking the complex conjugate of the original transformation. The
matrix thus resulting, which is called the adjoint matrix to <pk%> will be denoted
by the arc symbol (^). L We see that in the theory of unitary transformations
the symbol <p~~ l is dispensable. Since we are concerned here only with such
transformations, we shall consequently always express the reversed transfor-
mation immediately by the symbol #. Whereas we derive here condition (2)
from the definition of unitary transformations by the condition of orthogo-
nality (7), 9, condition (2) is frequently used, inversely, as the definition of
unitary transformations.
The question of iteration of transformations is to be asked as follows. We
have considered so far a transformation between the ^-system and the a-
system, given by the matrix p ik . Let us now consider a further transformation
between the a-system and a third system which we call the r-system, and let
a?* be the matrix effecting the transformation between the two latter systems.
We then have
(3)
(4)
1 Instead of this symbol, the symbol f is used in some other notations.
11. REVERSION AND ITERATION 59
From these relations we obtain the direct transformation between the $-
system and the r-system as follows :
2 S
-S-
(5)
Xmk
If (f> and co are orthogonal, so is x- This is shown as follows :
= X] XI CO w COnl* X) pfl
i I k
= YY
vv
We therefore can consider (5) as the expansion of \J/ k in the orthogonal set Xmk-
The iteration of two orthogonal expansions constitutes a new orthogonal
expansion.
The relation (6) determines the new set x as a function of the two given
sets co and #>; in this relation we meet the matrix multiplication of tensors,
which is asymmetrical in the subscript of summation. We can solve this
relation for either co mi or <p ik by multiplying, respectively, with <pi k * or co w j*, and
summing up on k or m; we thus obtain
<*ml = 2^ Xmkfplk* (8)
k
^V*"^ # /Q\
m
Using the sign of the reversed transformations we can write these relations:
(10)
<f>lk = ^UimXmk (11)
m
In this form of writing, these relations, too, assume the form of a matrix
multiplication.
These relations can be illustrated by a diagram, as given in figure 7, in
which the arrows indicate the transformations. The direction of the arrows
6o
PART II. MATHEMATICAL OUTLINES
is chosen in such a way that, for instance, <p is interpreted as a transformation
leading from <r to ^; this corresponds to (3), since this relation determines ^ for
a given <r. The arrow x is, geometrically speaking, the sum of the arrows w and
<f>; the corresponding transformation x is determined by (6) as the matrix
product of the two transformations w and <p*
The general form of the relation (6) is ex-
pressed by the following relation in which
a transformation like <p is represented by a
symbol of the form Trn(a$), and in .which
the cross expresses matrix multiplication:
Trn(r^) =
yr) X Trn(*#) (12)
>O
Tig. 7. Triangle of transformations,
The relations (10) and (11) result when we
reverse, respectively, the directions of the
arrows <p and o>; and they can be expressed
in a form corresponding to (12).
The relations holding for the triangle of transformations are summarized in
the following formulae :
-Y,T mXmk (18a)
T m =
Xmk =
(136)
(13c)
(13d)
(13/)
Relation (13d) admits of a further interpretation. Let us consider the sub-
script m as constant, and denote this by putting the m into brackets. (13d)
then represents the expansion of x<)* into the orthogonal set <pi k> with w (W ) as
expansion coefficients. Now xwk represents, for a variable m, not one function,
like fa, but a set of functions each of which has a particular value of m. We
see that the set w (m) < represents the corresponding expansion coefficients for an
2 In the diagram we use a yectorial representation in which a vector cannot be shifted
parallel to itself. This condition is necessary in order to make vectorial addition non-
commutative, corresponding to the multiplication of matrices.
11. REVERSION AND ITERATION 6l
expansion in the vut? Similar interpretations can be given for (13e) and (13/) . We
have written these equations in such a way that the first term on the right
hand side corresponds to the expansion coefficients, the succeeding term, to
the basic functions. Our result can be stated as follows: In a triangle of trans-
formations we can choose the direction of the transformations in such a way
that a given transformation can be considered as the set of expansion coeffi-
cients belonging to the expansion of the second transformation in terms of the
third. .
Our. results can be transferred to the continuous case. The reversed trans-
formation $(x,y) then is given by :
(14)
(15)
For the triangle of transformations we have here the relations:
iKx) = J <r(y)<p(y,x)dy = j r(z)x(z,x)dz (16a)
(*)flx,y)<fe (166)
r(z) =(x)%(xddx =*(yWyrfdy (16c)
x(z,x) = <*(z,
These relations correspond exactly to those of the discrete case given in (13).
In the heterogeneous case the situation is more complicated because the step
from \l/(x) to 0i is structurally different from the reversed step, owing to the dif-
ference between continuous and discrete variables. We therefore cannot define
8 This interpretation cannot be reversed, i.e., it would be wrong to say that we can
also consider w w fe in (13d) as basic functions and <?ik as the set of expansion coefficients.
The reason is that our expansions are so defined that the summation runs through the
first subscript. We can, however, transform (13d) into the form
*\> V * V-/ A
Xmk 2,1 <Pki <*im
i
In this form the function <p* represents the expansion coefficients, and w* the basic func-
tions.
PART II. MATHEMATICAL OUTLINES
a reversed transformation which is of the same structure as the original trans-
formation, and must leave the reversed step in the form (2), 9.
There will be differences, furthermore, in the iteration of transformations
according as we choose the transformation w as a discrete or as a heterogeneous
matrix. Using a discrete w we have for the triangle of transformations :
= I
J
Tm = I t
= / <T t -CO f ' m
J
X m (x)<pi*(x)dx
(17o)
(176)
(17c)
(17d)
(17e)
(IT/)
These relations correspond to those of the homogeneous cases, with the modi-
fication that the symbol <pi*(x) takes the place of a symbol for the reversed
T O
Fig. 8. Quadrangle of transformations.
transformation. As before, these relations are written in such a form that the
first term on the right hand side corresponds to the expansion coefficients, the
11. REVERSION AND ITERATION 6g
succeeding term, to the basic functions. The corresponding relations for a
heterogeneous w can easily be derived.
The given relations can be extended to the case of four different basic
functions, for which we construct a quadrangle of transformations (figure 8).
We shall write the relations only for the discrete cases. Thus, in figure 8 we
have, always following the rules of the arrows and noticing that the use of
the arc sign means reversion of the arrow :
<f>ik =
since
]T) XmkXkn = 8 kn (20)
*
We see that the relations of the triangle of transformations, which we have
assumed for the three triangles cr^r, ^rp, <r^p, hold also for the fourth triangle
<rpr. The same can be proved for the continuous and for the heterogeneous case.
We can use these results to establish a relation between the two diagonal
transformations # and x- With
(19) leads to
(22)
In the interpretation of the diagram this means that in the "sum" (19) we
have replaced the term w t - w by the "sum" (21). Relation (22) therefore states
that the train of arrows <p, x> i? is equivalent to the arrow #. Applying the form
of writing used in (12) we can write (22) :
Trn(<r,p) = Trn(<r,t) X Trn(t,r) X Trn(r,p) (23)
All these relations are easily transferred to the continuous and to the hetero-
geneous case. Let us write down for later use only the following relations,
which are taken from the quadrangle of transformations:
E *&(*) (24)
m
(25)
(26)
(27)
64 PART II. MATHEMATICAL OUTLINES
12. Functions of Several Variables and the Configuration Space
The relations presented can be transferred to the case of functions of several
variables. Let us begin with the continuous case (13), 9. We shall replace x
by a set of variables Xi . . . x n , and y by a set y\ . . . y n \ correspondingly the
integration is to be replaced by a multiple integration. Thus we have, instead
of(13)-(16),9:
... Xn) = /.../ <r(yi . . . y n )<f>(yi ...y n ,xi... x n )dyi ...dy n (1)
- 2/0 = / - - fo(%i - - - n)<P*(*/i . . . ffo, zi . . . x n )dxi . . . dx n (2)
. Xn) \*dx, . . . dXn =f. . .f \*<3ll . . . 2/n) | 2 <fyl . . . djfr (3)
/ . . . / ^(3/1 . . . y n , xi . . . z n )^*(zi . . . Zn, Xi . . . a: n )rf^i . . . dx n
= 5(2/1,21) *(y.s) (4)
The condition of orthogonality and normalization is here expressed in the
fictitious form introduced in (16), 9; we use in this case a product of 5-
symbols, which is different from zero only if for each of the symbols the cor-
responding values 2/i and Zi are equal to each other ; otherwise this product is = 0.
All theorems and proofs given for the case of one variable can be similarly
transcribed for functions with several variables. The theory of this generalized
expansion is therefore formally given with the theory of the simple expansion.
For the heterogeneous case (1), 9, we can develop similar formulae.
Instead of the coefficients d we then shall obtain coefficients o^ . . . <7 ln , and
instead of the functions <pi(x) we shall have functions #>, . . . <?i n (xi x n ). Since
the number of these constants and functions is denumerable, we can, however,
introduce a numeration running in one subscript i, and, instead of a multiple
summation, we then have simply one summation in i. We therefore obtain the
formulae:
. . . X n ) = <r&i(Xi . ..X n ) (5)
= / . . .l$(xi . . . x n )<pi*(xi . . . x n )dxi ...dx n (6)
.y^fo . . . X n ) \*dXi . . . dx n = 2 hi* (7)
/ . . J <f>i(Xl . . . X n )<f>k*(Xl . . . X n )dXi ...dXn = 8ik (8)
IS. SEVERAL VARIABLES 65
The discrete case (9), 9, need not be generalized, because a discrete func-
tion \l/ik can always be replaced by a discrete function in one subscript, ^-, at
least for the case in which only multiple summations in both subscripts are
used.
We now shall consider a second geometrical interpretation different from the
one we gave in 10, which is particularly appropriate for the case of several
variables, although it can also be carried through for one variable x. In this
interpretation we consider the n+1-dimensional space C built up by the n
variables x\ . . . x n and, in addition, a dimension for ^, which is usually called
configuration space. The expansion (1) then can be considered as a transforma-
tion in the space C.
As in the other case, we have here a passive and an active interpretation.
In the passive interpretation we consider the functions \f/(xi . . . # n ) and
<r (2/1 . - - 2/n) as identical; the transformation then is a transformation of coor-
dinates. In the active interpretation we identify the systems x\ . . . x n and
2/i . . . 2/ni the transformation then has the character of an operator and per-
forms a coordination between the surfaces ^ and cr. We may here also use
the version in which the space of the x\ . . . x n is distinguished from the space
i/i . . . y n ] the transformation then has the same operator interpretation.
We must now explain a fundamental difference between this geometrical
interpretation and the interpretation presented in 10. In the latter interpre-
tation the transformation has the character of a point transformation. If we
use the active interpretation, this means that each point in the function space
F determines a coordinated point in F. The interpretation in terms of the con-
figuration space C, however, does not lead to point transformations. The whole
surface <r determines a surface ^; but this coordination is not done point by
point, since a change in the shape of o- within a certain domain changes the
value of the integral in (1) and therefore changes the whole surface \l/. We
even cannot say which point on ^ is to be coordinated to a given point on <r.
We may speak here of a holistic transformation, since it is a coordination be-
tween the surfaces as wholes which is performed in (1).
The same feature is to be found in the passive interpretation; here it means
that the transformation of coordinates is not a point transformation, but a
holistic transformation, depending on the shape of the surface \l/ (or <r). Since
this is different from what is usually called a transformation of coordinates,
the active interpretation seems preferable for this case.
Since the relation between \l/ and a in (1) is expressed in terms of an integral,
the expansion (1) is also called an integral transformation. We see that an
integral transformation can be interpreted either as a point transformation in
the function space or as a holistic transformation in the configuration space.
For the discrete case and the heterogeneous case the interpretation in the
configuration space seems less appropriate, although, of course, it can be
carried through.
66 PART II. MATHEMATICAL OUTLINES
13. Derivation of Schrodinger's Equation from
de Broglie's Principle
In the preceding sections we presented relations belonging entirely to the do-
main of mathematics. In the present and in the following sections we turn to
physics. It is the quantum mechanical method of establishing relations be-
tween physical entities which we now must consider.
In order to understand this method we must realize that quantum mechanics
is constructed as a generalization of classical mechanics. This generalization
has been achieved by the establishment of a rule by means of which an equation
of classical mechanics is transcribed into an equation of quantum mechanics.
Since classical relations are causal relations, and quantum relations are proba-
bility relations, the rule of transcription is so constructed that it determines
probability laws by analogy with causal laws. We are bound to use such a
method of generalization, because classical mechanics represents the only start-
ing point from which we can construct the new domain of quantum mechanics.
On the other hand, it is clear that we have no purely logical directive for the
establishment of the method of generalization. All that is to be required logi-
cally is that the new relations be identical with classical relations for the limit-
ing case h = 0; but this requirement leaves the rules of generalization arbitrary
within wide limits.
The way towards the construction of these rules, therefore, could not be
found by logical reasoning. It was the instinct of the physicist which pointed
the way. It is true that the men who did the work felt obliged to adduce logical
reasons for the establishment of their assumptions; and it seems plausible that
this apparently logical line of thought was an important tool in the hands of
those who were confronted by the task of transforming ingenious guesses into
mathematical formulae. L. de Broglie 1 was led by the consideration that the
duality of waves and corpuscles discovered for light should be extended to hold
equally for the elementary particles of matter; Schrodinger 2 was guided by the
analogy of mechanics and optics which enabled him to construct his transition
from classical mechanics to wave mechanics on the model of the transition
from geometrical optics to wave optics; Heisenberg believed that, since state-
ments about the orbit of an electron inside the atom cannot be directly verified,
statements about transition probabilities expressed as relations between ma-
trices must include all that can be said about elementary particles. The
analysis, added in a succeeding period of criticism, showed that, although the
conclusions of these inferences were true, the inferences themselves could not
be considered as valid. What justifies us today in considering the conclusions
as a well-founded physical theory is the amazing correspondence of the mathe-
* Ann. d. Phys. (10), Vol. 3 (Paris, 1925), p. 22.
* Ann. d. Phys. (4), Vol. 79 (Leipzig, 1926), pp. 361, 489.
1 3- SCHRODINGER'S EQUATION 67
matical system obtained, with known observational results, in combination
with its predictive power manifested in the results of newly devised experi-
ments. The historical development of quantum mechanics, therefore, consti-
tutes an illustration of the distinction between context of discovery and context
of justification, a distinction which must be made for all kinds of scientific
inquiry. 3 The path of discovery runs through "series of inferences which are
deeply veiled by the darkness of instinctive guessing," if I am permitted to
apply here a phrase which Schrodinger once used in a letter to me, written
some years before his great quantum mechanical discoveries. once a theory
has been constructed, it is to be judged within the context of justification, i.e.,
by the inductive evidence conferred to it through its empirical success.
To make this distinction clear let us consider a way in which Schrodinger's
differential equation for waves of matter can be derived from the principles
introduced by Planck and L. de Broglie. This is not the way actually used by
Schrodinger; it is rather a logical short cut of this way, which has been con-
structed later and which we use because an exact analysis of Schrodinger's
ideas would lead us too far away from the purpose of a merely logical analysis
with which this book is concerned.
The introduction of particle concepts into the wave theory of light goes back
to Planck's introduction of the quantum h. According to his assumption, every
light wave of the frequency v is equipped with an energy quantum of the
amount 4
H = h - v (1)
Planck's assumption was supplemented by Einstein's idea that, in a similar
way, a momentum of the amount
P-^-J (2)
c \
can be coordinated to each light wave. The momentum differs from the energy
so far as it is an entity equipped with a direction; therefore it is to be repre-
sented by a vector with the components p it Similarly we have vectors with the
components X and c t -. Relation (2) holds only for the absolute amounts p, c, and
X of the vectors. For the components we have here the relation :
ft-x< (3)
These relations are greatly simplified when we introduce the vector wave num-
ber with the components
.-* (4)
8 Cf. the author's Experience and Prediction (Chicago, 1938), p. 7.
4 We follow the usual notation expressing frequency and wave length by the letters "v"
and "X", thus deviating from our rule of reserving Greek letters for complex values.
68 PART II. MATHEMATICAL OUTLINES
The absolute amount b of this vector measures the number of waves in a unit of
length in the direction of the wave motion and is given by 6 = - . Using (4),
A
we can write (2) and (3) in the form
p = h-b pi = h-bi (5)
The momentum relation is thus made analogous to Planck's relation (1), with
the formal difference, however, that the energy relation is a scalar relation,
whereas the momentum relation is a vector relation.
Einstein's assumption could be interpreted as endowing light waves with
the characteristics of corpuscles, and thus as a statement of a dualistic nature
of light. L. de Broglie made the decisive step of extending this dualism of
waves and corpuscles to particles of matter by coordinating a wave to each
mass particle. The frequency v of this wave was determined by (1) in terms of
the energy of the particle. L. de Broglie saw that the velocity w of these waves
must be different from the light velocity c and that, therefore, the momentum
of the wave must be written, by analogy with (2) and (5), in the form
hv h 7 , 7 j /?N
p = = - = h-b pi = h-bi (6)
w X
From the relativistic ｅxpression for the momentum of a free particle
P = ^ = ^ (7)
c 2 c 2
where v is the velocity of the particle, L. de Broglie inferred that for a free
particle the velocity of the wave (i.e., the phase velocity) must be given by
w = C - (8)
v
In contradistinction to (8), relations (1) and (6) are not restricted to free par-
ticles, but hold for particles in all conditions. Let us consider a particle moving
in a potential field V(q^ g 2 , 3), where the g as usual denote the position coordi-
nates of the particle. When we now use only the nonrelativistic form the rela-
tion which holds between energy and momentum of the particle is given by:
(9)
2m
The first ｅxpression on the right hand side denotes the kinetic energy of the
particle of the mass m, and the second ｅxpression denotes its potential energy
in a field of force U(qi,q*,qd ; H is the total energy of the particle.
Now let us first consider a simplified case, namely, the case that the poten-
tial U is constant, i.e., does not depend on the g. If, in particular, U = 0, we
have the case of a free particle; the case U = constant, however, is not essen-
i g. SCHRODINGER'S EQUATION 69
tially different from this case. Since the force acting upon a particle is given
by the derivative of U, the case U = constant represents a case in which no
field of force exists. Let us express the constancy of U by omitting the argu-
ments qi.
We now can introduce the relations (1) and (5) into the equation (9). We
thus arrive at the relation
h.v = -(bS + bf + bfi + U (10)
2m
We see that the Planek-de Broglie relations permit us to transcribe an equa-
tion holding between energy and momenta into an equation holding between
frequency and wave number. In other words: The Planck-de Broglie rela-
tions can be used to transcribe a particle equation into a wave equation. The
latter may be called a frequency wave-length equation, since it connects these
entities. It is also called a law of dispersionj since it can be transformed into a
law connecting frequency and velocity of the waves when the relation v = biWi
is used.
Now let us assume that the wave is given by the complex function
-*)
Here v and the 6 are constants whose values we assume to be connected by the
relation (10). The ｅxpression (11) then denotes a set of monochromatic plane
waves spreading in the direction of the vector with the components bi having
the velocity w = -and satisfying the frequency wave length condition (10).
b
The number of wave periods on a straight line of unit length crossing the
parallel wave planes aslant is given by the component & taken for the direction
considered. As usual, the wave is expressed by the use of an exponential expres-
sion with an imaginary exponent. This is a mathematical device used also for
waves of other kinds, for instance, sound waves; it represents a method of
abbreviating ｅxpressions written in terms of trigonometrical functions. By
superposing several wave sets of the form (11) possessing certain properties of
symmetry, the imaginary part of the wave ｅxpression is usually eliminated.
What distinguishes the waves to be considered here from other waves is the
fact that even in a superposition of sets like (11) the amplitude ^ is a complex
number; thus the imaginary part of the resulting ｅxpression cannot be elimi-
nated. The meaning of this feature, which is an essential part of wave me-
chanics, will be made clear in 20.
The assumption (11) permits us to express the derivatives of ^ as follows:
fc / ! *V in 1 ty ,
ss o k \L -- 1- = OfcV --- - = vw
d k * 2wi dt
These simple relations make it possible to transcribe the frequency wave-
70 PART II. MATHEMATICAL OUTLINES
length equation (10) into a differential equation for waves. For this purpose we
multiply every term in (10) by ^, and then substitute the values (12). We thus
obtain the relation :
h d\l/ h 2 /#V . d*\l/ , #V\ . TT f /10 v
- + 1 ) + U-Y (13;
'i 2 dq? dqtfj
2iri dt
If, in particular, U = 0, we arrive at the equation
h d* rt
/ay ay + ay
)* \dfli 1 dqf dqf
2wi dt 2m(2iriy i \dqi 2 dq 2 * dq,
This is the time-dependent Schrodinger equation for waves corresponding to free
particles.
Equation (13) is not essentially different from (14), since we assumed U to
be constant. A difference results only with respect to the law of dispersion (10),
relating the constants v and &; if the potential field U is not zero, the waves
possess a velocity w different from the velocity of the case U = 0.
Now let us turn to the more general case where the potential is not constant,
but is a function U(q\ y q^qz) of the spatial coordinates alone. The derivation
given does not cover this case, for the following reason. If we consider U in (10)
as a function of the q iy the fe cannot be constants; but then the relations (12) do
not hold. The method of derivation used here, therefore, does not tell us which
kind of differential equation we must use for the case of a variable potential f7.
It is here that the method of derivation was to be replaced by "the darkness
of instinctive guessing". Schrodinger saw that equation (13) can be extended,
its form unchanged, to the case of a variable potential, and that thus the waves
of the general case are controlled by the equation
h dt h 2
2iri dt 2m(2<jri) 2
The solutions of this equation, called the time-dependent Schrodinger equation
for waves corresponding to particles in a field of force, do not have the simple
form (11); this follows from the remarks just made. Schrodinger saw that,
instead, solutions of a more complicated kind exist and that these solutions
possess the mathematical properties required for a theory of the Bohr atom.
To show this, Schrodinger considered solutions of the form
27rt ,
The solution (11) has this form when we put
?(<7i,<Ma) - fcfl-*+*+* (17)
H
since v = . However, (16) is of a more general kind, since <p is not bound to
h
13. SCHRODINGER'S EQUATION 7 1
the special form (17). Solutions of (15) in the general form (16) do exist,
although, as we saw, there are no solutions satisfying (17). Let us assume that
such a solution of the general form (16) is given. Carrying out the differentia-
tion on the left hand side of (15) and canceling the exponential factor of (16),
which does not depend on g, in all terms, we arrive at the equation
ri + !r. + ri)
dqi 2 dq? dqf/
H * -
dqf
This is the time-independent Schrodinger equation for waves corresponding to
particles in a field of force. It establishes a particular rule for the spatial part v
of the ^-function. Schrodinger recognized that this equation, if some require-
ments of regularity are added, has, in general, solutions only for discrete values
of the constant H and therefore determines discrete energy values which cor-
respond to Bohr's energy levels of the atom.
We certainly do not disparage Schrodinger's work when we do not consider
the derivation given, or the more complicated derivations originally given by
Schrodinger, as a proof of the validity of the resulting wave equation. Such a
derivation and Schrodinger never meant anything else can be used to make
the wave equation plausible, and therefore represents an excellent guide within
"the context of discovery. In our exposition it is the extension of (13) to the case
(15) of a variable potential U(qi t qt t qs) which cannot be justified by deductive
reasoning. But even the derivation of the wave equation (13) for the particle
in a constant potential field includes assumptions which by no means can be
taken for granted. Thus we have no a priori evidence for the assumption that
equation (9), after the introduction of the Planck-de Broglie relations, will
hold strictly. Judging without further knowledge of quantum mechanics, one
might well suppose that waves are controlled by a more complicated relation
than (10), and that the introduction of particle concepts like position and
momentum is connected with simplifications which confer to (9) an approxi-
mative character. A proof that this is not the case can be given only by showing
that the consequences of this equation are verified by observations.
It is only after this proof has been given that we can consider as valid the
principles used in the derivation of the equation, thus evaluating the deriva-
tion in the reverse sense. We must limit, however, this evaluation to the wave
equation for free particles, since only the inferences used in the derivation of
this equation can be reversed. We then can state the bearing of the derivation
given as follows: A wave interpretation of free particles, using waves which
satisfy Schrodinger's equation, can always be translated through the Planck-
de Broglie principle into a corpuscle interpretation satisfying the energy-
momentum relation. In other words, Schrodinger's wave equation guarantees the
strict duality of wave and corpuscle interpretation for free particles.
Whereas the principle of duality has turned out to be a good guide for the
establishment of the wave equation, it appears advisable not to use this prin-
7* PART II. MATHEMATICAL OUTLINES
ciple for the presentation of the system of rules which is now generally used for
the solution of quantum mechanical problems. First, it is clear from the deriva-
tion given that the duality is limited to free particles, i.e., particles whose
energy is composed of kinetic energy and a constant amount of potential
energy. Particles in a field of force do not satisfy the energy-momentum rela-
tion (9), as will be discussed in 33. Second, it has turned out that the prin-
ciple is of no help so far as the meaning of the function ^ is concerned; the
rules translating this function into probabilities cannot be derived from the
principle of duality. Third, it seems appropriate to begin the presentation, not
with the general equation (15), but with the special equation (18), and to
introduce this relation in a form not restricted to the energy H, but holding
for all kinds of physical entities. The justification of the latter extension is
given, as before, by its success.
We therefore now shall turn to an exposition of the system of quantum
mechanical rules which is not governed by the desire to make these rules
plausible, nor by the intention to show the origin of these rules in physical
conceptions grown from older quantum physics. This exposition will not refer
to the derivation of the wave equation given in this section. We leave it to the
reader to recognize in the rules to be presented the path of the derivation given.
14. Operators, Eigen-Functions, and Eigen- Values
of Physical Entities
The rules of transcription which coordinate quantum mechanical laws to class-
ical laws are stated by means of operators. The term "operator" is used here in
the same sense as it has been used in our explanation of transformations
( 10, 12), namely, as meaning a rule which coordinates to a given function
\l/(g) another function x(q)> The operators used in quantum mechanics are
generally constructed out of two elementary operators: the differential quo-
tient , and multiplication by q. These ｅxpressions are operators because they
dq
coordinate to a given function $(q)j respectively, the functions an d q-\[/(q).
dq
The construction of more complicated operators is established in the following
way.
We assume that a mechanical problem is given, in the classical form, in the
canonical parameters gi . . . q n , pi . . . p n . A physical entity u is determined,
classically speaking, if it is given as a function of these parameters:
u = u(qi . . . q nj Pi . . . Pn) (1)
We may assume that u is a polynomial in the p -. We now coordinate
to qi the operator : multiplication by g<
14- OPERATORS 73
to/fe) the operator: multiplication by/fe)
7 jik
to pi the operator : . (2)
2-Tri d#i
A 2 d 2
to p 2 the operator : --
' p
where /(#) means any function of g, and operators of higher powers of pi are de-
fined tn continuation of the rule given for pf. Putting these operators in the
place of the coordinates g* and p t within the function u(qi . . . g n , PI . . pO, we
obtain a compound operator u op . We have thus coordinated to the entity u,
instead of the function u(qi . . . q n , pi . . . p n ) of (1), an operator u op .
For instance, let u be the energy H, and assume that the function H(qi . . . q ny
pi ... p n ) has the form (9), 13. Our rule (2) then coordinates to (9), 13,
the operator
Hop = -L (AY (j* + ^ + *\
P 2m\2<jri/ \d qi * dqj dqj/
where U(qi,q*,q$) is now an operator meaning multiplication with this function.
We say that the entity u is represented by the operator u op within the phys-
ical context stated by the equation (1) . It is important to realize that we cannot
speak without further qualification of an operator coordinated to an entity;
the operator is determined only if the entity occurs within a given context, i.e.,
if the entity is defined as a function of the pi and qi This context can only be
stated classically when we formulate it in ordinary language. If this reference
to classical mechanics is to be avoided, we must consider the operator as the
definition of the physical context.
The operator is a mathematical instrument which we use to coordinate to
the entity it represents: eigen-functions and eigen-values. This is done by means
of a differential equation constructed in terms of the operator, called the first
Schrodinger equation. It has the same form for all physical entities; the differ-
ences between the various entities find their ｅxpression only in the nature of
the operator. We therefore also speak of the eigen-functions and eigen-values
of the operator.
The first Schrodinger equation, which is also called the time-independent
Schrodinger equation, has always the following form:
Uo P <p(q) = u <p(q) (4)
Since this equation is required to hold in combination with certain require-
ments of regularity, such as the requirement that the function <p be finite for
all values of its argument, it has, in general, solutions <? only for certain discrete
values Ui of the constant u; these values u* constitute the eigen-values of the
entity u. We speak in such a case of a discrete spectrum of eigen-values. To each
74 PART II. MATHEMATICAL OUTLINES
value Ui belongs a function <pi(q) as a solution of the equation; these functions
>*(#) constitute the eigen-functions. We can write (4) for this case in the form
u OP <Pi(q) = Ui w(q) (5)
In other cases we have solutions for a continuous manifold of values u; we
then say that we have a continuous spectrum of eigen-values. The solutions <p(q)
contain, of course, the constant u\ they therefore have the form <f>(u,q) and
form a continuous set of orthogonal functions. The Schrodinger equation (4)
then assumes the form
Uo P v(u,q) = u - <p(u,q) (6)
We must realize that here the entity u has the character of a constant with
respect to the operator u op , since this operator, according to (2), contains only
operations in the variable q. This constant u takes the place of the subscript i
in (5) and classifies the solutions <p of (4) .
It may happen that, for one eigen- value w, (5) furnishes, not one, but n solu-
tions <pi(q) . This is called a case of degeneracy of the degree n.
In order to show the method at work we continue our example. If we insert
the operator (3) in the general equation (4), we obtain the differential equation
(18), 13. In combination with requirements of regularity the equation has
solutions only for a set of discrete values Hi, the eigen-values of the energy; the
coordinated solutions <p(q\,q*,qz) constitute the eigen-functions of the energy.
Examples of the latter functions, constructed for special forms of the potential
U(qi,q<t,qa), are given in the textbooks of quantum mechanics.
By means of the mathematical technique so far developed we can coordinate
to every physical entity within a given context a set of eigen-values and eigen-
functions. The mathematical importance of these concepts consists in the fact
that the eigen-functions defined by the first Schrodinger equation constitute an
orthogonal set, and that the eigen-values are real numbers.
The proof of this theorem requires some additional remarks about operators.
Using the geometrical mode of speech introduced in 9, we can regard an oper-
ator as a transformation in the active interpretation of this term; and we can
therefore say that the ｅxpression u op <p denotes a vector coordinated to the vec-
tor <f> by the deformation of the space. The Schrodinger equation (4), in this
geometrical interpretation, states the condition that the series of deformations
of the space expressed by the operator u op is of such a kind that ultimately the
vector <p is transformed into a simple multiple u <p of itself.
The operators of quantum mechanics are all of a special type : They are linear
and Hermitean. An operator u op is called linear if for any vectors <p and \ it
satisfies the relations :
x) = u op <f> + u op x 1
14. OPERATORS 75
where K is any complex number (not a vector). An operator is called Hermitean
if for any two vectors v and x the relation holds : l
P, x) = (<P, u>opx) (8)
where the parentheses denote the inner product, as above. It can be shown
that the operators defined in (2) are linear and Hermitean. If the latter condi-
tion is not automatically satisfied by the construction of further operators
through the rules (2), the form of the function (1) is interpreted in such a way
that the operator is made Hermitean. A corresponding qualification is to be
added to the rules determining the construction of operators. It can be shown,
furthermore, that the energy operator (3) is Hermitean.
We now can prove that the solutions of (4) constitute an orthogonal set, and
that the eigen-values are real numbers. To prove the latter condition, let us
start from the relation r \ / \ n m\
, <f>) - (<?, u op <f>) = (9)
which follows from (8) if we take the special case x = <f>- With (4) this can be
transformed into / \ t \ n fi<\\
(u<p, <p) - (<p, u<p) = (10)
Since u is a number (not a vector), we can apply (21), 10, and thus derive:
(11)
The division by (<p,<p) is permissible since (<?,<?) = holds only if the function
<t> vanishes, a case which need not be considered. The result means that u is a
real number. To prove the orthogonality of the set, let us assume two functions
W and w which constitute solutions of (4) ; we then have, because of (8) and (4) :
=
(12)
Since Ui and u* are assumed to be different from each other, the inner prod-
uct (<f>i,<pz) must vanish. This expresses the orthogonality of these functions.
The condition (p,p) = 1 can be easily satisfied by multiplying the function
by a suitable constant of normalization, since the definition of vectors requires
that (<P,<P) must be finite,
The given proof cannot be applied to the case of continuous eigen-values u
for the following reasons. Eigen-functions <p(u,q) do not possess a finite
/ I <p(u,q) | 2 dg, as is visible also from (16) and (17), 9, the $(z,2:)-symbol being
infinite for x = z. Such functions, therefore, do not constitute vectors in the
1 The name "Hermitean' ' is derived from the name of the French mathematician
Hermite. It can be shown that a Hermitean operator must be linear, whereas, of course,
a linear operator need not be Hermitean.
76 PART II. MATHEMATICAL OUTLINES
Hilbert space, and thus cannot be substituted for <f> and x in (8). The result
(^1,^2) = is therefore not derivable for them, in correspondence with the
fact that these functions are not reflexive (cf. 9). Functions of this kind are
2*i pq
the Fourier-functions const e h which represent the eigen-functions of the
momentum p. The eigen-value problem of such functions has been discussed
by J. v. Neumann; 2 it can be shown that also the continuous eigen-spectrum
of the Schrodinger equation constitutes an orthogonal set.
15. The Commutation Rule
We now must consider a further property of operators. If an operator u op is
applied to a function <p, it produces a new function; to this function we can
apply another operator v op and thus produce a third function. This iterated
application of operators is expressed in the form
V pUop<{> (1)
The combination v op Uop can be considered as a new operator which is called the
product of the two operators. Here the word "product" is, of course, used in a
generalized sense, similar to the term "relational product" of logistic. It is
obvious that, in general, the application of the two operators in the reversed
order will not lead to the same function; i.e., in general, the ｅxpression
V op Uop<f> UopV p<p = [VopUop UopV op ]<f> (2)
will not be equal to 0. The multiplication of operators, therefore, is, in general,
not commutative. There is, of course, nothing paradoxical in such a statement
if we realize the meaning of the word "product", or "multiplication", used here.
Usually the ｅxpression (2) will be different from zero for most functions ^?,
and equal to zero only for some special functions <p. When, for two particular
operators, the ｅxpression (2) vanishes for all functions ^>, this will represent a
specific property of these operators; they then are called commutative operators.
Similarly, the entities to which these operators belong are called commutative
entities. Regarding the ｅxpression in the brackets in (2) as one operator, which
we call the commutator, we can express this property symbolically by saying
that the commutator of the two operators vanishes, or satisfies the equality
VopU op UopVop = (3)
On the other hand, operators for which the ｅxpression (2) does not vanish
identically in the <p, or, what is the same, for which there are functions <p such
that (2) is not = 0, are called noncommutative. We characterize such operators
by saying that their commutator does not vanish, or that for it the inequality
VopUop UopVop 7*0 (4)
holds. The corresponding entities are called noncommutative entities.
2 Mathematische Grundlagen der Quantenmechanik (Berlin, 1932), II: 6-9.
15. THE COMMUTATION RULE 77
It can be shown that commutative operators or entities have the same sys-
tem of eigen-functions, although, of course, they have different eigen-values.
This case holds for entities which are functions of each other. For instance, the
entities u and u 2 have the same eigen-functions, but different eigen-values.
Noncommutative operators or entities, on the contrary, have different systems
of eigen-functions.
The proof follows when we apply the operator u op to both sides of (4), 14:
VopU p<f> = V op U<p
When the operators are commutative we can reverse their order on the left
hand side; on the right hand side, using (7), 14, we can put u before the oper-
ator. We thus obtain
UopV p<P = U V op <p
When we regard here the ｅxpression v op <p as a unit, this equation states that the
function v op <p satisfies (4), 14. Disregarding the case of degeneracy, we there-
fore can infer that v op <f> must be identical with <p apart from a constant factor
(since (4), 14, determines <p only to a constant factor). Calling this constant v }
we thus have
V OP <P = V(p
But this relation has the form (4), 14, for the operator v op , and therefore states
that <p is also an eigen-function of v op . It is easily seen that this proof can be
reversed; i.e., that starting with the assumption of identical eigen-functions,
we can show that the operators are commutative. It follows that noncommu-
tative operators have different eigen-functions. Let us add the remark that
the proof can be extended to include the case of degeneracy.
The distinction of these two kinds of operators finds an important applica-
tion with respect to the elementary operators p op and q op defined in (2), 14.
Applying these two operators (constructed for the same subscript i, i.e., for
coordinated values pi and #) to a function <p we have
hid. . d<p\ h x _ x
Po p q op <p - q op po P <P = <7- (ff ?) - ff f = ^-V W
2-n (dq dq) 2m
We write this in the form
Po p q op q OP po P = (6)
2m
This equation is called the commutation rule. It states the noncommutativity of
the momentum-operator and the position-operator. In contradistinction to this
case it can easily be shown that two operators of the same kind, such as the
operators q\ and q^ or the operators p\ and p^ are commutative. The two non-
commutative parameters pi and q> are also called canonically conjugated param-
eters or complementary parameters. Two parameters p and q k , which are not
canonically conjugated (i.e., which have different subscripts), are commutative.
78 PART II. MATHEMATICAL OUTLINES
The p and q are the fundamental noncommutative operators. There are also
other operators which are noncommutative with each other. An entity which
is a function of the p alone is commutative with the p, but not with the q\ and
vice versa a function of the q alone is commutative with the g, but not with
the p. Functions of both q and p, however, can be noncommutative with both
q and p ; an example is given by the angular momentum. 1
16. Operator Matrices
Before we turn to the application of the results so far obtained we shall first
explain a second method of determining eigen- values and eigen-functions.
The geometrical interpretation of operators suggests the supposition that
linear operators can be replaced by matrix transformations such as explained
in 9. This supposition can be shown to be true for the discrete case. An expres-
sion like u op <p then is replaced by a summation, or integration, in terms of a
matrix.
For this purpose we coordinate to the operator u op an operator matrix. We
call this matrix T^. The components of r& are defined by the application of u op
to the unit vector > (cf. 10). We define
TK = [u op e (i )]k (1)
It then can be shown that the application of the operator u op to a discrete
function x is equivalent to a summation in the following sense
The left hand side means the fc-th component of the function resulting from
the application of u op to x-
We can show that condition (8), 14, of the Hermitean character of opera-
tors is equivalent to the condition that the operator matrix m* satisfies the con-
dition * , N
TK = r ik * (3)
Writing (8), 14, as a summation according to (18), 10, we have with (2)
= (u op <f>, X) - (
k
= 2 ]^ (TK T ik *)<piXk* (4)
k i
If this is to hold for any two vectors <f> and x, the ｅxpression in the parentheses
in the last term must vanish. We therefore have TK = r t -**.
Whereas in the discrete case we can always coordinate an operator matrix
1 Cf. H. A. Kramers, Grundlagen der Quantentheorie (Leipzig, 1938), p. 166.
l6. OPERATOR MATRICES 79
to a linear operator, this is not always possible for the continuous case. If
there is such a matrix r(q',q) the application of the operator is equivalent to
an integration in the following sense :
J
x(q) r(q',q)dq (5)
The left hand side means the value of the resulting function at the place q'. It
can be shown, as before, that Hermitean character of the operator, defined
by (10)*, 13, is equivalent to the condition:
r(q',q) = r*(g,<z') (6)
The advantage of condition (8), 14, over (3) and (6) consists in the fact that
(8), 14, can always be applied, even if no operator matrices exist. 1 For opera-
tors of the heterogeneous type, i.e., operators which coordinate a discrete
function to a continuous function, or vice versa, Hermitean character cannot
be defined.
In cases where a continuous operator matrix does not exist, we can at least
introduce matrices which approximately possess the required properties. This
approximation is reached by means of the Dirac functions (cf. 9). It turns
*\
out that both the elementary operators q and require a treatment of this
dq
kind and have Dirac functions as their matrix functions. The matrix function
*\
of the operator q is the function q 8(q',q) ; that of the operator is the first de-
dq
rivative5'(g',g).
In order to construct the operator matrix of a physical entity, it is not
necessary to construct first the Schrodinger operator by means of the rules
(2), 14, and then to use (1). Starting from the classical function (1), 14,
one can directly proceed to the construction of the matrix T# of the entity u
when the matrices coordinated to the entities p and q are given; the matrix T#
then is built up of elements written as functions of the matrix elements of p
and q. We shall not give here the rules necessary for this construction of opera-
tor matrices, but refer the reader to the textbooks of quantum mechanics. A
further rule in this matrix mechanics is the commutation rule (6), 15; since all
other operator matrices are functions of the matrices p and q, this basic prop-
erty of the operators p and q finds an ｅxpression in the structure of all other
matrices.
1 It is possible to construct matrices which are at the same time Hermitean and unitary.
For such matrices we have with (2), 11, and (15), 11, and (3) and (6) :
wk ** <Pik~ l v(q',q) - <f>~ l (q',q)
The matrices of quantum mechanics, however, usually satisfy only one of the two con-
ditions. Matrices, or operators, coordinated to physical entities, are Hermitean; matrices
used as eigen-functions, or transformations, are unitary.
80 PART II. MATHEMATICAL OUTLINES
Once the operator matrices of physical entities have been constructed, they
can be used to determine eigen-values and eigen-functions of the entities. The
method used for this purpose is once more independent of the Schrodinger
method and thus does not make use of the Schrodinger equation; instead, it is
based on the following postulate.
Matrix postulate: Construct a unitary matrix <?ik which transforms nk into a
diagonal matrix Ui 5^, i.e., a matrix whose values are different from zero
only in the diagonal lines i = /fc; the values u> of this diagonal matrix are the
eigen-values of u, and the matrix <?ik represents the eigen-functions of u.
The eigen-values so defined are real numbers because of the Hermitean char-
acter of Tik and the unitary character of <pik] the latter property guarantees, at
the same time, that the <pik constitute a complete orthogonal system. The fun-
damental role which the commutation rule plays in this method is clear from
the following result: Although the elements Tik depend on the elements of the
matrices p and g, the eigen-values of nk, i.e., the diagonal matrix into which nk
is transformed, will not depend on the elements of p and q if only the latter
matrices satisfy the commutation rule. In other words: The latter rule intro-
duces a sufficient number of relations between the matrix elements of p and q
in order to enable us to eliminate these matrix elements from the eigen-values
of Tik. It is the commutation rule, therefore, which makes the eigen-values of
physical entities definite.
For continuous operator matrices r(u,g) a similar formulation can be given.
It was this matrix mechanics which was introduced by Heisenberg, Born,
and Jordan, 2 and separately by Dirac, 8 independently of the wave theory of L.
de Broglie and Schrodinger. Schrodinger, after having constructed the differ-
ential equation (4), 14, in the pursuit of wave conceptions, showed in a con-
secutive paper 4 that this differential equation is mathematically equivalent to
the postulate of matrix mechanics, since the eigen-values and eigen-functions
of the operator, defined by (4), 14, are identical with those defined by the
matrix postulate. Mathematically speaking, the Schrodinger method has the
advantage that it is easier to solve the differential equation (4), 14, than to
find a unitary matrix ^ of the required properties; and actually the construc-
tion of such unitary matrices is usually given by the use of the Schrodinger
equation. This is the reason that now the matrix form of quantum mechanics
is usually replaced by the operator form, i.e., a form in which the equivalence
(2), or (5), between operator and matrix summation is not used. The eigen-
values and eigen-functions of operators are then defined by the Schrodinger
equation (4), 14, which makes the use of the matrix postulate unnecessary.
2 Zeitschr.f. Phys., 35 (Berlin, 1926), p. 557.
3 Proc. Roy. Soc. London, (A), Vol. 109 (1925), p. 642.
* Ann. d. Phys. (4), Vol. 79 (Leipzig, 1926), p. 734.
17- DETERMINATION OF DISTRIBUTIONS 8l
17. Determination of the Probability Distributions
Eigen-values and eigen-functions characterize physical entities within a given
physical context. We must, however, distinguish the physical context, as the
totality of structural relations, from the physical situation, which includes, in
addition to the structure, a determination of the probability distributions of
the entities involved.
Classically speaking, this distinction refers to the functional relations of a
physical problem on the one side, and the numerical values of the entities on
the other side. The totality of functional relations determines the context; but
the situation is known only if, in addition, the numerical values of the entities
are given. In quantum mechanics the functional relations are replaced by the
structure of the operators which, in combination with the first Schrodinger equa-
tion, determine the eigen-values and eigen-functions. The indication of numer-
ical values, on the other hand, is replaced by the indication of the probability
distributions of the entities. This corresponds to the transition from causal
laws to probability laws, which expresses the outstanding feature of quantum
mechanics.
The characterization of the physical situation therefore requires a mathe-
matical instrument which, in addition to the structure expressed in terms of
eigen-values and eigen-functions, indicates the probability distributions. This
instrument is the state function ^. Postponing the question how this function
is to be ascertained, we shall now explain the way in which it determines the
probability distributions.
The state function ^ is a function of the coordinates g; in addition, it will,
in general, depend on the time t. It therefore has the form \l/(q\ . . . q n , t), or
$(q,i). The second term may be considered as an abbreviation, in which we
write, as before, q for q\ . . . q n . We shall turn to the analysis of the time
dependence of \p later ( 18) ; for the present we shall therefore omit the argu-
ment t, and simply write \I/(q). We may understand this as meaning that we
choose a special value of t, and consider the relations then ensuing between the
so-specialized function $ and the probability distributions; the same relations
will hold for every value of L
A general requirement concerning \l/ is that this function is quadratically
integrable and normalized, i.e., that it satisfies the relation
= 1 CD
According to our statement concerning time dependence, this condition is sup-
posed to hold for every value t. The meaning of this condition, which is closely
connected with the probability interpretation, will turn out presently.
Before we determine the probabilities of observable values we must first
82 PART II. MATHEMATICAL OUTLINES
state which are the possible values. This is necessary because in quantum me-
chanics, in contradistinction to classical physics, not all numerical values are,
in general, physically possible values. The desired statement is given by the
following rule:
I. Rule of the eigen-values. The eigen-values constitute the possible values of the
entity within the given context.
An example of a discrete spectrum of eigen-values is given by the energy
levels in an atom, which are determined by the Schrodinger equation (5), 14,
written for the operator H op . The sensational effect of Schrodinger's discovery
was due to the fact that with his equation Bohr's discrete stationary states of
the atom were shown to be interpretable as an eigen-value problem. only if
the spectrum of eigen-values is continuous and unlimited, does the list of pos-
sible values correspond to the classical case. Of such kind is the eigen-value
spectrum of the position coordinates of a free particle. In most cases the eigen-
value spectrum is partly discrete, partly continuous. In the hydrogen atom,
for instance, discrete spectra are produced by transitions from one orbit to
another; the continuous spectrum corresponds to the capturing of free elec-
trons, or to the reversed process, i.e., ionization.
Whereas the eigen-values have an immediate meaning with respect to obser-
vations, the eigen-f unctions have not; they constitute only a mathematical
instrument necessary for the derivation of the probability distributions. These
distributions, to which we now must turn, are determined by the following rule,
which has been introduced by Born, and which we have already mentioned in
a special form in 2:
II. Rule of spectral decomposition. Expand the function \f/(q) into the eigen-
functions of the considered entity; the squares of the expansion coefficients cr<, or <r(u),
then determine the probability that an eigen-value Ui, or u, is to be observed.
This rule is expressed by the following formulae :
Discrete case:
(2)
P(s >Ui ) = H (3)
Continuous case:
ff(u)<p(u,q)du (4)
J ff
P(,)=K)| (5)
The symbol P stands for "probability". Since these probabilities are relative
to the physical situation s characterized by the function ^, we have put the
symbol "s" into the first term of the probability ｅxpression. Instead of using
17- DETERMINATION OF DISTRIBUTIONS 83
the logical symbol P, we can also express a probability by means of the mathe-
matical symbol "d", which denotes a probability function. We then have, in-
stead of (3) and (5), the relation:
d(ui) =|<r,| (6)
<*(")- k(t0| (7)
The function d depends on the situation s; if necessary this can be expressed
by a subscript, in the form d&(u). (3) and (6) are probabilities; (5) and (7) are
probability densities, i.e., they are turned into probabilities by integration
over u, between any two limits u\ and u*.
From the exposition of the theory of expansions given in 9, it is clear that
the probabilities (3) and (5) do not depend upon q y since a does not. Because
of (1) and (3), 9, and (15), 9, the probabilities satisfy the condition
/
<r(i*)|<fti = l (9)
This condition is necessary because one of the eigen-values must be realized.
For the case that an eigen-value is degenerate (cf . p. 74) the expansion (2)
assumes a somewhat more complicated form. Let <f>i.(q) be the n eigen-functions
belonging to the eigen-value w; then the term o-^(g) of the expansion (2) is re-
placed by a term
The <r; g satisfy the condition
i>.i 2 =i
=i
n
The ｅxpression <*%&%,(& can be formally treated like a term Vi(q). We there-
fore shall not introduce a special indication for the possibility of degeneracy
in the following formulae. An illustration of degeneracy is given by a Bohr
atom in which the same total energy Hi can result with various arrangements of
the electrons in different orbits. Each such arrangement constitutes a determi-
nate physical state and is characterized by one of the eigen-functions <pi t (q).
To one eigen-value Hi there correspond here different states <p i9 (q) ; the proba-
bilities of these states are given by | ffiff it \ 2 .
For the continuous case (5), (7), and (9), we must add the qualification that,
if u is not suitably chosen, a density function r(u) according to (22), 9,
will appear; we then have
k(u)|Mu) (12)
)du = 1 (13)
84 PART II. MATHEMATICAL OUTLINES
The density function r(u) can always be eliminated if we introduce, instead of
u t a suitable function u r of u.
We now must add a rule determining the probability distribution of the
entities used as arguments of the ^-function. We have here the rule mentioned
already in 2:
III. Rule of the squared ^-function. The probability of a value q (or of a set of
values qi . . . q n ) which is an argument of the \f/-f unction is given by the ｅxpression
This rule makes clear the meaning of the normalization (1). The integral
of | ^ | 2 taken over all values q means the probability of observing any value q
at all, and this probability must be = I. 1
Rules I-III hold for every quantum mechanical system, and for every time
point t. These rules connect the abstract mechanism of formulae with observa-
tional entities. It is this fact from which they derive their significance. Since
the probabilities are derived always as the squares of complex functions, these
latter functions are sometimes called probability amplitudes.
It can be shown that rule III can be formally included in rule II. The eigen-
functions <p(u,q) of the entities u considered contain the entities u and the
coordinates q as their arguments; they therefore represent relations between
u and q. It is formally possible also to introduce eigen-f unctions of the q\ these
functions then represent relations in which the entity q occupies both places of
the function. We express these functions by <f>(q'jq). By putting the operator
q op in the place of u op we can even use the Schrodinger equation (5), 14, to
define these functions; the equation then furnishes the Dirac functions 5 (#',<?)
as representing the >(<?',#). Expanding \f/(q) into these eigen-functions we have
the trivial relation (cf. (14), 9) :
(15)
We see that rule II, applied to this expansion, immediately furnishes (12),
i.e., rule III.
Rule II in combination with the expansion of \l/(q) in terms of the <p and the
Schrodinger equation (5), 14, permits us to derive a simple relation for aver-
ages of physical entities, which is frequently used in quantum mechanics.
According to the rules of probability, the average of an entity u with the distri-
bution d(u) is given by the ｅxpression
r
Av(u) = lud(u)du (16)
1 For certain physical problems condition (1) is abandoned, and a nonquadratically
integrable ^-function is used. This means that |^| 2 is interpreted, not as a probability,
but as an instrument of determining ratios of probabilities.
1 8. TIME DEPENDENCE 85
With (7), (4), and the complex conjugate relation of (11), 9, we have
Av(u) = Ju d(u)du = Ju | ff(u) \ z du = j<r(u)u <r*(u)du (17)
q)dudq (18)
when we change the order of the integrations. Now we have with (4), and be-
cause oiF the linearity of u op (note that a(u) has the character of a constant
with respect to u op ) :
u pt(q) = u op j<r(u)<f>(u,q)du = I u op (r(u)v(u,q)du
= / cr(u)u p<p(u,q)du = / <r(u)u <p(u,q)du (19)
For the last step we have used the Schrodinger equation in the form (6), 14.
With (19) we can write (18) in the form
Av(u) =*(q)u op t(q)dq (20)
Since the average of an entity is sometimes called expectation value, this formula
is often called expectation formula. Comparing (20) with the last ｅxpression
in (17) we see that, formally speaking, the introduction of $(q) in the place
of <r(u) is accompanied by the replacement of u by u op .
18. Time Dependence of the ^-Function
We now turn to considerations which concern the changes of the physical
situation within the course of time, and which therefore deal with the de-
pendence of ^ on t. Just as the numerical values of variables may change in
classical physics, such as the position of a mass point in motion, or the angular
deviation of a pendulum, the probability distributions of the entities can
change in quantum mechanics.
It is here that we come to use the general Schrodinger equation introduced
in (15), 13, since this equation controls the time dependence of \f/. Whereas
(15), 13, is given for a special form of the energy, we shall now present the
general form of this equation. Let us state here, as before, that the justification
of this equation is given only by its empirical success. The second Schrddinger
equation, or time-dependent Schrddinger equation, has the form
H^(q,t)=^^(q,t) (1)
&TC Ot
Hop means the operator of the energy. It is important that, whereas the first
Schrodinger equation can be equally applied to operators belonging to all
86 PART II. MATHEMATICAL OUTLINES
kinds of physical entities, the second Schrodinger equation is constructed only
in terms of the energy operator. 1
For an illustration of (1) we may use the special form (3), 14, of the
Hamiltonian operator H op . (1) then assumes the form (15), 13.
The form of the general equation (1) makes it necessary that the function \l/
practically always depends on time. For a time-independent ^ the right hand
side of (1) vanishes. Now the energy operator H op , in all practical cases, is so
constructed that the left hand side of (1) does not vanish. 2 Hence V' must
depend on time.
The significance of equation (1) consists in the fact that it determines the
form of \ls(q,f) for every time t, if the form of ^ at the time fo, i.e., the function
), is given. This property follows, because (1) is a differential equation of
\
the first order in . The second Schrodinger equation therefore formulates
dt
the law which controls the change of the state function ^ with time.
We now must consider the consequences resulting from the time dependence
of ^ in an expansion of the form (2), 17, or (4), 17. If \l/ depends on t,
we must introduce the argument t also on the right hand side of the expansion.
This can be done in two ways: either the eigen-functions <p (case 1) or the
coefficients a (case 2) can be considered as depending on t. Both cases occur in
quantum mechanics. They may also be combined such that both $ and a
depend on t (case 3) .
The simplest case of time dependence is given by a ^-function of the special
form (16), 13, i.e., by a function
2iri A
^(<7) *"**' (2)
which splits into a factor depending on q alone, and an exponential factor con-
taining t and the eigen-value Hk of the energy, but not q. The inferences
leading from the time-dependent equation to the time-independent one, given
in 13 for a special form of the energy, can be repeated for the general form (1).
Introducing the solution (2) into (1) we obtain
H-Afofl = fftffcfofl (3)
This is the first Schrodinger equation (4), 14, written for the operator H op .
We now can cancel the exponential factor on each side; on the left side this
is possible because the exponential factor, which does not depend on #, has
the character of a constant with respect to the linear operator H op . We thus
(4)
1 Originally Schrodinger had developed also his first equation only in terms of the
energy operator. The extension to other operators was introduced later.
2 If, in (9), 13, the potential U is =0, the left hand side of (1) vanishes for a function ^
which is linear in q. But such a function cannot satisfy the condition of normalization
(1), 17. We therefore must have a time-dependent ^ also in this case.
1 8. TIME DEPENDENCE 87
We see that, as before, the time-independent factor <p k (q) is determined by the
first Schrodinger equation. If <p k (q) is chosen in correspondence with this equa-
tion, i.e., if it represents an eigen-function of this equation, the function
fa(q,t) of (2) will be a solution of the second Schrodinger equation (1).
On the other hand, the function fa(q,t) of (2) will then also be a solution of
the first Schrodinger equation; this is stated in (3). We see that the first
Schrodinger equation leaves its solutions undetermined with respect to a
factor which does not depend on g, but may depend on t. It follows that we
can consider also the function $k(q,t) as an eigen-function of the energy.
We may therefore consider (2) as a special case of an expansion in eigen-
functions of the energy, in which the coefficients a m vanish except for <r&
which is = 1. In this interpretation, (2) represents case 1. on the other hand,
while keeping the idea that all <r m = f or m 7* k, we can also consider <p k (q)
as the eigen-function of the energy, and then interpret the exponential factor
in (2) as representing the coefficient o^. In this interpretation, (2) represents
case 2. We comprise both cases in saying that (2) represents a ^-function which
is given through an eigen-function of the energy.
Although fa in (2) depends on , the square | fa \ 2 is independent of , since
the exponential factor has the absolute value 1 and drops out when we multiply
fa by \l/k*. Furthermore, in both interpretations |<7A:| 2 is = 1. The probability
distributions d(q) and d(H) are therefore time independent. We have a sta-
tionary case] the ^- waves represent stationary oscillations which leave observ-
able entities unchanged. Such a case corresponds to one of Bohr's stationary
cases of the atom, in which each electron moves in its orbit and the total
energy H has a sharp value H k .
The second Schrodinger equation is so constructed that it follows the prin-
ciple of superposition. This means that every linear combination of solutions
of it is also a solution. We can use this property in order to construct a
function \l/(q,t) of a more general form than (2). This is achieved when we write
the expansion
(5)
in which the coefficients <r k are independent of t, and fa has the meaning (2).
The bundle of waves of different frequencies, given in (5), is a solution of (1),
because each fa is a solution of (1).
In the form (5) the expansion represents case 1, since it is constructed in
time-dependent eigen-functions. With the abbreviation
*k'(()=<r k e- 2j ir Hkt (6)
we can write (5) in the form
> (7)
88 PART II. MATHEMATICAL OUTLINES
In this form the expansion represents case 2, since we have time-dependent
coefficients and time-independent eigen-functions. This double interpretation
is possible because the time enters only in the form of an exponential factor
which can be included either in the coefficient or in the eigen-function. Since
the exponential factor has the absolute amount 1, such an inclusion neither
violates the condition (1), 17, of normalization, nor the condition (2), 9,
of orthogonality.
Both the coefficients <rk and <rk(t) are so constructed that their squares |<r fc | 2
and |(rj/(0 1 2 are independent of the time t. We therefore have a probability
distribution d(H) of the energy which does not vary with time. It is different
with the function \\lt(q,() 1 2 . This function depends on time, because, when we
multiply the sum in (5) or (7) with its complex conjugate, the exponential
factor furnishes terms e~ T (H *~ //w > ) ' which do not drop out. We thus have here
a probability distribution d(q) which varies with time. The ^-function (5),
or (7), therefore does not represent a stationary case, but a case in which, in
the language of Bohr's model, transitions between stationary states occur.
The mentioned exponential term indicates, through its exponent, that these
TJ _ TT
transitions are combined with oscillations of the frequency - ~,incor-
h
respondence with Bohr's rule for the emission of radiation. This nonstationary
state, however, is represented in (5) as a superposition of stationary states, a
conception unknown in the older quantum theory.
Reversing the inferences given above, we can prove that if the function
$(q,t) is expanded in the time-independent eigen-functions <?k(q) of the energy,
i.e., if it has the form (7), the coefficients (r k '(t) must have the form (()), and
thus will contain the time only within an exponential factor of the amount 1.
In other words, the general situation can always be interpreted as a superposi-
tion of stationary states, at least if it is a situation for which time-independent
eigen-functions of the energy can be defined. The proof is as follows. From (7)
we infer the two relations, using in the second line the first Schrodinger
equation for the last step :
ih d\l/ \-^ih d<rk l , \
(9)
According to (1) the terms of both lines are equal. Comparison of the coeffi-
cients of the expansions furnishes
ih dffk trr
<T ~^7 ^ ** Hk
2w dt
This differential equation determines a-/ as being of the form (6).
l8. TIME DEPENDENCE 89
This result can be stated in the form: Whatever be the form of the function
\ls(q,t), the probability distribution of the energy is time independent. 8 This
theorem can be considered as the quantum mechanical generalization of the
principle of the conservation of energy. Similar results obtain for other entities
which are commutative with the energy, and which, classically speaking, are
time-independent integrals of the equations of motion.
If we expand the function ^(g,0 in terms of the eigen-functions Xm(q) of any
other efitity v, which is not commutative with the energy, we obtain
m
Here the expansion coefficients r m (t) do not have the special form (6), i.e.,
they do not consist of a constant multiplied by an exponential factor of the
amount 1. Their squares | r m (f) | 2 will therefore be time dependent. The prob-
ability distribution d(v) of an entity v, which is not commutative with the
energy, will therefore, in general, vary with time. only in the stationary case
(2), i.e., in a state where ^ is given through an eigen-function of the energy,
will the probability distributions d(v) of other entities v be time independent.
This follows because the expansion of (2) in the eigen-functions Xm(q) of v can
be constructed by expanding only <pk(q), and can therefore be written, with
constant coefficients r m , in the form:
2iri rr
m
2iri ,
The exponential factor can here, as before, be included either in the coefficient
or in the eigen-function. In both cases the squares of the coefficients are time
independent.
The results of this section can be summarized as follows. The state function
\l/ is always time dependent, i.e., has the form t(q,t). Its dependence on time is
controlled by the second Schrodinger equation (1). In stationary cases ^(#,0
consists of a time-independent part <f>k(q), which is an eigen-function of the
energy, and an exponential factor of the amount 1 containing L In nonsta-
tionary cases of the form considered, the function t(q,t), if expanded in terms
of the eigen-functions of the energy, has the form (5) or (7), and therefore leads
to constant expansion coefficients 0-*., or coefficients <n!(t) which, according to
(6), contain the time only in an exponential factor of the amount 1. The proba-
bility function \<T k \ 2 or |o-j/(0| 2 is therefore time independent. If the non-
stationary ^(g,0 is expanded into the eigen-functions of an entity v which is
noncommutative with the energy, the coefficients r k (f) will contain the time
8 This result is limited to a time-independent energy operator. We shall not enter here
into the discussion of more general cases.
90 PART II. MATHEMATICAL OUTLINES
in a form different from (6), and the probability distribution | T k (f) | 2 , there-
fore, will depend on time.
In the following sections we shall not indicate the variable t in the function
^, but simply write \l/(q). All relations to be explained then will be understood
to hold for every value t.
19. Transformation to Other State Functions
We have so far used the function $(q) as state function, i.e., as the function
describing the probability state of a physical system. In so doing we have dis-
tinguished the coordinates q from other physical entities, since \f/ is formulated
in terms of the q. The same distinction is expressed in the form of the Schro-
dinger equation (5), 14, which determines eigen-functions <p(u,q) as rela-
tions between any entity u and the coordinates q. We shall now show that this
distinction can be eliminated, and that we can use other systems of physical
entities in the place of the coordinates q.
Let us consider the continuous case. The considerations to be given then are
based on the symmetry of the functions \f/(q) and v(u) as explained in 9.
Applying (16), 11, we can expand a(u) in terms of the <p(g,w), the expan-
sion coefficients being given by the function \l/(q) :
<y(u) = I t(q)v(q,u)du $(q,u) = <p*(u,q) (1)
We can interpret this as meaning that a(u) is the state function and that the
tp(q,u) are the eigen-functions of q with respect to u. Using this conception and
applying rule II to the expansion coefficient ^(#), we immediately obtain (14),
17, i.e., rule III.
Using the geometrical language of the F-space (10), in the passive inter-
pretation, we may say that instead of using the ^-system as the system of
reference we have now introduced the <r-system as the system of reference.
The eigen-functions of the entity q in the cr-system are the complex conjugates,
with reversed arguments, of the eigen-functions of u in the ^-system. When
we now introduce a third entity v, with a state function r(v), we must make
use of the triangle of transformations (11). If the eigen-functions of v in
the ^-system are x(v,g), we cannot use these functions as the eigen-functions
of v in the cr-system; instead we have to introduce eigen-functions co(t;,te)
which are determined by (16e), 11, as being
w(t>,w) = I x(v,q)v(q,u)dq (2)
We see that we must revise our terminology of eigen-functions so far used. We
must not speak simply of eigen-functions of a physical entity, but of eigen-
functions of a physical entity with respect to a given entity used as argument
of the state'function.
510. DETERMINATION OF THE ^-FUNCTION Ql
The transformation of the state function involves some further changes.
The second Schrodinger equation, which determines ^ in its dependence on
time, must be transformed into a similar equation determining the dependence
of <r(u) on time. We refer the reader to textbooks on quantum mechanics. 1
With these considerations the absolutism of the coordinates q is to a great
extent eliminated and replaced by a relativity of parameters. There is, how-
ever, a residue of absolutism in the fact that the transition from the classical
function (1), 14, to the operator of the entity u is defined only for the q
and p as arguments.
20. Observational Determination of the ^-Function
We saw in 18 that the second Schrodinger equation determines the ^-function
at any time t, if the ^-function at the time to is given. In order to make use of
this procedure we must therefore know the ^-function at the time fa. This
determination cannot be made by mere theoretical inquiry; it requires, in
addition, experiment and observation.
An analogy with classical mechanics may be used to clarify these considera-
tions. Schrodinger's second equation corresponds to the causal law which
determines the change of the values of physical entities in the course of time.
In order to apply such laws we must know the initial values of the entities
involved; this knowledge is attained by means of experiment and observation.
The corresponding knowledge in quantum mechanics is expressed in the
^-function at the time to.
The difficulties in determining the ^-function originate from the fact that
the complex functions of quantum mechanics do not represent immediately
observable entities; it is only their squares which symbolize such entities.
These squares express probabilities and are therefore accessible to statistical
determination. Now the square of a complex function does not determine the
function, since many different complex functions may have the same square.
The reason is that a complex function represents a pair of two real functions,
whereas its square represents only one real function. In order to determine a
complex function we must therefore know two suitably chosen real functions
of a statistical character.
The results of 17 suggest an attempt in the following direction. If the
function \l/(q) is given, rules III and II determine the probability distributions
d(q) and d(p) of position and momentum; the first is given as |^(#)| 2 , the
second as \a(p) | 2 , where <r(p) represents the coefficient in the expansion into
eigen-functions. We might suspect that if d(q) and d(p) were given, it should
be possible to determine both \l/(q) and o-(p), since the eigen-functions of the
expansion are known; therefore each one of the two functions $(q) and v(p)
is determined by the other.
1 Cf., for instance, H. A. Kramers, Grundlagen der Quantentheorie (Leipzig, 1938), pp.
145, 153.
9* PART II. MATHEMATICAL OUTLINES
It turns out that this supposition is wrong. V. Bargmann, of whom the
author asked this question, has shown through the construction of an example
that a class of several differing functions ^(g), and, therefore, also a correspond-
ing class <r(p), is compatible with the same distributions d(q) and d(p). It
would, of course, be irrelevant if the ambiguity left in \l/(q) were reduced to the
occurrence of an arbitrary phase factor, since two functions \l/(q) differing only
with respect to a constant phase factor, i.e., a complex factor of the amount 1
which does not depend on q, furnish the same probability distributions for all
physical entities. The example constructed by Bargmann, however, .shows
that the ambiguity in \f/(q) is of a more general kind. Given d(q) and d(p), it
is possible to construct two different functions fa(q) and ^ 2 (g) compatible with
the given distributions, but leading to different distributions di(u) and d 2 (u)
of a third noncommutative entity w. 1
We therefore must say that the function $(q) represents the pair d(q) and
d(p) plus some further knowledge. It does not seem possible at the present
time to state by what kinds of observation this particular additional knowledge
can be acquired.
There exists, however, another way which is free from these ambiguities
and which enables us to determine \l/(q) by means of statistical observations.
This way has been opened by E. Feenberg 2 who has shown that \l/ can be de-
termined if the time dependence of \l/ is included in these statistical considera-
tions. As we have pointed out in 17, the function $(q) must be regarded as
the special value of a time-dependent function \fr(q,t) for a certain time t.
Similarly, the function | $(q) | 2 represents a special value of a function | t(q,t) | 2 ,
and we have, therefore, also a time-dependent probability distribution d(q,t).
We shall indicate the special functions resulting for the time t by writing *
for the argument. Now Feenberg has shown that ^(g,J ) is determined to a
constant phase factor if the two real functions d(q,t Q ) and d(qfa) are given,
__ _ dt
1 Bargmann assumes the function <r(p) to be of the special form<r(p) = <r( p); he then
shows that, if <?(p,q) is the Fourier function -JL e***, the two relations hold:
V27T
m I ff
ff(p)<f>(p,q)dp
**(l>)*(p,tidp
This follows, since for the Fourier function <f>*(p,q) <p( p,g), through the relations
__
The function \J/(q) then also satisfies the relation $(q) ^(- g), but can be arbitrarily
chosen apart from this restriction. It follows that the given probability distributions
d(q) and d(p) can be combined with the couple if, <r as well as with the couple ^*, <r*; these
two couples, however, lead to different probability distributions d(u) of other entities u.
Starting from this simple example, Bargmann has constructed even more complicated
forms of ^-functions holding for given d(q) and d(p).
* Feenberg's proof is presented in E. C. Kemble, Fundamental Principles of Quantum
Mechanics (New York, 1937), p. 71.
2O. DETERMINATION OF THE ^-FUNCTION 93
the latter ｅxpression meaning the time derivative of d(g,) taken at the time Jo-
The proof, which requires some mathematical methods not represented in this
book, makes use of the time-dependent Schrodinger equation. This equation
establishes, as is seen from (1), 18, a relation between the shape of the
^-function in the configuration space and its time derivative, since its left
hand side contains space derivatives of \f/ and its right hand side represents
the time deriyative. The knowledge of a time derivative, even if it concerns
only the time derivative of the function | \l/(q,t) | 2 , leads therefore to some
determination of the shape of the function ^(g,0 in the configuration space,
which in combination with the value of | $(q,t) \ 2 leads to a complete determina-
tion of t(q,t), apart from a constant phase factor.
f\
We see that the pair of the two real functions d(q,t<!) and d(q,t Q ) is supe-
dt
rior to the pair d(q) and d(p), or d(q,t Q ) and d(p,to), so far as only the first pair
leads to a determination of the ^-function. The reason is that the time-
dependent Schrodinger equation restricts the choice of ^-functions compatible
with the pair d(q,t Q ) and d(q,to) ; if we start from the pair d(q) and d(p) , there
dt
is no such restriction narrowing down the class of possible ^-functions. Since
the momentum p is determined by the velocity, and the latter represents the
time derivative of q, we can state our result as follows: The time derivative
of the distribution d(q) states more than the distribution of the time deriva-
tive of q.
The practical way of determining the function ^ can now be described, at
least in principle, as follows. We start, not from one system, but from an
assemblage A of systems in equal physical conditions; these conditions must
be so specified that they assure the same, though unknown, form of the func-
tion ^(g,) for each of the systems, or, in other words, that they assure the
existence of one ^-function for the assemblage (cf. 23). We now select at
random a subclass B of the systems and measure at the time to the value of q
in each of the systems of the subclass B. The results will vary with the indi-
vidual systems, and we thus obtain a probability distribution d(q,t Q ). We then
assume that the so-obtained distribution holds equally for the remainder of
the class A, i.e., we assume that we would have obtained the same distribu-
tion if we had measured q in all systems of A. This assumption represents the
usual inductive inference without which, of course, no physical statements
could be constructed.
We now make an important logical step. The systems of the subclass B,
since a measurement has been performed on them, have been disturbed, and
they are therefore excluded from the assemblage. These systems now have
new ^-functions which do not interest us; the ^-function whose square is
represented by the obtained distribution d(q,t^ is the ^-function of the re-
mainder assemblage A'.
94 PART II. MATHEMATICAL OUTLINES
At the time ti shortly after t Q we select another subclass B' from A', and now
make measurements of position on all the systems of B f . The resulting distri-
bution d(q,ti) is then considered as determining the square of the function
\fs(q,ti) of the remainder assemblage A", resulting from A' by the exclusion
of B'. The difference d(q,ti) - d(g,*o), divided by ti - , then represents a nu-
a *\
merical approximation for the derivative d(q,to), or I \l/(q,t Q ) 1 2 .
dt dt
We now must determine a function \K(?^o) which satisfies the two given nu-
merical data | ^(4,60) I 2 and |^(g,o)| 2 , and which at the same time satisfies
dt
the time-dependent Schrodinger equation. This determination, since it is to
lead only to approximative results, can be carried out by numerical methods.
We thus finally arrive at a function t(q,to) which we consider, not only as the
^-function of the assemblage A n ', but of any assemblage of the kind A, i.e.,
of any assemblage produced under the physical conditions by which A was
defined.
The value of this consideration, although it cannot always be carried
through for practical reasons, consists in the fact that it shows the way to an
observational determination of the ^-function. The state function $ can be
ascertained in terms of observational data; this is what we want to show here.
Once this is made clear it appears advisable to look also for other ways of
determining the ^-function. A frequently used method is to guess the form of
the ^-function from the general physical conditions of the arrangement, and
to check its numerical constants from observational results derivable from the
assumed function. Such methods are permissible after it has been made clear
to what extent the observational data used determine the ^-function.
We now see the reason that complex functions are used in quantum me-
chanics. The complex function ^ can be considered as a mathematical abbrevia-
tion which stands for pairs of statistical sets. Thus, we can consider ^(g,<o) as
the representative of the pair d(qfy) and d(g, ); or, since the time deriva-
dt
tive is determined by the distribution at an immediately following time t\, as
the representative of the pair of two consecutive distributions d(g,2 ) and
d(q,ti) ; or, as the representative of the pair d(g,o) and d(p, ), with the qualifi-
cation that \l/ contains some more indications than the latter pair. The value
of the mathematical algorism of quantum mechanics consists in the fact that
with the knowledge of the complex function \l/(q) we know, not only the two
original statistical sets from which \[/(q) was derived, but also the statistical
distributions of all other entities of the system at the same time; and, further-
more, the statistical distributions of all these entities at any later time. The
procedure by which this is accomplished is given in the rules I and II, 17,
and in the time-dependent Schrodinger equation.
2 1 . THEORY OF MEASUREMENT 95
A remark of a logical nature must be added. Since the determination of a
^-function is achieved by statistical methods, a ^-function must be ascribed,
not to an individual system, but to an assemblage. Now the same individual
system may belong to different assemblages; therefore, ｅxpressions like "the
^-function of this system" are, strictly speaking, meaningless. In spite of this
difficulty it is permissible to use such ｅxpressions when it is clear, from the
context of the consideration, to which assemblage the system is referred. A
similar difficulty is known for statements about the degree of probability of a
single fease; since probability is a property definable only for classes, such
statements, too, are meaningful only when the case under consideration is
conceived as incorporated into a certain reference class. 3 When this class is
not specifically named it must be required that it be evident from the context.
Similarly, the term "^-function of a certain physical system" will be considered
by us as meaningful when it is clear which reference class is understood.
21. Mathematical Theory of Measurement
We now shall turn to an analysis of measurement. We have made use of meas-
urements of the values of entities in our statistical consideration; and we must
discuss what is to be meant by a measurement.
When in classical physics we distinguish between a measured and an un-
measured entity, the difference of the two cases can be analyzed as follows. An
unmeasured entity is not known; but that does not exclude that it can be pre-
dicted with a certain probability. So long as this probability is smaller than 1,
we must say that the entity is unknown. In order to know the entity, we have
to perform further physical operations by which the entity is measured. After
these operations have been made, the uncertainty of the prediction has been
overcome; this means we now can predict with certainty the result we would
obtain if we were to repeat the measurement.
We can use this idea for a definition of measurement: 1 A measurement is a
physical operation which furnishes a determinate numerical result and which in
an immediate repetition furnishes the same result.
The addition of the term "immediate" is necessary, because, if we wait too
long, the entity may change its value. only for time-independent entities can
this qualification be omitted.
It is obvious that the use of the term "measurement" in classical physics is
covered by our definition, in the sense that every classical measurement has
the property required in our definition. Usually we combine with the word
"measurement" in classical physics some further requirements, such as the
requirement that the entity be not produced by the measurement but be in
existence before it, etc. Let us postpone the discussion of such questions and
3 Cf . the author's Experience and Prediction (Chicago, 1938), 33-34.
1 This definition of measurement has been very clearly formulated by E. Schrodinger,
Naturwissenschaften 23 (1935), p. 824.
96 PART II. MATHEMATICAL OUTLINES
keep to the given definition, if for no other reason than that, for our present
analysis, it frees us from all such questions.
It is an advantage of the given definition that it can be applied also in
quantum mechanics. In order to show this we must first give a translation of
our definition into the mathematics of quantum mechanics.
Since the probability state of a physical system is expressed through the
^-function, our definition requires the introduction of a ^-function which
permits us to make strict predictions, i.e., predictions relative to which the
probability distribution d(u) of an entity u degenerates into a concentrated
distribution. We use the term concentrated distribution in the discrete case for
a distribution such that d(ui) = 1 for the subscript i, but d(uk) = for all other
subscripts k ^ i. For the continuous case the concentrated distribution can
only be defined in terms of approximations; the term then means that d(u) can
be considered as a Dirac function 8(u,u\), where u\ is the existing value of the
entity u.
Now it is easy to see that, in the discrete case, d(u) is a concentrated distribu-
tion when the function \l/(q) is identical with an eigen-f unction <f>i(q) of the entity
u. In the term "identical" we here include the possibility that \(/(q) and <pi(q)
differ by a constant phase factor of the amount 1. The theorem stated follows,
because the expansion of \l/(q) in terms of the <?k(q), namely,
ffk
furnishes in this case the relations
1^1 =1 <r k = for k^i (2)
According to rule II, 17, this means
P(a,ui) = I P(s,u k ) =0 for k T* i (3)
For the continuous case we have here some mathematical complications.
Since continuous basic functions are not reflexive (cf. 9), the function \l/(q)
cannot be identical with one of the eigen-functions of u. Using, however,
methods of approximation, we can carry through considerations analogous to
those of the discrete case. Instead of a sharp value Ui we consider here a small
interval u\ it , which we abbreviate as u e . The condition that d(u) be a con-
centrated distribution then is given by the relations expressing an approxi-
mate equality
P(SjU t ) ~ 1 P(s,u) ~ for u outside u e (4)
This condition is satisfied if
/"*14l-4-
/ k(u)|
/ t*i e
2 du ~ 1 a(u) ~ for u outside u< (5)
2 1 . THEORY OF MEASUREMENT 97
The expansion of $ then is practically given by
rui+t
lKff)~l <r(u)<p(u,q)du (6)
J Wl-
This means that \f/(q) is composed practically of a bundle of contiguous eigen-
functions <p(u,q) corresponding to values u within the interval u e . We shall say
that in this case \(/(q) is a practical eigen-function of the entity w. 2 These results
enable us to give the following definition.
Quantum mechanical definition of measurement: A measurement of an entity u
is a physical operation relative to which the ^-function of the physical system is
represented by one of the eigen-functions of u, or by one of the practical eigen-
f unctions of u.
This definition must be considered as one of the fundamental principles of
quantum mechanics; it expresses the part allotted to measurement in a world
of probability connections. That there are physical arrangements which are
measurements in the sense defined, is a matter of fact; the description and
construction of such arrangements constitute that part of physics which we
call the technique of measurement. We shall call a measurement defined in
terms of eigen-functions, as presented in the above definition, an ideal measure-
ment. It follows from the definition of practical eigen-functions that in the con-
tinuous case we have only more or less ideal measurements, i.e., measurements
that can be made only to a certain degree of approximation. In the discrete
case an ideal measurement can be made in principle; but in practice this will
not be possible, and even in this case we shall therefore have a ^-function
which is only a practical eigen-function of the entity measured, i.e., which will
be composed of a bundle of contiguous eigen-functions. In both these cases we
shall call the system definite in w. 3
We now can turn to the discussion of the role played by measurements with-
2 It should bo realized that the values of the coefficients a(u) inside the interval ?Y are
left completely undetermined by the requirement (5), and that therefore the shape of the
function \f/(q) is not determined. What we call a practical eigen-function is therefore not
to be imagined as a function similar to an eigen-function. It is true that (6) can be written
approximately as , Ml+e
<r(u)du
But the integral in this ｅxpression will converge toward zero with smaller e, whereas
<f>(u,q) is finite; this convergence is necessary because \f/(q) is normalized according to
(1), 17, whereas the corresponding integral for <f>(u,q) is infinite. \f/(q) then will also con-
verge toward zero, whereas the condition of normalization remains satisfied. This means
that the standard deviation Ag occurring in Heisenberg's inequality (2), 3, goes toward
infinite values. Since the order of magnitude of $(q) will then be the same as that of the
unexactness of the above relation, it is not possible to say that \f/(q) is proportional to
<f>(u,q). It follows that the process of going to the limit e does not determine a definite
form of the ^-function. The continuous case differs here essentially from the discrete case.
8 It follows from the considerations given in the above footnote that if a system is defi-
nite in u its ^-function is not yet determined, and that therefore the probability distri-
butions of other entities will not be determined. only if we have the discrete case and
know that \f/(q) is strictly equal to an eigen-function <pi(u) are the probability distributions
of all other entities also determined. In general, we need two probability distributions,
as explained in 20, in order to determine $(q).
98 PART II. MATHEMATICAL OUTLINES
in physical operations. We shall base our discussion on the ideal definition of
measurement given above. We shall see that this definition leads immediately
to the relation of indeterminacy and to a formulation of the disturbance
emanating from the measurement. For the ideal measurement this disturbance
will represent a minimum; actual measurements will always produce greater
disturbances. This fact justifies a restriction of the discussion to an analysis
of ideal measurements; we then determine the minimum of disturbance which
any measurement can produce.
The relation of indeterminacy concerns the question of different measure-
ments made at the same time, i.e., by the same physical arrangement. If such
an arrangement is to determine the exact values of different physical entities,
these entities must have the same eigen-f unctions; otherwise the ^-function
of the arrangement can correspond only to the eigen-function of one of the
entities. We showed above that entities with commutative operators have the
same eigen-f unctions; therefore, such entities can be measured simultaneously.
Entities with noncommutative operators, however, have different eigen-
f unctions; it is therefore impossible to measure such entities by the same
physical arrangement. We saw in (6), 15, that, in particular, momentum
and position are noncommutative entities; it follows that there exists no phys-
ical arrangement which measures simultaneously position and momentum.
If these entities, however, cannot be measured simultaneously, is it not
possible to measure one after the other? We must turn to the answer of this
question. For the sake of simplicity we shall deal here only with the hetero-
geneous case; the continuous case can be reduced to the same results by ap-
proximative methods such as explained before.
If a measurement of u has been made with the result Ui, the ^-function of the
system is identical with <f>i(q). In order to determine the probability of measur-
ing, in such a system, a value of an entity v, we must expand this ^-function
into the eigen-functions of v. Since v is assumed to be noncommutative with u,
these eigen-functions Xk(q) are different from the ^ t (g). Now the expansion of
<?i(q) in terms of the Xk(q) is determined by the triangle of transformations
( 11) ; and (17/), 11, furnishes
We now can apply rule II, 17. Since the situation s of the system is a situa-
tion produced by a measurement of u with the result u^ we can replace s by Ui
in the probability ｅxpression, and write, by analogy with (3), 17 :
P(u i>Vk ) = |w*| 2 (8)
This is the probability of finding the value v k after the value Ui has been meas-
ured. We see that we have here, not a concentrated distribution, but a proba-
bility spectrum extending over all possible values of v. Predictions concerning
2 1 . THEORY OF MEASUREMENT QQ
the measurements of v after a measurement of u has been made, can therefore
be made only with determinate probabilities.
Now imagine that after the measurement of u with the result u it a measure-
ment of v is actually made, and furnishes the result Vk. We then have a new
physical situation s with a new ^-function, namely, a ^-function which is
identical with Xk(q)- We know that, if we were to repeat the measurement of v,
we would once more obtain Vk] this is shown by considerations similar to those
used before. But if we want to make, instead, a new measurement of w, we
can no'longer predict its result with certainty. Rather, we have a probability
spectrum for u, given by the expansion of the ^-function Xk(q) in terms of the
eigen-functions <pi(q). This expansion is determined by (17d), 11:
Xk(q) = ]T) <ki<f>i(q) (9)
We therefore have
i) = K| 2 (10)
With (2), 11, we have
|&*| 2 = |oW| 2 = |co w | 2 (11)
and therefore
(12)
Relation (10) expresses the fact that after the measurement of v we have only a
probability spectrum for u. The symmetry of this spectrum with that existing
for v after the measurement of u t expressed in (12), does not eliminate the
fundamental fact that we cannot get rid of probabilities here. Before the
measurement of v we knew with certainty that a second measurement of u
would have furnished the value n, and we knew with the probability (8) that a
measurement of v would produce the value v^ After the measurement of v with
the result v k we know with certainty that a second measurement of v would
furnish v^ and we know with the probability (10) that a measurement of u
would produce the value Ui.
It is this fact which must be interpreted as a disturbance of the object by
the measurement. There are measurements in quantum mechanics; this
means, there are physical operations with numerical results which, if repeated,
lead to the same result. But if two measurements of an entity u are separated
by a measurement of a noncommutative entity v, the second measurement of u
is not bound to reproduce the previous value Ui, and its result can be foretold
only with the probability (10). The measurement of v must therefore have
effected a physical change in the conditions of the system.
Mathematically speaking, the influence of the measurement finds its expres-
sion in the commutation rule, which (6), 15, formulates for the fundamental
operators p and g, and which holds similarly for a series of other param-
eters. It is the noncommutativity of the operators which makes it impossible
to create a physical situation which represents a simultaneous measurement
100 PART II. MATHEMATICAL OUTLINES
of such parameters. The commutation rule expresses, therefore, in a general
form, the same idea which we formulated above as the principle of inverse
correlation. It thus can be considered as a second formulation of Heisenberg's
principle of indeterminacy, (2), 3, the latter inequality being derivable from
the commutation rule, as is shown in the textbooks of quantum mechanics.
If h were = 0, all parameters would be commutative, and we would have no
disturbance by the measurement and no indeterminacy.
Let us add a remark about the fundamental parameters p and q. The rela-
tion canonically conjugated, or complementary, divides these parameters into
two classes such that each class contains only commutative parameters; but
this division can be given in various ways. Thus, we can put into one class all
the q^ into the other class all the p<. We can, however, also put into the first
class some of the position parameters and some of the momentum parameters,
if the latter are not complementary to the first. Thus, for a mass particle with
three position coordinates q\, q%, qz> and the momentum parameters p\, p z , pz,
we can put into the first class the entities qi, p 2 , <? 3 , and into the second class
the entities p\, q^ PS. For each such class a compound measurement can be
devised such that the parameters of this class are determined; then the pa-
rameters of the complementary class remain undetermined.
In a similar way, as holds for momentum and position, energy and time are
coordinated to each other as complementary parameters. Formally speaking,
there is a difference so far as time is not considered in quantum mechanics as a
physical entity of the same kind as other physical entities; thus, there is no
time operator to be used in quantum mechanical equations. In consequence
of this fact a commutation rule of the form (6), 15, cannot be established
for energy and time. It can even be shown that, if an attempt were to be made
to introduce a time operator, a commutation rule of the form (6), 15, for
energy and time would lead to contradictions. 4 In spite of that, the uncertainty
relation (3), 3, holds; but it possesses no analogue in a commutation rule.
A measurement of the energy is, in general, incompatible with a measure-
ment of position or momentum; thus, if the energy operator has the form (3),
14, it is noncommutative with the position operator as well as with the
momentum operator. A measurement which determines the maximal number
of parameters that can be ascertained in one measurement is called a maximal
measurement. Of course there are also measurements which determine a smaller
number of entities; for instance, we can measure qi alone.
22. The Rules of Probability and the Disturbance
by the Measurement
The influence of the measurement can also be stated by means of another con-
sideration. Let us assume that first a measurement m u of an entity u is made,
4 Cf. W. Pauli, "Die allgemeinen Prinzipien der Wellenmechanik," Handbuch derPhysik,
Vol. XXIV, 1 (ed. by Geiger-Scheel, 2d ed., Berlin, 1933), p. 140.
22. DISTURBANCE BY THE MEASUREMENT 1O1
then a measurement m v of an entity v, and then a measurement m w of an
entity w. We then have probabilities
P(ui,v k ) and P(v k ,w m ) (1)
determining the results of the measurements. The probability of obtaining a
value w m then depends only on the value v k obtained, but is independent of the
value Ui which preceded the measurement of v, i.e., we have
P(u i .v h) w m ) = P(v k ,w m ) (2)
if the measurements are made in the order which we stated. The period sign
within the parentheses denotes the logical "and". (2) follows, because, with the
measurement of ;, the ^-function has assumed the form xk(q) which contains
no reference to the preceding measurement of u. We therefore have, according
to the rules of probability, 1 starting with the general theorem of multiplication:
P(Ui,V k .W m ) = P(u iy Vk} ' P(Ui.V k ,W m )
- P(v k ,w m ) (3)
where P(v,i,v k .w m ) means the probability that after finding Ui we shall find v k and
then w m . Applying a theorem which in the calculus of probability is known as
the rule of elimination, we can calculate the probability that Ui is followed by
any value of v, and then by w m , as
Pfa, [ vi V V* V . . . ] .w m ) = ]T P(ui,v k ) - P(v k ,w m ) (4)
k
The sign V denotes the logical "or". The left hand side is the same as
(5)
since one of the values vi, v% . . . must occur. The first term on the right hand
side of (4) is given by (8), 21. In order to interpret the second term we must
realize that we have here a quadrangle of transformations (figure. 8, 11) in
which \[/ represents the entity q, a the entity u, r the entity v, and p the entity
w. Using the triangle \l/rp (or qvw) we therefore have, as stated in (27), 11:
(6)
m
and therefore
(7)
1 The notation which we use here for probability ｅxpressions and the rules of operations
with such terras are presented in the author's Wahrscheinlichkeitslehre (Leiden, 1935).
The symbol W used there is here replaced by P, in adaptation to the English language.
The same replacement has been carried through in the author's paper, "Les fondements
logiques du calcul des probabilite*s," Ann. de VInst. Henri Poincart, tome VII, fasc. V,
pp. 267-348. This paper contains a summary of the author's construction of the calculus
01 probability.
IO2 PART II. MATHEMATICAL OUTLINES
Consequently (4) can be written
ft*! 1 - w (8)
On the other hand, we can also omit the measurement of v and pass directly
from the measurement of u to that of w. In the mathematical algorism this
means that we expand the function $(q) = <f>i(q) in the eigen-f unctions m (q) of
the entity w. The triangle <rp^ (figure 8, 11) furnishes for this expansion the
formula (26), 11, which we write here in the variable q:
_ (9)
m
We therefore have :
P(u ij w m ) = \# im \* (10)
According to (19), 11, the triangle crpr determines # t - m as
Q X""^ X /I 1\
* m = 2-/ ** ' ^ km ' '
A;
and therefore we have
(12)
The relation (12) contradicts the relation (8), since, in general, the square of
the sum is different from the sum of the squares. Stating it more precisely, we
may say : For most entities w the relation (12) contradicts (8) .
Now the rules of probability are tautological if the frequency interpretation
of probability is assumed; 2 therefore we cannot exclude the use of (4), the rule
of elimination. The rules of logic cannot be affected by physical experiences.
If we express this idea in a less pretentious form, it means: If a contradiction
arises in physical relations, we shall never consider it as due to formal logic,
but as originating from wrong physical interpretations. In our case the mistake
is given in the ｅxpressions used on the left hand sides of (8) and (12) ; these two
ｅxpressions do not mean the same, and should therefore be distinguished by a
suitable notation.
For this purpose we must indicate the fact that a measurement has been
made. Let us denote a measurement of u by m u . Instead of (8), 21, we then
shall write: D/ \ i^ 12 /10 \
P(m u .Ui.m v ,v k ) = I co fc I 2 (13)
Here the order of the terms connected by the period sign expresses the time
order. 3 The relation (12), 21, will be written in our new notation:
P(m u .Ui.m v ,v k ) = P(m v .v k .m u ,Ui) (14)
2 Cf. the author's Wahrscheinlichkeitslehre (Leiden, 1935), 18.
8 We are therefore using in this notation an asymmetrical "and". This can be avoided
by the use of superscripts expressing the time order.
22. DISTURBANCE BY THE MEASUREMENT 10$
For (7) we now have :
P(m v .v k .mw,w m ) = hfcml 2 (15)
For (8) we shall now write:
P(m u .Ui.m v .m w ,w m ) = ^|w t ^| 2 |i7*m| 2 (16)
and (12) assumes the form
P(m u .Ui.m w ,w m ) =
ife ' 1km
2
(17)
k
Instead of a contradiction we then simply derive the inequality
P(m u .Ui.m v .m w ,w m ) -? P(m u Mi.m W} w^ (18)
This inequality states clearly the influence of the measurement: The proba-
bility of obtaining a value w m does not depend only on results of preceding
measurements of other entities, but also on the fact of such measurements.
The term m v entering into the first place of the probability ｅxpression changes
the probability, even if no value Vk occurs in this ｅxpression.
The amount of this influence of the measurement m v can be exactly calcu-
lated so far as probabilities are concerned; this is expressed in (16). That, in
spite of the various forms in which an entity can be measured, such a general
calculation of the influence of the measurement can be given is due to the fact
that the theory is concerned only with ideal measurements (cf. p. 97). For
actual measurements (16) will furnish only approximative results.
The influence of the measurement is even more apparent if we choose as
third entity w the first entity u, i.e., if we repeat the first measurement.
Without an intervening measurement of v we then have (cf. (3), 21) : 4
P(m u .Ui.m u ,u m ) = d im (19)
This means that we have a concentrated system; the probability of getting the
same value as observed before is = 1, and the probability for any other value
is = 0. However, when we intercalate a measurement of v we have, with (4),
and using (8), 21, and (10), 21,
(20)
k
This is a general distribution, as though there had been no preceding measure-
ment of u. The intervening measurement of v has destroyed the concentrated
distribution existing for u before, and we arrive at the inequality correspond-
ing to (18) : P(m.Ui.m v .m u ,Um) * P(m u .Ui.m u ,Um) (21)
The physical disturbance by a measurement is of such a kind that every
4 (19) follows also from (12) by the substitution of w^ for W OT and application of (2), 11,
and the orthogonality condition for w in the form (7), 9.
1 04 PART II. MATHEMATICAL OUTLINES
measurement creates a new situation in which the influence of the preceding
situation is no longer recognizable. This fact is expressed by the two relations
P(s.m v .v k .m w ,w m ) = P(m v .v k .m w ,w m ) (22)
P(m u .Ui.m v .v k .m w ,w m ) = P(m v .v k .m w ,w m ) (23)
which state that the term s, and also terms like m u .Ui, can be dropped in the
first place of a probability ｅxpression if they are followed by terms like m v .v k .
These relations take the place of the incorrectly written relation (2).
The inequalities (18) and (21) constitute a very clear formulation of the dis-
turbance of the object by the measurement. We can use this formulation for a
proof of our statement that the principle of indeterminacy (cf. p. 17) is not a
logical consequence of the disturbance by the observation. For this purpose
we must show that a physical world is possible in which the measurement
disturbs the object, but in which a strict prediction of the result of the meas-
urement can be made. Such a world can be constructed as follows. Imagine
that the relation holds:
v ,Vk) = 5i+i,fc (24)
This means that the measurement always changes the system in such a way
that the eigen-value with the next higher subscript is produced. Let us assume
that this relation holds also if we put ra tt for m v , and u k for v k , i.e., if the
measurement of the entity u is repeated. We then obtain with the rule of
elimination:
P(m u .iii.m v .m wj w m ) = ^P(m u .u t .m v ,v k ) P(m v .v k .m w ,w m )
k
= 2^J $i+l,k * Sfc+i.w ^ 5+2,w (25)
k
whereas we have with (24)
P(m u .Ui.m W) w m ) = 6i+i,m (26)
This result shows that, as before, the inequality (18) holds, and that we
therefore must speak here of a disturbance of the object by the measurement.
(24), however, shows that the result of each measurement can be strictly
predicted.
(24) is, of course, not valid in quantum mechanics. We use this formula only
for the formal purpose of proving that the principle of indeterminacy is not a
logical consequence of the disturbance of the object. It is rather the reverse
relation which holds. If there were no disturbance by the measurement, we
could make strict predictions; reversing this implication by negating implicans
and implicate, 6 we come to the result that the principle of uncertainty implies
a disturbance of the object by the measurement.
6 This is the logical principle of contraposition formulated by us in (9), 31.
23- NATURE OF PROBABILITIES 105
23. The Nature of Probabilities and of Statistical
Assemblages in Quantum Mechanics
Although the difference between the right hand sides of (16), 22, and (17),
22, and of (19), 22, and (20), 22, has always been correctly interpreted as
expressing the disturbance through measurements, this result has sometimes
been obscured by an inappropriate terminology. The phrase "interference of
probabilities" has been used to express the fact that complex numbers are
used, in the derivation of probabilities in (17), 22, in a way resembling the
mathematical treatment of the interference of waves. This phrase, however,
can be understood as meaning that quantum mechanics uses a calculus of
probability different from the classical one. We see that this is not the case.
It is probability amplitudes, not probabilities, that interfere. We know from
the considerations of 20 that such amplitudes represent, not probability
functions, but pairs of probability functions; we therefore should not be sur-
prised if laws combining such pairs of probability functions are structurally
different from laws governing individual probability functions. Furthermore,
since each such pair characterizes a specific physical situation, the laws con-
trolling the relations of such pairs determine the probability values for various
kinds of physical situations; they therefore include statements about the in-
fluence of physical situations on the probability values pertaining to them.
Once a statement about probabilities is derived in quantum mechanics it has
the same meaning as ordinary probability statements, and nowhere is there
any deviation from the classical rules of probability. A correct notation makes
this clear; in following the rules of the calculus of probability it will point out
the physical content of quantum mechanical statements. What distinguishes
quantum mechanics from classical physics is the inequality (18), 22, which
states that the probability of obtaining a certain result of a measurement
depends on the fact of other measurements. This means a difference in physics,
but not in the calculus of probability.
The origin of such misinterpretations is perhaps to be found in a misinter-
pretation of the calculus of probability, which, throughout the history of this
calculus, has greatly contributed to the growth of erroneous conceptions about
the nature of probability. The calculus of probability is concerned only with
the derivation of probabilities from other probabilities; it can therefore be
applied for practical purposes only if some probabilities are given. The deter-
mination of these initial probabilities is not a mathematical, but a physical
problem. In most cases this determination is given simply by the use of the
statistical inference also called induction by enumeration, or a posteriori de-
termination of probabilities; we then count a frequency within an observed
sequence of events and assume this frequency to persist for further prolonga-
tions of the sequence. We speak here of statistically inferred probabilities. There
106 PART II. MATHEMATICAL OUTLINES
are, however, other cases in which we introduce probabilities by means of
theoretical assumptions which are not immediately based on statistical con-
siderations. Such assumptions may concern individual cases, or they may be
established within the frame of physical theories; i.e., they may have the form
of physical laws stating that under certain physical conditions a probability
has this or that value. Examples of such theoretically introduced probabilities
are given by the statement that the probability of a given face of a die is - ,
6
or that all arrangements of molecules in a Boltzmann-Maxwell gas are equally
probable. Such statements sometimes have been considered as derivable a
priori by means of a "principle of no reason to the contrary", which was
assumed to constitute a part of the calculus of probability. This is, of course,
an untenable conception. The justification of theoretically introduced proba-
bilities can be given only in the way in which any other physical assumption
is verified: by testing its observable consequences. The quantum mechanical
probabilities, defined in rules II and III, 17, as resulting from probability
amplitudes, are theoretically introduced probabilities in the sense defined.
The probability amplitudes, however, do not belong to the calculus of proba-
bility, but to physics. The justification of this derivation of probabilities is
given, as in all other cases of derivations of observable numerical values, by the
success of the assumption. The hypothetical manner of their introduction,
however, does not make them probabilities of another kind. ｅxpressions like
"potential probabilities", sometimes used to distinguish these probabilities
from other probabilities, must appear inadequate. All probabilities are poten-
tial so far as they determine the results of future observations. The meaning
of probabilities, whether they are introduced by the statistical inference or by
way of physical assumptions, is always given in a statement about the limit
of a frequency in a sequence.
Keeping to the results of these general considerations, we now can under-
stand a physical distinction which must be made with respect to statistical
assemblages occurring in quantum mechanics, known as the distinction be-
tween the pure case and the mixture. The nature of this distinction may be
made clear by the following consideration.
Imagine we have a great number of systems in which we make a measure-
ment of u\ some systems will furnish a value u^ others a value u k , etc. Let us
assume that the eigen- values Ui,Uk, . . . are not degenerate, i.e., that each of
these eigen-values corresponds to a specific eigen-function. We then collect in
one class C all those systems in which we obtained Ui\ this assemblage is said to
constitute a pure case. It is so called because if we were to repeat our measure-
ment of u in this class we should find Ui in each of the systems. Now imagine
we make a measurement of a noncommutative entity v on all systems of C;
we then shall obtain in one system the value Vk, in another the value v m , etc.,
the probabilities of these values being determined by (13), 22. After this
2g. NATURE OF PROBABILITIES 1<>7
measurement the assemblage C no longer constitutes a pure case, but a mixture,
since it contains systems with different values of v. If we now repeat the meas-
urement of u, we shall find different values of u in the various systems; the
percentage of a value u m in C is then determined by the probability (20), 22.
The pure case, once destroyed, therefore cannot be reestablished through the
performance of measurements alone; in addition, we must make a new selec-
tion out of C in order to construct a new pure case.
These considerations will serve to clarify a terminology frequently used with
respect to individual systems. When we say that a measurement on an indi-
vidual system creates a pure case, or reduces the ^-function of the system to an
eigen-function of the measured entity, we mean that such will be the ^-
function of an assemblage constructed by putting into one class the system
considered and a number of other systems with the same result of the measure-
ment. The latter class constitutes the reference class (cf. the remarks at the
end of 20) which is understood in the ｅxpression "^-function of an individual
system after a measurement". This phrase, therefore, means the same as
'^-function of a class of systems after a measurement of u with the result m".
Whereas the assemblage representing a pure case consists of systems each
having the same ^-function, namely, a ^-function given by the eigen-function
(f>i(q), the subscript i being the same for all systems, the mixture includes
systems of various forms of ^-functions, namely, functions <?k(q) with various
subscripts &. Each function <pk(q) will be realized in a certain percentage Ck
of cases. Now both assemblages are only statistical assemblages, since even in
the pure case the results of measurements of an entity v having an eigen-
function different from the ^-function can only be statistically predicted. The
question arises: Is it possible to characterize the mixture, by analogy with the
pure case, in terms of one function vK(?)? This function, of course, would have
to be different from each of the eigen-f unctions <?k(q) .
The answer is negative; i.e., it is not possible to characterize a mixture in
terms of one ^-function. We prove this by assuming that there is one such
^-function and showing that this assumption leads to contradictions of the
kind explained in 22. Let us denote the mixture by S. If there is a ^-function
of the mixture /S, it can be expanded in the eigen-functions <f>k(q) :
(1)
According to rule II, 17, the value 1 0k \ * represents the probability of measur-
ing the value Uk of the entity u; since this result will be found in all those cases
in which the individual system is represented by the eigen-function <t>k(q) 9
we have
P(S.m u ,u k )= W 2 = c* (2)
Now let us consider the question of measurements, not of the entities u or v,
10)8 PART II. MATHEMATICAL OUTLINES
but of another entity w with the eigen-functions fmfa). Writing ^ in these
eigen-f unctions we have with (24), 11 :
(4)
and therefore with the use of (25), 11 :
P(Sm wy w m ) = |pm| 2 =
On the other hand, this probability can be determined as follows. Writing
in the eigen-functions f w , we have with (26), 11:
Since the probability of having the eigen-function <f> k (q) in the mixture S is
given by c*, we have
P(S.m wy w m ) = ' I***! 8 = M 2 |^| 2 (6)
We are led here to the same kind of contradiction as presented by (8), 22,
and (12), 22. Although there are exceptional matrices & km which satisfy
both (4) and (6), most unitary matrices &km will not do so; in other words: It
is always possible to construct a unitary matrix #A, or, what is the same, to
define an entity w, for which (4) and (6) are not compatible.
It follows that it is not permissible to consider the assemblage S as charac-
terized by one ^-function. The assemblage S can be characterized only by a
mixture of ^-functions. Using the notation introduced in the probability ex-
pressions (16), 22, and (17), 22, we can characterize the assemblage S by
saying that measurements of w in S have the form m u .Ui.m v .m w , and not the
form m u .Ui.m w . Formally speaking, we can say that we can replace only the se-
quence "m u .Ui" in probability ｅxpressions by a small "s"; the sequence
"m u .Ui.m v " must be replaced by a capital "&", i.e., by a symbol representing,
not one ^-function, but a mixture of ^-functions. The meaning of this formal
rule is that, if a measurement is not merely a repetition of preceding measure-
ments, the assemblage created by it irrespective of its results is a mixture.
We see that the concept physical situation, although determining, not one
system, but a statistical assemblage, does not apply to all kinds of statistical
assemblages, but is restricted to special forms of such assemblages. These are
the assemblages which can be characterized by one ^-function.
The question arises whether the term physical situation is wider than the
term pure case. This would be so if there were ^-functions of such a shape that
they could not be considered as eigen-functions of any physical entity. It is,
however, not necessary to make such a distinction. We can always define a
physical entity of such a kind that a given ^-function is one of the eigen-func-
tions of this entity. Mathematically speaking, this means simply that we can
*3- NATURE OF PROBABILITIES 1OQ
always incorporate a given function in an orthogonal set, and that we can
construct an operator such that the constructed set represents the solutions
of the Schrodinger equation of this operator. The physical entity so defined
will, in general, be none of the entities to which names have been given; it
may, for instance, be something like the product of energy and position divided
by the square root of the momentum. Furthermore, we do not know whether
there exists a physical method of measuring such an entity, i.e., a method of
producing physically a system of that kind of ^-function. But formally speak-
ing, any ^-function can be considered as an eigen-function of a physical entity.
We therefore can identify the two terms pure case and physical situation, or
assemblage characterized by one ^-function. When we distinguish, in probability
ｅxpressions such as (22), 22, and (23), 22, a general situation s from a
situation created by a measurement m u with the result Ui, we do so only to indi-
cate a difference of practical realization; in principle these two situations are
of the same kind.
We therefore distinguish only two kinds of statistical assemblages: those
which are pure cases s, or consist of systems in the same physical situation,
and those which are mixtures S. 1 With this distinction we do not introduce
differences concerning the application of the calculus of probability; the laws
of probability are the same in both cases. What we introduce is a classification
of the form in which the values of probabilities are determined. The pure case
allows us to comprise the values of the probabilities into one complex function,
whereas the mixture requires, in addition, the individual enumeration of the
probabilities with which the individual ^-functions occur in the assemblage.
This shows a physical peculiarity of the pure case assemblage which distin-
guishes this assemblage from others. The peculiarity derives from the fact that
the pure case represents the utmost degree of homogeneity attainable in quantum
physics; this case may therefore be considered as defining one specific physical
situation.
The present section concludes our short exposition of the mathematical
methods of quantum mechanics. Summarizing, we may enumerate as the
basic principles of quantum mechanical method the following principles:
1) The characterization of physical entities in a given context
by operators, eigen-functions, and eigen-values, determined by
the first Schrodinger equation ( 14-16).
2) The characterization of the physical situation by a function
\f/ which, through the rules I-III, 17, determines the probability
1 It can be shown that every mixture S can be conceived as a mixture of ^-functions
which constitute an orthogonal set. Using the considerations given above, we therefore
can say that for every mixture there exists an entity x such that the mixture can be con-
ceived as a mixture of pure cases corresponding to measurement results of the entity x.
Cf. M. Born and P. Jordan, Elementare Quantenmechanik (Berlin, 1930), 59.
1 10 PART II. MATHEMATICAL OUTLINES
distributions, and is inversely determined by these distributions
(20).
3) The law of the time dependence of ^, given in the second
Schrodinger equation ( 18).
4) The quantum mechanical definition of measurement ( 21).
The principle of indeterminacy and the disturbance of the object by the
measurement are not included in these basic principles; they coi^stitute
derivable principles, since they are derivable from the basic principles.
Part III
INTERPRETATIONS
24. Comparison of Classical and Quantum Mechanical Statistics
In Part II we have explained the mathematical methods by which quantum
mechanics derives the probability distributions of physical entities. We now
shall turn to a logical analysis of these methods.
In classical physics a situation is determined if the parameters q\ . . . q n and
Pi . . . p n are given. Let us write, as before, qforqi... g n , and pforpi... p n
Values at the time t are indicated by the subscript 0. We then can summarize
the derivative relations of classical physics in the following table :
Given:
1) the values of q and p at a time t
2) the physical laws
Determined :
3) the value u of every other entity concerned at the time t
4) the values q t and p t of q and p at a time t
5) the value u t of any entity u at a time t
The derivation of the values 3-5 is achieved by means of the following mathe-
matical functions, in which we use the symbol / to indicate a function in
general, without specification of its individual form:
o) (1)
Pt = Mqo,po) (2)
m=/te,P) (3)
These relations constitute the causal laws of the problem.
Now let us consider a generalization which may be called classical-statistical
physics, and which consists in a combination of causal laws with statistical
methods. In this case we consider the causal laws (l)-(3) as still valid; we
introduce, however, a method which takes into account the fact that the values
q and p cannot be exactly measured. The specification of these values is there-
fore replaced by the statement of their probability distributions. We write
these distributions at the time t in the form d (q), d (p), and d (q,p), the
latter function giving the probability of combinations of values q and p. The
1 13 PART III. INTERPRETATIONS
table of derivative relations of classical-statistical physics then can be written
as follows:
Given:
1) the probability distribution d (q,p) at the time t
2) the physical laws
Determined:
3) the probability distributions d (q), d (p), and the probability dis-
tribution d (u) of every other entity concerned, at the time-
4) the probability distribution d t (q,p) at the time t
5) The probability distributions d t (q), d t (p), and the probability dis-
tribution d t (u) of every other entity concerned, at the time t
Comparing this table with that of the strictly causal case, we see that the
occurrence of probability distributions involves a further distinction which
originates from the distinction between d (q,p) on the one side, and d (q) and
d (p) on the other side: The first distribution appears in the category of what
must be given, the latter two distributions appear in the category of what is
determined. The derivation of the latter two distributions is achieved through
the relations: ^
d (q) =Jdo(q,p)dp (4)
do(p) = I d (q,p)dq (5)
It is important to realize that this determination cannot be reversed, i.e.,
that the distribution d (q,p) of the combinations is not determined merely by
the individual distributions. only in case we know that q and p represent
independent entities can we write
do(q,p) = d (q) d (p) (6)
But the application of the special theorem of multiplication, expressed in (6),
presupposes some definite physical knowledge about the independence of q
and p. Without such additional knowledge the function d (q,p) must be di-
rectly given; and in general it will not have the special form (6). on the other
hand, if neither such additional knowledge nor a statement of the function
d (q,p) is available, the knowledge of the distributions d (q) and d (p) alone
will not be sufficient to determine the corresponding functions d t (q) and dt(p) at
a later time t.
The function d (q,p), therefore, may be said to determine the probability
state of the system, or its physical situation, so far as it can be stated in statis-
tical terms; for, if d (q,p) is known, all the other probability distributions are
determined. As to the distributions d (q) and d (p), this determination is
expressed in (4) and (5) ; the determination of the other distributions included
24- CLASSICAL AND QUANTUM STATISTICS llg
in 3-5 is based on the following ideas. With d (q,p) we know the probability of
every combination of values q and p. Now the causal law (3) coordinates to
every such combination a value u\ the probability distribution d (u) is there-
fore determined. Similarly, the causal laws (1) and (2) coordinate to every
combination of values q and p a value of g, or p, at a later time ; the probability
distributions of these values are therefore determined.
We now shall show the way in which these derivations are carried through.
Since q and p stand for sets qi . . . q n and pi . . . p n , the function d (g,p) stands
for 1
JL\Ji tin n n n\ /PT\
d(q\ - qn , PI ... p n ) (7)
In order to simplify our notation we shall omit the subscript , and use the
symbols q\ . . . q n , pi . . . p n for the values of these variables at the time t.
Furthermore, the general symbol /, standing for function, may be specified
individually; we then can write for (1) :
. . . Pn)
(8)
Pn =
'1 ... q n , Pi . . . Pn)
Solving (8) for the variables QI . . . p n , we can introduce the variables qi . . . p n
in (7). Since a probability distribution involves integrations in its variables,
we must use here the rules for the introduction of new variables in integrals;
we therefore must multiply with the functional determinant
A =
and we have
dt(q t p) = d(qi . . . q n , pi . . . p n ) =
dqi
dp n
dqi
dqi
dqi
dp n
dp n
dp n
(9)
|A| (10)
For canonical parameters q and p, such as position and momentum, Hamilton's
equations lead to the result that A = 1 ; this is the theorem of Liouville.
In a similar way dt(u) is determined. When we omit the subscript t, (3) can
be written: -/ x /--v
u=f(qi...q n ,pi...p n ) (11)
1 We use here a way of writing in which the symbols g, p denote, not constants, but
variables. The values of q and p at the time t then are considered as forming a set of
variables, different from the set of variables constituted by the values of q and p at a later
time. This way of writing is preferable for the present purpose.
114 PART III. INTERPRETATIONS
Starting from the function d(qi . . . q n , pi . . . p n ), we can now substitute for
one of its variables, for instance for qi, the variable u\ for this purpose we
solve (11) for qi and put the obtained ｅxpression in the place of qi. The resulting
function may be written d'(u,qz . . . q n , pi . . . p n ). This is not yet a probability
distribution; in order to make it so we must multiply it by the functional de-
terminant which reduces here to the value . We then obtain the probability
distribution
d(u,qz . . . q n , pi . . . Pn) = d'(u,qi . . . q n , pi . . . p n )
(12)
du
When we now reintroduce the subscript t and write d t (u) for d(u), the function
dt(u) is obtained by integrating (n 1) times:
/r
, . . n 1 . . . I d(u,q 2 ... g n , Pi ... Pn)dq 2 . . . dq n dpi . . . dp n (13)
In order to state our results symbolically, we shall use the symbol f op to
express an operator, without specification of the individual form of the opera-
tor, which is different in each of the following equations. We then can write:
(14)
(15)
(16)
d t (u) = fo P [d t (q,p)] =f op U[do(q,p)} (17)
The meaning of the operators f op is given by the procedures explained above,
which include the application of the causal laws (l)-(3) .
Whereas (14) has no equivalent in the strictly causal case, we see that the
relations between the first and the last term in (15)-(17) represent direct
generalizations of the causal case (l)-(3). The only formal difference is that
the values p, g, u are replaced by the functions d(p), d(q), d(u) y or d(<?,p),
and that the function / is replaced by the operator f op . This means that in the
statistical case, relations between statistical functions take the place of the
relations between variables of the classical case.
Now let us consider quantum mechanics. The analogue of the probability
distribution d (q,p) is here the complex function $o(q), if we write this term
for ^(g,k), i.e., the form of the ^-function at the time to. Like d (q,p), $ (q)
determines the probability state of the system, and with this it determines the
physical situation of the system as completely as it is possible in quantum
24- CLASSICAL AND QUANTUM STATISTICS 1 15
physics. We therefore can summarize the derivative relations of quantum me-
chanics in the following table :
Given:
1) the complex function il/ (q) at the time t
2) the physical laws
Determined:
. 3) the probability distributions d (q), d (p), and the proba-
bility distribution d (u) of every other entity concerned, at
the time t
4) the complex function if/ t (q) at the time t
5) the probability distributions d t (q) y d t (p), and the probabil-
ity distribution d t (u) of every other entity concerned, at
the time t
The physical laws mentioned in 2 include, with respect to points 3 and 5,
the rules for the construction of operators and of the first Schrodinger equa-
tion, and, in addition, the rules I-III, 17, determining the probability dis-
tributions; with respect to point 4 we include the second Schrodinger equation
in the physical laws.
Using the operator form of writing applied in (14)-(17), we can write the
derivative relations of quantum mechanics in the form:
(18)
(19)
(20)
*() = /*[*() ] = f w [*.() J (21)
These relations show the analogy with the classical statistical case (14)-(17).
The function d(q,p) is here replaced by ^(g). The role which the causal laws
(1) and (2) play with respect to the establishment of (14) is assumed by the
time-dependent Schrodinger equation with respect to (18).
As in the classical statistical case, the two individual distributions d (q) and
d (p) are not sufficient to determine the probability state, since the latter dis-
tributions do not determine the function \l/ (q) (cf. 20). The function \ff (q)
therefore resembles the function d (q,p) so far as it states more than the indi-
vidual distributions. on the other hand, we notice here a remarkable difference.
The complex function $ (q) is determined by a pair of two other probability
distributions, for instance, by the pair of two consecutive functions d (q) and
di(q) (cf. 20). Whereas, therefore, the complex function $ (q) represents a
pair of two probability distributions in one variable, the function d (q,p) is,
in general, superior to such a pair. Apart from special cases such as (6), it is
impossible to determine a function of two variables in terms of two functions
1 1 6 PART III. INTERPRETATIONS
of one variable. The same holds when we replace q and p by the sets q\ . . . q n
and pi . . . p n ; the function d (q\ . . . q n , Pi . . . Pn), which has 2n variables, is,
in general, superior to a pair of two functions of n variables each. We see that
the probability state which we call a quantum mechanical situation, or a pure
case, is of a special kind; if the number of parameters of the classical problem is
2n, the probability state of the quantum mechanical problem is expressible in
terms of two probability distributions of n variables each. It is this funda-
mental fact which is expressed in the use of a complex function ^.for the
characterization of a quantum mechanical situation. only for a mixed case,
i.e., for an assemblage not describable in terms of one ^-function, do we need
a function of 2n arguments comparable to d (q\ . . . q n , Pi . . . p w ).
The physical situations of quantum mechanics represent, therefore, statis-
tical cases of a special type which may be compared with classical statistical
cases of the independence type (6), i.e., cases in which the function d (q,p)
splits into two functions d (q) and d (p). This is, however, not more than an
analogy. The quantum mechanical situation cannot be conceived as a situa-
tion in which the parameters q and p are independent, because, as we saw in
20, d (q) and d (p) do not determine ^ (q). Such a statement must even be
considered as being beyond the reach of quantum mechanics. There is no way
of deriving any statement concerning a distribution d (q,p) in a quantum
mechanical situation; therefore, a statement with respect to a special multipli-
cative form of this distribution cannot be made either. The reason that
quantum mechanics does not need a function d(q f p) is given in the fact that
the transformations of quantum mechanics in the configuration space are
holistic transformations, not point transformations (cf. 12). A transition
from the function \f/(q) to the function <r(p), which is such a holistic transfor-
mation, thus does not determine any coordination of values q to values p.
Quantum mechanics therefore does not include any statements about simul-
taneous values of q and p.
The intervention of the function d(q,p) in classical statistical considerations
is due to the fact that these methods make use of causal laws which coordinate
to a combination p, g, the later values of these entities, or values of an entity
u } in the sense of a mathematical function. If such laws are necessary for the
determination of those values, a knowledge of the probability of the combina-
tion p,q cannot be dispensed, with. The fact that the quantum mechanical
method does not use a function d(q,p) therefore proves that this method does
not include any assumption of such laws.
This can be understood in the following way. A causal law like (1) or (2)
is to be defined, as explained in 1, as the limit of probability relations of the
form (14) or (15), resulting when both distributions d(q) and d(p), or the
distribution d(g,p), approach concentrated distributions. The probability dis-
tributions then degenerate to the values at the center of the concentrated
distributions, and the operator f op degenerates to a function/. Now the relation
24- CLASSICAL AND QUANTUM STATISTICS 117
of indeterminacy shows that it is not possible to find physical situations in
which both the distributions d(q) and d(p) are concentrated. It is not possible,
therefore, to verify causal laws of the form (l)-(3); more precisely: It is not
possible to verify such laws to any desired degree of approximation. There is
a limit to this approximation drawn by the relation of indeterminacy. This is
the reason that quantum mechanics does not use such laws.
Must we therefore insist that it be impermissible to speak of such laws?
Such an attitude would mean the introduction of a restriction for which there
is no need. We can as well assume the opposite attitude and permit the estab-
lishment of such laws in the sense of definitions, if it is not possible to introduce
such laws as verifiable statements. The condition which such a liberalism would
demand is that any law be permissible if only it establishes a connection be-
tween observable distributions corresponding to the numerical results of
quantum mechanics. Instead of one law determining an entity u as a function
of q and p we then have a class of equivalent laws. It is therefore the theory of
equivalent descriptions (5) which we can use in order to fill in the blank
between the observable distributions.
For this reason we now must inquire into the various forms of supplementa-
tion by interpolation left open, for such a logical liberalism, by the established
results of quantum mechanical method. The question of causal laws then will
be turned into the question of whether there is a supplementation in which
there are causal laws. The answer will depend on the meaning assumed for the
term "causal law". This term usually includes two distinct conditions. First,
it is required that the cause determine the effect univocally; second, it is
demanded that the effect spread continuously through space, following the
principle of action by contact. The latter condition is frequently expressed
by the term causal chain. Anticipating our results, we may say here that our
answer will be negative, that we shall find it impossible to interpret quantum
mechanical relations completely in terms of causal chains. We then shall
attempt to generalize the conception of causality by retaining only the second
condition, while replacing the first by a condition of probability relations be-
tween cause and effect. In this sense we shall speak of probability chains; the
precise definition will be given later. We shall be led to the result, however,
that it is even impossible to interpolate such probability chains, by way of
definitions, between the observable data of quantum mechanics. This negative
result, which represents a statement about the whole class of equivalent de-
scriptions, excludes the introduction of causality, in any sense, into the world
of quantum mechanical objects, and thus establishes the principle of anomaly
formulated in 8.
1 1 8 PART III. INTERPRETATIONS
25. The Corpuscle Interpretation
Every interpretation which coordinates to a measured value #, one simul-
taneous value p, and vice versa, may be called a corpuscle interpretation. The
coordination may be established in various ways. We shall outline one way
which is distinguished by its relative simplicity and may be considered as the
prototype of the corpuscle interpretation.
It is clear that the values of unobserved entities can be introduced only in
the sense of definitions. We therefore cannot be required to prove that our
statements about unobserved entities are true] all that is to be required is that
our definitions be admissible. An interpretation consisting of several admissible
definitions connected with each other will be called an admissible interpretation.
Referring to our exposition in 22, we can show that an admissible interpreta-
tion can be given in a rather simple way.
Our first definition in this interpretation contains the idea that the entity
which is measured is not disturbed by the measurement; only entities non-
commutative with it, according to this idea, are disturbed. We formulate
this idea as follows :
Definition 1. If the value u r of an entity u has been observed in a measurement of
u, this value Ui means the value of u immediately before and immediately after the
measurement.
If u is a time-independent entity, it follows that the same value Ui is valid
also for a longer time before and after the measurement. only an intervention
from outside, such as the measurement of a noncommutative entity v, then
will destroy the value u^
Using the terminology of 22, we can express definition 1 in terms of proba-
bility relations as follows :
P(s,uJ = P(s.m u ,Ui) (1)
This relation states that the value Ui exists even before the measurement m u .
That Ui exists also after the measurement m u need not be specially expressed,
because our way of writing expresses this idea by the rule that the terms follow
each other in the order of time.
If a measurement of u has been made, we can express our convention with
respect to a measurement of v :
P(m u .Ui,v k ) = P(m u .Ui.m v ,v k ) (2)
Formally we can express the relations (1) and (2) by the rule: The term m v
can be dropped in the first place of a probability ｅxpression if it is immediately
followed in the second place by a term Vk-
We now shall show that our definition leads to a determination of simul-
taneous values of noncommutative entities. Imagine we first make a measure-
25- THE CORPUSCLE INTERPRETATION 1 1Q
ment of w, obtaining Ui, and then a measurement of the canonically conjugated
entity v, obtaining Vk. Let both entities be time independent. Since, according
to our convention, the value u i exists after the measurement of u, and the
value Vk before the measurement of v, we now know two values, Ui and Vk, exist-
ing at the same time, namely, between the measurements m u and m v . This
knowledge can be symbolized by writing
Ui m u Ui Vk m v Vk (3)
in which diagram time runs from left to right. We see that our definition makes
it meaningful to speak of the combination of the canonically conjugated
values Ui and Vk, existing simultaneously, since Ui as well as Vk exist throughout
the time from m u to m v .
If we want to extend this method of determination to time-dependent
entities, we must make the two measurements m u and m v immediately after
each other; we then shall know the simultaneous values Ui and Vk between these
two measurements. If only v, but not u } is time dependent, the two measure-
ments, with respect to time, need not be close together; but the results then
determine only the combination of simultaneous values existing immediately
before m v . Such cases occur, for instance, when u means the velocity of a free
particle not subject to forces, and v its position.
The knowledge of simultaneous values attained by definition 1 is of a rather
restricted kind, since we acquire the knowledge of a combination Ui, v k , only after
one of these values has been destroyed. once the measurement m v has been
made, the value Ui exists no longer; we would have to make another measure-
ment of u in order to know the then-existing value of u. But with this measure-
ment of u the value Vk would be destroyed, etc. This result has an important
consequence. In 21 we stated the principle that repetition of a measurement
yields the same value as the preceding measurement. This principle does not
apply to a combination of simultaneous values. We have no means of producing
the same combination of values Ui and Vk a second time; only by chance can it
happen that the same combination results once more. We therefore cannot use
the knowledge of simultaneous values for the prediction of the results of future
measurements; the future remains undetermined, since the knowledge acquired
is no longer valid for the state of affairs existing at the time of its acquisition.
We may add the remark that we cannot use our knowledge for an inference
into the past either, since we cannot tell from the observed combination ut and
Vk which value of v existed before the measurement m u . 1
Definition 1 determines values of unobserved entities only between two
1 We may, of course, know this value from a previous observation whose result is re-
ported and therefore known at a later time, whereas the results of future measurements
cannot be known at earlier times. This shows the existence of a direction of time. But the
use of reports involves manipulations which cannot be performed on a closed system. So
far as closed systems are concerned, it seems that quantum mechanics does not distinguish
a direction of time. This follows from (14), 22.
120 PART III. INTERPRETATIONS
measurements. In order to know something about such values in other situa-
tions, we must introduce a second definition. It is sufficient for our purposes
if we define for the general case, not the value of the unobserved entity, but
the probability of such a value. We shall use here the special rule of multiplica-
tion, thus constructing our interpretation by analogy with the form (6), 24,
of the classical-statistical case. This is done by the following definition:
Definition 2. The probability of a combination VkW m relative to a physical situa-
tion s is given by the product of the individual probabilities of Vk and w m relative
tos. Insymbols: n/ \ n/ \ n/ \ " //<N
J P(s y v k .w m ) = P(s,v k ) P(s,w m ) (4)
Combining this relation with (1) we have
P(s,v k .Wm) = P(s.m v ,v k ) P(s.m w ,w m ) (5)
Definition 2 is assumed to hold also in case ^ is replaced by an eigen-function
of u, i.e., if \l/ characterizes a situation resulting from a measurement of u.
We then have
i,v k .Wm) = P(m u .Ui,v k ) P(ra u .Wt,Wm) (6)
= P(m u .Ui.m v ,v k ) P(m u .u i .m wj w n ^ (7)
Definition 2 states the independence of the values v k and w m . Instead of
expressing this idea by means of the special rule of multiplication (4), we can
also express it as a statement that the relative probability of w m with respect
to Vk is equal to the "absolute" probability of w m . This means we can replace
(4) and (6), respectively, by the formulae: 2
P(s.v k ,w m ) = P(s,O (8)
P(m u .u l .v kj w m ) = P(m u .Ui,w M ) (9)
Applying here (1) and (2), first on the left hand sides, then on the right hand
sides, we obtain:
P(s.v k ,w m ) = P(s.v k .m wj w m ) = P(s.m w ,w m ) (10)
m} = P(m u .Ui.v k .m w ,w m ) = P(m u .Ui.m w ,w m ) (11)
In the substitution of (1) on the left hand side of (8) we have considered v k as
belonging to the state s, and have therefore placed the term m w to the right of
v k . The same holds for the application of (2) to the left hand side of (9). We
can also consider ｅxpressions of the form:
P(s.m w .v k ,w m ) (12)
2 (8) is derivable from (4) only for the case P(s,Vk) & 0, whereas inversely (4) is deriv-
able from (8) also for this case. We shall therefore consider (8) and (9) as the definition
of independence, and thus as replacing definition 2, whenever the case P(s,Vk) is to be
included.
25- THE CORPUSCLE INTERPRETATION 121
Such an ｅxpression cannot be derived from P(s.v k ,w m ) by means of (1). We
can, however, determine the value of (12) by the following method, using,
besides (1), the general rule of multiplication of probabilities, the rule of elimi-
nation corresponding to (4), 22, and (22), 22:
- k.u>m) __ P(s.m w ,w m ) P(s.m w .w m ,v k )
.m w .v k ,w m ) -- - - --- - - -
P(s.m W) Vk) P(s.m w .m v ,v k )
_ P(s.m w ,w m ) P(m w .w m .m v ,v k ) ( .
P(s.m w ,Wi) P(m w .Wi.m v ,v k )
The relations (8) and (9) can be expressed by the following rule, which is
so formulated that it does not apply to terms like (12) : a term like v k may be
dropped in the first place of a probability ｅxpression if it is neither preceded
by the term m v , nor by a term m w which is followed by w m in the second place.
The admissibility of definition 2 is shown by the following consideration
which proves that the rule of elimination is tautologically satisfied:
P(s,w m ) = P(*,[t;iVffcV. . . ].w m )
P(s.V k) Wm)
= P(8,W m ) P(8,V k ) = P(8,W m ) (14)
For the last line we have used (8), and the relation ^ P(s,v k ) 1. The same
can be shown for ｅxpressions of the form (13) . k
Furthermore, we can show that our rule (9) does not contradict our previous
rules for relative probabilities. The relation (9) states that the value w m is
independent of an unmeasured value v k . The relative probabilities previously
introduced, such as (15), 22, or (22), 22, state a dependence of w m on a
measured value v k .
With the definitions 1 and 2 we have given an interpretation which is
exhaustive, i.e., in which statements about unobserved entities, including
simultaneous values, can be made. Using a terminology introduced in the
general theory of meaning, 3 we can say that in this interpretation statements
about simultaneous values of noncommutative entities have truth meaning for
situations between two measurements, and probability meaning for general
situations. This distinction expresses the fact that for the latter case we have
defined only the probability of the values of the entities (definition 2), whereas
in the first case the values themselves are defined (definition 1).
8 Cf. the author's Experience and Prediction (Chicago, 1938), 7.
1 2 2 PART III. INTERPRETATIONS
26. The Impossibility of a Chain Structure
After the corpuscle interpretation has been established as an admissible inter-
pretation, we can ask the question of the consequences of this interpretation.
It is, in particular, the question of causality which interests us. We saw that in
the classical-statistical conception there are causal laws which connect the
states g,p with future states, and with other entities u. Are there -similar
laws introduced with our corpuscle interpretation?
A short analysis shows that this is not the case. Let us assume a sequence of
values corresponding to (3), 25, in the form
q* m q qi p k m p p k (1)
and let u be an entity which is neither commutative with q nor with p. We
then have no way of measuring u between the measurements m q and m p
without disturbing the situation. We can only introduce a probability
According to (9), 25, this probability is equal to
P(m q .qi.p kl u m ) = P(m q .qi,u m ) = P(m q .qi.m u ,u m ) (2)
This means that u does not depend on the value of p existing simultaneously
with q it and that, on the other hand, the dependence of u on q is so constructed
that to one value q there is coordinated a spectrum of possible values u. We
therefore cannot introduce a causal function
(3)
such that the statistical relations of quantum mechanics are reduced to the
classical-statistical case.
The latter case is constructed in such a way that it possesses causal chains
which establish the connection between d(q) and d(p) on the one hand and
d(u) on the other hand. Our considerations therefore show that such causal
chains cannot be constructed in the interpretation of definitions 1 and 2.
Generalizing our investigation in the direction indicated at the end of 24, we
shall ask whether in the interpretation given by these definitions at least
probability chains can be constructed. The meaning of this term must now be
defined precisely.
For this purpose let us consider a structure in which a relative probability
function
d(q,p;u) (4)
takes the place of the causal function (3) and establishes the connection be-
tween the distributions d(q) and d(p) on the one hand, and d(u) on the other
hand. Relative probability functions correspond to relative probabilities;
2 6. CHAIN STRUCTURE 123
they are integrated only over the variable following the semicolon and deter-
mine the probability of this variable relative to values of the variables before
the semicolon. These functions are a generalization of causal laws into proba-
bility laws. 1 We shall say that the interpretation so introduced is given in
terms of probability chains if the function (4) is independent of the functions
d(q) and d(p). The term chain structure therefore characterizes a structure in
which the probability of a value u is determined if the values q and p are given.
We can interpret this feature as meaning that the connection between d(q)
and d(p) on the one side, and d(u) on the other side, goes through the values
q and p.
Our results show that we cannot even construct probability chains in the
sense defined. Because of (2), the function (4) degenerates into a function
d(q;u) (5)
of q alone. Whereas, in this case, the value of p has no influence on u, it is
different after a measurement of p has been made; we then have a probability
function ,, N //%x
) (6)
which shows u to be independent of q. This means that the probability of a
value u, in the given interpretation, depends sometimes on q alone, and some-
times on p alone. In the first case, therefore, the value of p, which we assumed
as existing, though not being measured, has no influence on u; whereas in the
second case the value of q has no influence on u. Now the two cases differ with
respect to the probability distributions d(q) and d(p), since in the first case
d(q) is a concentrated distribution, and d(p) is not, whereas in the second case
the converse holds. We thus find that the dependence of u on q or p varies
with the distributions d(q) and d(p) ; we therefore do not have a function (4)
which is independent of d(q) and d(p). In consideration of the requirement
stated above, the existing relations, therefore, cannot be interpreted as a chain
structure.
This result is made clearer if we start from a general situation s. We then
have, according to definition 2,
d(?,p;i) = d.(u) (7)
where d s (u) is the distribution of u relative to the situation s. This means that
the probability of a value u does not depend on the particular combination of
values q,p, nor on either of these values. Instead, it depends directly on s,
and with this on d(q) and d(p). The fact that these latter distributions,
although they determine the situation s to a great extent, do not completely
determine s (cf. 20), has no influence on our result; it proves only that, be-
sides d(q) and d(p), there are further factors determining s. We therefore do
not have a chain structure in the sense defined.
1 Cf. the author's Wahrscheinlichkeitslehre (Leiden, 1935), 44.
124 PART III. INTERPRETATIONS
The question arises whether this negative result is confined to the interpre-
tation of definitions 1 and 2. We shall show that this is not the case, and that
also with other definitions of the values of unobserved entities we cannot intro-
duce a chain structure. In order to show this we shall proceed by steps,
choosing more and more general forms of definitions.
We begin with the following set of definitions. We leave definition 1 un-
changed, but retain definition 2 only for q and p, whereas other entities u are
considered as being dependent on p and q even when u is not observed. We
ask whether this dependence of u on p and q can be so constructed -that a
function (4) exists. If this interpretation is to represent a chain structure, we
must demand, as explained above, that the same function (4) holds for all
possible physical situations s. Now it can be shown that it is impossible to
define a function (4) of the required property. This is proved by the following
considerations.
Let d (p) be a distribution compatible with a rather exact measurement of g;
thus, d (p) will represent a flat curve. We divide the axis q into small intervals
Ag corresponding to the greatest exactness with which q can be measured when
p has the distribution d (p) ; let <fr be the mean value of an interval A<?i. Further-
more, let di(q) be a distribution which is practically entirely within the interval
Ag; when we have such a distribution we can say with practical certainty that
the value of q is #. A ^-function which unites the two distributions d (p) and
di(q) may be written ^(q) . Since our definition leaves di(q) open within the limits
of the interval Ag t - (and besides for the reasons explained in 20), there is, not
one, but a class of functions \l/i(q) satisfying this condition. Let us assume that
a rule which determines one ^-function of this class for every qi has been intro-
duced ; this function may be written ^(q^q) . The function d l (q) = \ ^(q^q) \ 2 then
is approximately given by a Dirac function. The class of situations character-
ized by the functions $(qi,q), for any interval A<ft, may be called the class A.
If, in a situation of the class A, we measure an entity u not commutative
with q, the probability of obtaining a value u is determined by the coefficients
<r of the expansion
(8)
Since ^(q^q) represents a measurement of q with the result q iy we can write the
probability under consideration in the form
P(m q .qi.m u ,u) = |er(#,w)| 2 (9)
Applying definition 1 in the form (1), 25, we can drop the term m u on the left
hand side, and thus arrive at
(10)
2 6. CHAIN STRUCTURE 125
Using the d-symbol, we can write the left hand side in the form di(qi*,u) 9 and
therefore have for all systems of the class A :
*(;*)= kCfetOI 1 (11)
Now let us consider a class B of systems which all possess the distribution
d (p) previously used, but which have any of the distributions d(q) compatible
with d (p). This class will include the systems of class A, but will furthermore
include systems with rather flat distributions d(q). For each system of class B
we haVe, according to the rule of elimination holding for probabilities, the
relation: ^
d B (q;u) =Jd B (q;p) d(q,p;u)dp (12)
Here we have given no subscript B to the second term under the integral
because this function, introduced in (4), is supposed to be the same for all
situations. Now applying definition 2, 25, to q and p, and using the definition
of class B y we have
d B (q;p) = d B (p) = do(p) (13)
We therefore infer
\u) = / d (p) - d(q,p;u)dp (14)
X
This means that the distribution d B (q]u) is the same for all systems of the class
Bj since the right hand side of (14) has the same value for all systems of B. We
can now determine the function d B (q]u) as follows. Since the systems of the
class A belong to the class 5, we have, using any of the intervals A<ft:
d B (qi]u) = di(qi\u) = \<r(qi,u)\* (15)
Since this relation holds for all i, we can omit the subscript i from g t , and
thus arrive at , . >. , f >., 2 , 1fiN
d B (q\u) = |cr(g,w)| 2 (16)
Now the meaning of d B (q\u) is given by
P(*B.q,u) = d B (q;u) (17)
where S B represents any situation of the class B. Using (9) we derive now from
(16) and (17) the relation
P(s B .q,u) = P(m q .q.m u ,u) (18)
Applying (1), 25, to the left hand side and (22), 22, to the right hand side,
we can write this in the form :
P(s B .q,u) = P(s B .q.m v ,u) = P(s B .m q .q.m u ,u) = P(s B .m q .q,u) (19)
when we apply (1), 25, once more in the last step. Using considerations such
126 PART III. INTERPRETATIONS
as presented in 22, we can show that this result leads to contradictions when
we use the rule of elimination. When we apply the latter rule formally to the
term on the left hand side in each of the following relations, we have
P(s B .q,u)dq (20a)
P(s B -m q ,u) = I P(sB.m q ,q) P(8 B .m q .q,u)dq ^ (206)
Now the first terms under the integrals are equal because of (1), 25; the
equality of the second terms is stated in (19). We therefore derive, using (1),
25, for the second line :
P(8 B ,u) = P(8 B .m q ,u) (21a)
P(sB-m u ,ii) = P(sB.m q .m u ,u) (216)
The latter equation contradicts the inequality (18), 22, when we replace,
in the latter relation, the ｅxpression m u .Ui by SB, the term w by u, and the term
v by q. More precisely: For all those entities u for which the inequality (18),
22, holds, (216) will be false. It follows that there are entities u such that it
is impossible to use the same function d(q,p;u) for all situations of the class B.
We may try to avoid the contradiction by abandoning definition 2, 25,
which we omitted already for w, also for the entities q and p. In this case
q and p are no longer independent of each other, and we have a function rf(g;p)
such that
(22)
Since p is now dependent on #, the requirement of a chain structure must
be applied to the function d(q\p). This means that for all systems of the class B
the function ds(q',p) is the same. But then (12) leads, as before, to the result
that ds(q]u) is the same for all situations of the class B; and with this the same
contradictions as before are derivable.
It may be questioned whether we are entitled to extend the postulate of a
chain structure to the function d(q;p). Although the pi in this case are depend-
ent on the qt, one might argue that this is not to be regarded as a kind of
causal dependence; i.e., this dependence should not be construed as a physical
law, but as being due to accidental conditions determining the probability
situation and varying from case to case. The g and p- should, rather, be con-
ceived as the independent parameters of physical occurrences in the sense that
they can be arbitrarily chosen, and causal dependence in the sense of a chain
structure should be assumed to hold only for other entities u with respect to
the g and p^ If this objection is maintained, we argue as follows. If the proba-
bility situation varies from case to case, all kinds of situations should occur;
mathematically speaking, this means that the three probabilities d(q), d(p),
2 6. CHAIN STRUCTURE
d(q;p), represent the fundamental probabilities which can be assumed arbi-
trarily. We therefore can make the following assumption.
Assumption F: Among the systems of class B there is a subclass B' of sys-
tems having the same distribution d (q',p)j and for every ^-function in B there
is a number of systems having this ^-function and belonging to B', i.e., having
at the same time the distributions d (p) and d (q;p).
This means that we can coordinate a distribution d (q\p) to a situation hav-
ing ther distributions d(q) and d (p), and that there must be systems having
these three distributions. It is true that we have no means of determining
observationally the latter systems, i.e., the systems of the class B'. Assump-
tion T has therefore the character of a convention. But should it be impossible
to carry through this convention, i.e., should the possibility of defining such
systems be excluded, this would mean that the probability d(q\p) depends on
the distribution d(q), since the distribution d (p) is the same for all systems
of B. In other words, this would mean that the relative probability of having
a value p when a value q is given depends on the distribution of the values q.
The probability that a particle at the place q has the momentum p then would
depend on where the other particles are situated. This would not only contra-
dict our assumption according to which the q and p represent the independent
parameters, but would introduce a kind of dependence of the p on the q which
contradicts the requirements of a chain structure. Using assumption F, we
now write (12) for the class 5', in the form:
7;^) = Ja
d B '(q',u) = Jd B '(q;p) d(q ) p' J u)dp (23)
Since ds'(q]p) is now the same for all systems of B 1 ', namely, = d (q]p), it fol-
lows that also dB'(q\u} is the same for all systems of J3'. With this result the
same contradictions as before are derivable, since, according to assumption F,
B' includes a corresponding subclass A' of A, namely, a class A! of systems
with functions $(qi,q).
We have carried through this consideration for continuous variables q and p
because position and momentum are, in general, represented by variables of
this kind. Since a sharp measurement of continuous variables is impossible,
our result must be stated, more precisely, in the form: However small the
intervals Ag are chosen, it is always possible to find an entity u for which the
numerical contradictions arising from (21b) are sufficiently large. We do not
want to say by our proof that the classes A, A', B, B 1 must remain the same
for smaller A; this is obviously not the case. A statement, therefore, of what
happens in case A = is not derivable; but such statements are in any event
beyond the limits of assertability in quantum mechanics. If we were to use
discrete variables <fr and pk, such statements could be made; but in this case
our proof can be given in a simpler way. If q is exactly measured as being = q^
we have a function d(qi' 9 pk) ; the assumption that in a general situation s, p is
158 PART III. INTERPRETATIONS
independent of g, i.e., that d a (qi;pk) = d a (pk), then contradicts our definition of
a chain structure, since then d 8 (qi',pk) depends on d(q) and d(p), namely, is dif-
ferent according as s represents, or does not represent, a measurement of q. If,
on the other hand, we put d 9 (qi*,pk) = dfe ;p*)> we can construct, using (12) with
the subscript s instead of B, the same contradictions as above. For continuous
variables q and p the logical situation is different, because here sharp values do
not exist, and for every measurement of q within the exactness A we can assume
that d(qi',pk) = d(pk), without arriving at contradictions if only d(pk) is within
the limits given by the Heisenberg relation. For this case we therefore need a
special proof showing that contradictions arise for other entities u, as given
above.
We now must ask whether our negative result can be evaded by the aban-
donment of definition 1, 25. In order to answer this question we must enter
into some general considerations concerning a definition of the values of unob-
served entities.
If our system included no definition of unobserved entities, it would be easy
to carry through a chain structure. We then would assume that there were
unknown values q and p which were connected with the unknown value u in
such a way that a function (4) existed; we even could assume that this function
was degenerated into a causal function (3), 24. Furthermore, we could
give arbitrary values to the unobserved entities for each experiment in such
a way that the functions (4), 24, or (3), 24, are satisfied. The chosen values
could never be shown to be invalid if only we assume that every observation
disturbs the values in an unknown way. Such an interpretation, however,
cannot be called a permissible supplementation of observations because the
unobserved values are not connected with the observed values by means of
general rules. It therefore must be considered as a requirement of an interpo-
lated chain structure that there be general rules which determine the un-
observed values as functions of the observed values, and that these rules be
the same for every situation s. The latter qualification is necessary because
otherwise the disturbance by the observation would depend on the probability
distributions of the measured entities; in this case, the disturbance, which in
itself represents a causal occurrence, would not conform to our definition of a
chain structure.
Now we know that in a situation s we can make only one observation, either
of q, or of p, or of u; a further observation then will start from a new situation
created by the preceding observation. The required rules therefore must be so
given that they determine the unobserved value of any entity as a function
of the observed value alone. Let u be the observed value of an entity; then we
can introduce two functions fb(u ) and f a (u ) which determine the values
before and after the measurement, so that we have
U b = f b (Uo) U a = /a(tto) (24)
27- THE WAVE INTERPRETATION 1*9
Now this assumption leads to the following difficulties. When we have a situa-
tion which is definite in u , such as exists after a measurement m u with the
result u , and then make a measurement of u, the value after this measure-
ment must be the same as before this measurement. Otherwise the value of u
would change between the two measurements of u\ this would be a change
without cause, contradicting the principle of strict causality as well as the
principle of a probability chain structure. For such a situation the two func-
tions/6 and f a must therefore be the same. But then these two functions cannot
be different in any other situation. Since such a different situation is distin-
guished from the previous case only by a difference in the probability distribu-
tions, an assumption stating that the two functions /& and f a depend on the
situation s would mean that the disturbance by the measurement depends,
not only on the values of the entities, but also on the probabilities with which
these values occur. This is the case we excluded above as contradicting the
requirements of a chain structure.
It follows that for all situations we must have
/6=/a (25)
But then the generalization introduced by the function / is trivial, and does
not alter our previous results. Instead of considering the observed value u c
as the value before and after the measurement, we then would regard a value
f(uo) as the value before and after the measurement; and the preceding con-
siderations leading to contradictions could then be carried through in the same
way as before.
The considerations given represent a general proof that it is impossible to
interpret the statistical relations of quantum mechanics in terms of a chain
structure. This includes a proof that these relations cannot be interpreted in
terms of causal chains, since the latter constitute a special case of probability
chains. With these results the proof of the principle of anomaly, to which we
referred in 8, is given.
The demonstrations of this section are illustrated by the analysis of the
interference experiment which we discussed in 7. We saw there that we
cannot introduce a probability P(A.Bi,C) which is independent of the occur-
rences at the place B%. This is a special case of our general theorem stating
that we cannot introduce an invariant function (4); the function (4) will
always depend on the whole distributions d(q) and d(p) y i.e., on the situation s.
27. The Wave Interpretation
The duality of wave and corpuscle interpretation is based on the equivalence
of the function \l/(q) on the one side, and pairs of suitably chosen probability
functions on the other side. In the corpuscle interpretation the probability
IgO PART III. INTERPRETATIONS
functions are considered as directly physically meaningful, whereas the func-
tion \f/ appears only as a mathematical abbreviation standing for those
functions. In the wave interpretation it is the function \f/(q) which is considered
as directly physically meaningful. This function, however, is not interpreted as
a combination of two probability functions coordinated to particles, but as
the description of a physical state spread out over the whole space, i.e., of the
wave field. Whereas a probability function d(q) determines physical reality in
terms of an "or", meaning that there is a particle in the place gi, or q^ or etc.,
a wave function \l/(q) determines reality in terms of an "and"; it states that
there is a physical state in the place q\, and in the place g 2 , and etc. The nu-
merical values of the function d mean probabilities ; the numerical values of the
function \f/ mean amplitudes of the field.
This conception is introduced by a definition concerning unobserved entities
differing from definition 1 with respect to the rule which establishes the values
existing before the measurement. This is the following definition.
Definition 3. The value of an entity u measured in a situation s means the value
u existing after the measurement; before the measurement, and thus in the situation
Sj the entity u has all its possible values simultaneously.
By "possible values" we mean values which in the situation s can be expected
with a certain probability greater than to be the result of a measurement. If,
therefore, the eigen-values of u are discrete, only these discrete values represent
possible values; furthermore, if s is given by an eigen-function <p l (q) of u, only
one value is possible, namely, the value w f . The latter case, however, is excep-
tional; there will be, in general, a spectrum of possible values which are all
considered as existing simultaneously in the situation s.
The logical reason that this definition can be carried through is given by the
fact that statements about unobserved entities are un verifiable ; the given defi-
nition is therefore compatible with all observations, as is any rule defining one
of the possible values as the actual one. That, moreover, the given definition
leads to a conception of a field, in the technical sense of the word, has a mathe-
matical reason grounded in the form of the time-dependent Schrodinger equa-
tion : It is the fact that the function \l/ is additive. This means that if we have
two different possible descriptions ^i and ^ 2 of physical states, their sum
^i + fa is also a possible description of a physical state. We can interpret this
additivity of amplitudes as meaning the superposition of two wave fields. This
fact is frequently expressed by saying that the ^-function follows the principle
of superposition.
If we use the corpuscle interpretation, the additivity of ^ means a rather
complicated causal dependence between the physical states. Since the functions
\f/i and fa express the probability situations holding for these individual states,
the function ^i + fa expresses the probability situation of the state given by
the superposition. The latter probability situation, however, is not given by the
addition of the probability functions, but constructed out of these functions
27- THE WAVE INTERPRETATION
in a rather complicated way. If we use Feenberg's rule for the determination
of the ^-function, we can write these relations as follows:
=fop\
= Mr)
d(q) =
<
ot
at
of
(1)
The last two lines state that the two functions d(q) and d(q) of the resulting
dt
situation arc determined by the corresponding functions of the individual situ-
ations. The distributions of all other entities, for instance, the distribution d(p) ,
are similarly determined. The operators f op used here represent the mathe-
matical operations explained in 20. These relations take the place of an
additivity of probabilities. As we have shown in the discussion of the inter-
ference experiment, this means that the situation resulting from such super-
position is constructed in terms of causal interactions which may be of such
a kind that they violate the principle of action by contact.
It is an advantage of the wave interpretation that it satisfies the principle
of action by contact, since the value of the resulting amplitude in a place is
determined only by the individual amplitudes in the same place. The disadvan-
tage of the wave interpretation, on the other hand, consists in the fact that
this interpretation is restricted to interphenomena] as soon as we are concerned
with phenomena, we must return to the corpuscle interpretation by replacing
$(<j) by probability functions. Observable entities always show a punctiform
character; their spatial description is written in terms of the "or", not of the
"and". This fact finds its ｅxpression in definition 3 which coordinates to the
entity u after the measurement only one value and thus abandons the wave
interpretation for this case. We see here the reason that the wave interpreta-
tion cannot reintroduce determinism in the physical world. It is true that the
laws controlling the ^-function, i.e., the Schrodinger equation, have the form
of causal laws if the ^-function is considered as representing a wave field; but
the determinism so established is restricted to interphenomena, and breaks
down in the realm of phenomena. 1 The indeterminism holding for phenomena
finds its ｅxpression in the fact that the numerical results of observations in
iCf.alsop.31.
132 PART III. INTERPRETATIONS
general cannot be strictly predicted; such predictions are bound to be proba-
bility statements.
If we interpret ^ as a wave, we must do the same with | ^ | 2 ; we therefore
must consider also the function | \l/(q,t) | 2 as spread out over the g-space. In the
cases where | ^ | 2 is independent of t, i.e., for stationary systems such as (2),
18, |^| 2 then represents a static field. It also happens, however, that |^| 2
is time dependent, and in such a way that it represents oscillations. An illus-
tration is given by the nonstationary case of (5), 18. The function |^| 2
is here determined as
I 2 = *M?,fl*V**ta,0
(2)
We have dropped here the argument q in <p. This can be written in the form of
a Fourier expansion in real functions with real coefficients a , a*, bkm, which
depend only on q:
a ** cos 2 " Vkmt + *"? sn 2jnf
k m>k
1 / x H k -H
Q>km = CLkm + Omfc bkm = 7 (dkm ttmfc)
/i
m* (3)
The fact that we have here a double summation is irrelevant; we can easily
introduce a numeration in one subscript by using one of the methods of
enumerating a two-dimensional lattice of whole numbers. The duality of inter-
pretations is now carried through as follows. Interpreting | \ls(q,f) | 2 as a proba-
bility function d(q y t) we shall say that here the probability d(q,t) performs
oscillations which are given by a superposition of elementary oscillations of the
TT _ TT
frequency vkm = * - - . In the wave interpretation we shall speak of waves
h
given by a superposition of such frequencies; this corresponds to Bohr's inter-
pretation of his atom model in which the Vkm represent the frequencies of the
waves emitted by the transition of an electron from the energy level Hk to
the energy level H m . We see that in this case waves are given, not only by the
function ^, but also the function | ^ | 2 .
The duality of wave and corpuscle interpretation is sometimes presented as
a parallel to the transformation of the state function (cf. 19). The use of
the state function \l/(q) then is considered as the wave interpretation, whereas
the use of a state function <r(p), or <?(#), is conceived as a corpuscle interpreta-
tion. This conception appears even more suggestive if the latter state function
is a discrete function cr*, since the discreteness seems to symbolize corpuscles.
5} 7- THE WAVE INTERPRETATION 133
Such a conception, however, is based on a misunderstanding of the question
of interpretation. The duality, or multiplicity, of state functions has no rela-
tion to the duality of wave and corpuscle interpretation. Each state function,
whether it is the function $(q) or the function o^, is capable of a dual interpre-
tation. If we consider \l/(q) as a wave, and correspondingly | \l/(q) j 2 as a static
field, or also as a wave, we conceive these functions as spread out over the
whole g-space; in the corpuscle interpretation, on the contrary, we conceive
$(q) as, a probability amplitude, and \\l/(q) | 2 as the probability of finding a
particle at the place q. Likewise, we have two interpretations for the state
function <?& Let us assume that the <r k are the coefficients of the expansion in
eigen-functions of the energy H . The corpuscle interpretation then is given by
the idea that there exists only one energy state Hk, although we do not know
which is this value Hk] the probability for each such state is given by l^l 2 .
The wave interpretation is represented by the assumption that all states Hk
exist simultaneously, and that .the product \<r k \ 2 'Hk determines the relative
amount of energy contributed to the total energy H by each state H k . The sum
H = ^\<r k \*H k (4)
then represents this total value, whereas in the corpuscle interpretation it
represents the average value.
Both cases of state functions are coupled in such a way that if we use the
wave interpretation for \l/, we must also use the wave interpretation for o^;
and if we decide for the corpuscle interpretation of ^, we must also apply the
corpuscle interpretation to <r k . This coupling is necessary because, if ^ is a real
field extended over the 0-space, its components ^ are as real as \f/, and exist
simultaneously.
We see that the decision for wave or corpuscle interpretation is independent
of the choice of the state function. It is true that each state function describes
the physical situation completely; but each state function is capable of both
interpretations.
Let us now consider the reason why a swarm of n particles can be considered
equally as a field of waves. By a "swarm" we understand a number n of
particles so far distant from each other that their mutual interaction can be
neglected. In this case the energy operator H op splits into a sum of operators
H op = H op w+H op + ... (5)
such that each operator H op (m) concerns only one particle, i.e., contains only
the three coordinates of the mth particle. Let <pk m (m} be the eigen-functions of
the operator H op (m) ; they will then satisfy the first Schrodinger equation
#op (m V* m (w) = H km ^<> (6)
1 34 PART III. INTERPRETATIONS
Then the Schrodinger equation of the whole system
H op <pk = H k <pk (7)
is satisfied by the solution
<Pk = <f>kfa. . . k n = <?*, * 9k, ' * ' <(>k n H k = H kl + H kt + . . . #* n (8)
This means that the eigen-function <p kl . . . fcft of the whole swarm is the product
of the eigen-functions of the individual particles. The proof is easily given by
putting for H op in (7) its value from (5) :
.+//*)% **> (9)
We can therefore consider a ^-function of the form
which splits into a product of ^-functions of the form
as the definition of a physical situation which can be interpreted as a swarm of
particles in a stationary state. Correspondingly, the general nonstationary
state of a swarm of particles is characterized by a linear superposition of
functions \l/ kl . . . km (g,0 of the form (11), i.e., by a ^-function
*(ff,0 = E <^..*^ ' '^n ' - T <**+" -^ (12)
* 4 . . . A- n
The possibility of a corpuscular interpretation of a ^-function of this kind,
as a swarm of particles, is based on the corpuscular interpretation of the indi-
vidual functions \l/ kl k^ - .of which the total ^-function is composed. If, on the
other hand, these individual ^-functions are interpreted as waves, the swarm
obtains a wave-character. Whereas H k in the corpuscle interpretation repre-
sents the energy of a particle, it determines in the wave interpretation the
TT
frequency through the relation v k = . In the corpuscle interpretation the
h
square |^(g,) | 2 means a probability density and thus determines the relative
number of particles for a given place q at the time /; in the wave interpretation
the same amount represents the intensity of the field.
Conditions similar to the kind described hold in electro-magnetic fields,
although this case is combined with further mathematical complications which
we cannot present in this book. Suffice it to say that the corpuscle interpreta-
tion of light waves is analogous to the duality of interpretations existing for a
swarm of electrons.
2 7- THE WAVE INTERPRETATION 135
There are two difficulties frequently mentioned in combination with the
wave interpretation. one is that the values of ^ are complex numbers. This,
however, cannot be considered as an obstacle to a physical interpretation. A
complex number is as good for the description of physical entities as is a real
number; the complex function is to be conceived as an abbreviation standing
for two real functions. We know from the considerations of 20 that these two
functions represent, indirectly, two probability distributions. If we consider ^
as a wave, its complex value can be interpreted as meaning that the physical
wave field is characterized, not by one scalar function, but by two. Wave fields
of such a dual character are known from the electro-magnetic fields which
include both an electric and a magnetic vector.
The other difficulty is more serious; it consists in the fact that the ^-waves
are functions in the configuration space of the n parameters gi . . . q n , whereas
ordinary waves are functions in the 3-dimensional space. If we wish to in-
terpret these n-dimensional waves as waves of the 3-dimensional space, we
are led to wave fields of a very complicated structure, Let us begin with a
case of n = 6 parameters q corresponding to the Cartesian coordinates of
two particles. Translating the 6-dimensional function \l/(qi . . . g 6 ) into 3-
dimensional functions, we then must start from the fact that the function
$(q\ . . . q&) coordinates an amplitude, not to each point of the 3-dimensional
space, but to each pair of points. This can be interpreted as meaning a set of
wave fields such that there is a wave field, filling the whole space, for each
point as starting point. This set of wave fields cannot be replaced by the indi-
cation of one wave field conceived as the superposition of the set. Such a
superposition can be mathematically constructed by integrating \l/(q\ . . . q&)
over three of its variables ; but even if we thus construct two 3-dimensional func-
tions, by integrating first over #1, q Zj q* and then over g 4 , # 5 , q 6 , the resulting
two functions are not equivalent to the function \l/(qi . . . # 6 ). We are therefore
compelled to describe the situation in 3-dimensional space as an infinite set
of wave fields.
Furthermore, we must be aware of the fact that a function resulting from
partial integration of a ^-function does not have the properties of a ^-function.
This means that such a function does not describe a physical situation, and
that its square does not determine a probability distribution. Thus the function
(13)
does not represent the probability of observing the particle number 1 at the
place qi, q*, #3. This probability is rather determined by the function
= /
(14)
which is different from (13) .
136 PART III. INTERPRETATIONS
For a greater number of variables the 3-dimensional description is even more
complicated. Thus, for three particles with nine Cartesian coordinates, we have
a value ^ coordinated to every triplet of points. This means that we have a
set of wave fields such that for every pair of two points there is a 3-dimensional
wave field. In the general case of r particles we have a set such that for any
combination of (r 1) points there is a 3-dimensional wave field. If the
q\ . . . q n contain other than Cartesian coordinates, for instance, parameters
of rotation, we must introduce sets of wave fields characterized by values of
these parameters.
These considerations show that the ^-waves in general do not have the char-
acter of ordinary wave fields; this is the case only when the g-space reduces
to three dimensions, i.e., when single particles are concerned. Of this kind are
problems of swarms of particles for which the interaction between the indi-
vidual particles can be neglected. This is the reason that a wave interpretation
in the usual sense of the word can be carried through for light rays and
electron beams.
28. Observational Language and Quantum
Mechanical Language
In the preceding sections we have seen that every exhaustive interpretation
leads to causal anomalies. We now shall inquire whether these causal anomalies
can be avoided by the use of restrictive interpretations. The answer will turn
out to be in the affirmative.
Before we can present such interpretations we must enter into closer analysis
of the logical nature of interpretations, and inquire into the conditions under
which the world of unobserved entities can be called complete.
Problems of epistemology are greatly simplified when we turn from the
consideration of physical worlds to the consideration of physical languages. 1
Questions concerning the existence of physical entities are thus transformed
into questions of the meaning of propositions. This has the great advantage
that they can be discussed soberly as questions of logic outside the atmosphere
of metaphysical preconceptions. For our problem a linguistic analysis can be
carried through in the following form.
We have an observational language and a quantum mechanical language. The
observational language contains terms such as "Geiger counter", "Wilson
cloud chamber", "black line on a photographic film", "indication of a dial",
etc.; the phrases "measurement of u" and "the result of the measurement of
u" are defined in terms of these elementary ｅxpressions. Similarly, a physical
situation s can be defined in observational terms; a way in which this is done
is indicated in 20 with the observational determination of the ^-function.
1 The importance of this method for philosophical analysis has been brought to the fore
in recent years by R. Carnap, Logical Syntax of Language (London, 1937).
28. OBSERVATIONAL AND QUANTUM LANGUAGE 137
The quantum mechanical language contains terms like "position q of an
electron" and "momentum p of an electron". Between the two languages
there exists the following relation : The truth and falsehood of statements of
the quantum mechanical language is defined in terms of the truth and false-
hood of statements of the observational language. We say, for instance, "the
electron has the position <?", when we know that the statement, "a measure-
ment of position has been made and its result was g", is true.
This relation between truth values of statements in the two languages can
be conceived as a relation of meanings. The meanings of statements of the
quantum mechanical language are definable in terms of the meanings of state-
ments of the observational language. Usually this relation is conceived as an
equivalence of meanings. In this interpretation a quantum mechanical state-
ment A has the same meaning as the set of observational statements ai . . . a n
which verify A. This is not strictly correct; the relation of meanings is more
complicated because we have no absolute verification. We can only say: A
is highly probable if the set ai . . . a n is true. For the present purpose, how-
ever, we need not enter into a consideration of this question; we may neglect
the probability factor in the correlation of A and ai . . . a n and consider the
meaning of A as given by the meaning of ai . . . a n .
It is important to realize that such a statement about meaning, with or
without consideration of the probability factor, is always based on definitions.
We define A in terms of a\ . . . a n . Without such definitions a quantum me-
chanical language could not be established. The definitions 1 and 2, 25,
and definition 3, 27, constitute examples of such definitions. It is therefore
no objection to our exhaustive interpretation of 25 that it uses definitions;
any interpretation will do so.
Judged from the standpoint of observational predictions, no interpretation
and therefore no quantum mechanical language at all is necessary; we then
need not speak of electrons and their speeds and positions, but can say every-
thing in terms of instruments of measurements. We shall say, for instance,
"if a measuring instrument of a certain kind is used in such and such observa-
tional conditions, its dial will show this or that number". only if we wish to
introduce statements about microcosmic entities must we use definitions.
Now let us turn to an analysis of the observational language coordinated to
quantum mechanics. As to this question, the results of 24 lead to important
consequences. We saw in 24 that, so far as predictable statistics of individual
entities are concerned, quantum mechanics is equal to classical statistics. A
difference is to be found only in the way of derivation; classical statistics uses
here, and must do so, an assumption about th& distribution of combinations
q and p, whereas quantum mechanics does not use such an assumption. Let us
ask what the omission of this assumption means when we judge it in terms of
the observational language.
When we say that, in a general physical situation s, simultaneous values of
138 PART III. INTERPRETATIONS
q and p cannot be ascertained, this statement of the quantum mechanical
language is to be translated into observational language as follows. The
operation measurement of the entity q, abbreviated m q , is defined in terms of
macrocosmic manipulations by the use of devices like Geiger counters, X-ray
tubes, dials, etc. ; in a similar way the operation m p is defined. From these
definitions we can infer by considerations remaining entirely within the obser-
vational language that the two operations m q and m p cannot be performed
simultaneously. It is true the observational definition of a measurement is
chosen with respect to quantum mechanical considerations; this is the reason
that, for instance, a measurement of the momentum is defined in terms of
macrocosmic devices which exclude the presence of extremely short waves.
But once the observational definition is given, the incompatibility of m q and m p
is a merely macrocosmic affair.
It follows that within the observational language we cannot raise a question
with respect to what would happen if the observations m g and m p were made
simultaneously. Such a question is as unreasonable as, for instance, the ques-
tion "what would happen if the sun's temperature were at the same time one
hundred degrees, and one million degrees?" The absence of statements about
simultaneous values of entities does not lead, therefore, to blanks in the
observational language.
Combining this result with the results of 24 according to which the statis-
tics of individual entities are fully determined by quantum mechanics, we
arrive at the statement that the observational language of quantum mechanics
is statistically complete. By "statistically complete" we mean that for every
possible situation, defined in observational terms, the observational result of
a measurement can be predicted with a determinate probability. We can ex-
press the given statement, therefore, also by saying that the predictive methods
of quantum mechanics are statistically complete in observational terms. If we were
able to know causal laws and to apply classical statistics to given probability
distributions of initial conditions which satisfy the relation of uncertainty, our
predictions so constructed would not be superior to quantum mechanical
predictions so far as observations are concerned; every observational result
that could be predicted by classical-statistical methods would be equally pre-
dictable by quantum mechanical methods.
This result has an important bearing on the question of interpretations.
Whatever interpretation we give, i.e., however we construct the quantum me-
chanical language, it will always be statistically complete in observational
terms. When we call one interpretation incomplete, or less complete than
another, this word "complete" must have a meaning other than "complete in
observational terms".
This is indeed the case. When we consider a statement about simultaneous
values of q and p, or at least about the statistical distribution d(q,p), as nec-
essary for a complete description, the reason is given in considerations con-
2Q. RESTRICTED MEANING 139
cerning the micro-world. Here the entities q and p are interpreted as being
constituents of the space-time description of a physical state. If the operations
m q and m pt observationally defined, are incompatible, we wish to have other
ways of knowing simultaneous values of q and p; this desire is satisfied by
suitable definitions which determine these values by other means than meas-
urements. An interpretation which does not include such definitions is statis-
tically incomplete with respect to space-time description, although it will be
statistically complete with respect to testability.
In order to avoid the ambiguities of the word "complete", we have intro-
duced in 8 a terminology which distinguishes between exhaustive and re-
strictive interpretations. An interpretation like that of 25, in which the values
of unobserved entities are completely defined, is exhaustive; interpretations
in which those values remain undefined are restrictive. The preceding results
make it clear that a restrictive interpretation will always be complete in
observational terms.
Restrictive interpretations are introduced with the intention of eliminating
the causal anomalies ensuing for exhaustive interpretations. The results of 26
have made it clear that such anomalies will always arise if we give a complete
set of definitions like definitions 1 and 2, 25, for the values of unobserved
entities. If we want to exclude statements involving causal anomalies from the
assertions of quantum mechanics, we must therefore use an incomplete set of
definitions. We thus restrict the domain of quantum mechanical assertions;
this is the reason that we speak of a restrictive interpretation.
The results of 24 make it clear that restrictive interpretations will at least
be individually complete in the statistical sense. We mean by this term that they
are statistically complete with respect to statements about every individual
entity; what is excluded are only statements about the combinations of values
of entities, namely, noncommutative entities.
Restrictive interpretations can be given in two ways. In the first way the
undesired statements are ruled out of quantum mechanical language by a
definition of meaning which makes such statements meaningless. We shall
therefore speak here of an interpretation by a restricted meaning. In the second
way the undesired statements are excluded, not from quantum mechanical
language , but from quantum mechanical assertions; this is done by ascribing
to those statements a third truth value, called indeterminate. This interpreta-
tion may therefore be called an interpretation by a restricted assertability.
We now shall turn to an analysis of these two restrictive interpretations.
29. Interpretation by a Restricted Meaning
The interpretation by a restricted meaning which we shall present here formu-
lates, on the whole, ideas which have been developed by Bohr and Heisen-
berg. We therefore shall call this conception the Bohr-Heisenberg interpretation
140 PART III. INTERPRETATIONS
without intending to say that every detail of the given interpretation would
be endorsed by Bohr and Heisenberg.
This interpretation does not use our definitions 1 and 2, 25. For the values
of measured entities it uses the following definition :
Definition 4. The result of a measurement represents the value of the measured
entity immediately after the measurement.
This definition contains only the common part of our definitions 1, 25,
and 3, 27; a statement concerning the value of the entity before the measure-
ment is omitted. In this interpretation we therefore can no longer say that
the observed entity remains undisturbed; both the observed and the unob-
served entity may be disturbed. on the other hand, such a disturbance of the
measured entity is not asserted- this question is deliberately left unanswered by
definition 4. The restrictive interpretation neither asserts nor denies a dis-
turbance by the measurement. The interval "immediately after" of definition
4 is determined by the time dependence of the ^-function; in a stationary state
this interval may be chosen as long as we wish.
It is an immediate consequence of the restriction to definition 4 that simul-
taneous values cannot be measured. The considerations which we attached to
definition 1 are no longer applicable, and if we measure first q and then p, the
obtained values of q and p do not represent simultaneous values; only q
represents a value existing between these two measurements, whereas p repre-
sents a value existing after the second measurement, when the value q is no
longer valid. only if a measurement of an entity u is followed by a measure-
ment of the same kind, the value u obtained in the first measurement repre-
sents the value existing before and after the second measurement; but it is
valid before the second measurement only because it represents, jointly, the
value after the first measurement. only a measurement following an equal
measurement, therefore, is known not to disturb the entity measured.
The reason we interpret the measured value, in definition 4, as the value
after the measurement, and not before the measurement, is as follows. We know
that a repetition of the measurement produces the same value; therefore, if the
obtained value holds for the time before the second measurement, it must also
hold for the time after the first. Thus, if we replace the term "after" in defini-
tion 4 by the term "before", we are led to the consequence that the measured
value holds before and after each measurement; this means that we have
introduced definition 1. The asymmetry of time with respect to measurements,
expressed in this consideration, is a characteristic feature of quantum me-
chanics (cf., however, fn. 1, p. 119).
We said that if q has been measured, we do not know the value of p. This
lack of knowledge is considered in the Bohr-Heisenberg interpretation as
making a statement about p meaningless. It is here, therefore, that this inter-
pretation introduces a rule restricting quantum mechanical language. This is
expressed in the following definition.
29' RESTRICTED MEANING
Definition 6. In a physical state not preceded by a measurement of an entity u,
any statement about a value of the entity u is meaningless.
In this definition we are using the term "statement" in a sense somewhat
wider than usual, since a statement is usually defined as having meaning. Let
us use the term "proposition" in this narrower sense as including meaning.
Then, definition 5 states that not every statement of the form "the value of
the entity is u", is a proposition, i.e., has meaning. Because it uses definition 5,
the Bohr-Heisenberg interpretation can be called an interpretation by a re-
stricted meaning.
We must add a remark concerning the logical form of definition 5. Modern
logic distinguishes between object language and metalanguage' the first speaks
about physical objects, the second about statements, which in turn are referred
to objects. 1 The first part of the metalanguage, syntax, concerns only state-
ments, without dealing with physical objects; this part formulates the struc-
ture of statements. The second part of the metalanguage, semantics, refers to
both statements and physical objects. This part formulates, in particular, the
rules concerning truth and meaning of statements, since these rules include a
reference to physical objects. The third part of the metalanguage, pragmatics,
includes a reference to persons who use the object language. 2
Applying this terminology to the discussion of definition 5, we arrive at
the following result: Whereas definition 4, and likewise definitions 1-2, 25,
and definition 3, 27, determine terms of the object language, namely, terms
of the form "value of the entity u", definition 5 determines a term of the meta-
language, namely, the term "meaningless". It is therefore a semantical rule.
We can express it in table 1. We denote the statement "a measurement of u
is made" by m u , and the statement "the indication of the measuring instru-
ment shows the value u" by u. These two statements belong to the observa-
tional language. Quantum mechanical statements are written with capital
letters; and U expresses the statement "the value of the entity immediately
after the measurement is w". 3 Table 1 shows the coordination of the two
languages.
1 A means of indicating the transition from object language to metalanguage is in the
use of quotes; similarly, italics can be used. We shall use, for our presentation of logic,
italics instead of quotes for symbols denoting propositions, in combination with the rule
that symbols of operations standing between symbols of propositions are understood to
apply to the propositions, not to their names (i.e., autonomous usage of operations, in
the terminology of R. Carnap). Thus, we shall write "a is true", not " 'a 1 is true n , and
"a.b is true", not " 'a.b' is true". All formulae given by us are therefore, strictly speak-
ing, not formulae of the object language, but descriptions of such formulae. For most
practical purposes, however, it is permissible to forget about this distinction.
2 This distinction has been carried through by C. W. Morris, "Foundations of the
Theory of Signs, 11 International Encyclopedia of Unified Science, Vol. I, no. 2 (Chicago,
1938).
1 More precisely: "immediately after the time for which the truth value of the state-
ment m u is considered". If m u is true, this means the same as "immediately after the
measurement": and if m u is false, we thus equally coordinate to U a time at which its
stipulated truth value holds. (The ｅxpression "after the measurement 11 then would be
inapplicable.)
142 PART III. INTERPRETATIONS
A meaningless statement is not subject to propositional operations; thus,
the negation of a meaningless statement is equally meaningless. Similarly, a
combination of a meaningful and a meaningless statement is not meaningful.
If the statement a is meaningful, and 6 meaningless, then a and b is meaning-
less; so is a or b. Not even the assertion of the tertium non datur, b or non-b, is
meaningful. The given restriction of meaning therefore cuts off a large section
from the domain of quantum mechanical language. With this, all statements
referring to causal anomalies are cut off (cf . pp. 40-41).
The only justification of definition 5 is that it eliminates the causal anoma-
lies. This should be clearly kept in mind. It would be wrong to argue that
statements about the value of an entity before a measurement are meaningless
TABLE l
Observation
m u
al language
u
Quantum
mechanical
language
U
true
true
true
true
false
false
false
true
meaningless
false
false
meaningless
because they are not verifiable. Statements about the value after the measure-
ment are not verifiable either. If, in the interpretation under consideration,
the one sort of statement are forbidden and the other admitted, this must be
considered as a rule which, logically speaking, is arbitrary, and which can be
judged only from the standpoint of expediency. From this standpoint its
advantage consists in the fact that it eliminates causal anomalies, but that is
all we can say in its favor.
It is often forgotten that the Bohr-Heisenberg interpretation uses definition
4. This definition seems so natural that its character as a definition is over-
looked. But without it, the interpretation could not be given. Applying the
language used in Part I we can say that definition 4 is necessary for the transi-
tion from observational data to phenomena; it defines the phenomena. It is
therefore incorrect to say that the Bohr-Heisenberg interpretation uses only
verifiable statements. We must say, instead, that it is an interpretation using
a weaker definition concerning unobserved entities than other interpretations,
and using a restricted meaning, with the advantage of thus excluding state-
ments about causal anomalies.
We now must consider the relations between noncommutative entities. We
know that if a measurement of q is made, a measurement of p cannot be
2Q- RESTRICTED MEANING 143
made at the same time, and vice versa. Statements about simultaneous values
of noncommutative entities are called complementary statements. With defini-
tion 5 we have therefore the following theorem :
Theorem 1. If two statements are complementary, at most one of them is
meaningful; the other is meaningless.
We say "at most" because it is not necessary that one of the two statements
be meaningful; in a general situation s, determined by a ^-function which is
not an eigen-function of one of the entities considered, both statements will
be meaningless.
Theorem 1 represents a physical law; it is but another version of the commu-
tation rule, or of the principle of indeterminacy, which excludes a simultaneous
measurement of noncommutative entities. We see that with theorem 1 a
physical law has been expressed in a semantical form; it is stated as a rule for
the meaning of statements. This is unsatisfactory, since, usually, physical
laws are expressed in the object language, not in the metalanguage. Moreover,
the law formulated in theorem 1 concerns linguistic ｅxpressions which are not
always meaningful; the law states the conditions under which these ｅxpressions
are meaningful. Whereas such rules appear natural when they are introduced
as conventions determining the language to be used, it seems unnatural that
such a rule should assume the function of a law of physics. The law can be
stated only by reference to a class of linguistic ｅxpressions which includes both
meaningful and meaningless ｅxpressions; with this law, therefore, meaning-
less ｅxpressions are included, in a certain sense, in the language of physics.
The latter fact is also illustrated by the following consideration. Let 17(0 be
the prepositional function "the entity has the value u at the time t". Whether
U(t) is meaningful at a given time t depends on whether a measurement ra
is made at that time. We therefore have in this interpretation propositional
functions which are meaningful for some values of the variable t, and meaning-
less for other values of t.
The question arises whether it is possible to construct an interpretation
which avoids these disadvantages. An interesting attempt to construct such
an interpretation has been made by M. Strauss. 4 Although the rules under-
lying this interpretation are not expressly stated, it seems that they can be
construed in the following way.
Definitions 1-2, 25, and definition 3, 27, are rejected. Definition 4 is
maintained. Instead of definition 5, the following definition is introduced.
4 M. Strauss, "Zur Begriindung der statistischen Transformationstheorie der Quanten-
physik," Ber. d. Berliner Akad., Phys.-Math. Kl., XXVII (1936), and "Formal Problems
of Probability Theory in the Light of Quantum Mechanics, " Unity of Science Forum }
Synthese (The Hague, Holland, 1938), p. 35; (1939), pp. 49, 65. In these writings Strauss
also develops a form of probability theory in which my rule of existence for probabilities
is changed with respect to complementary statements. Such a change is necessary,
however, only if an incomplete notation is used, such as is done by us in the beginning of
22. In a notation which uses the term m u in the first place of probability ｅxpressions, such
as introduced in (13), 22, a change of the rule of existence can be avoided.
144 PART HI. INTERPRETATIONS
Definition 6. A quantum mechanical statement U is meaningful if it is possible
to make a measurement m v .
It follows that all quantum mechanical statements concerning individual
entities are meaningful, since it is always possible to measure such an entity.
Only when U stands for the combination of two complementary statements
P and Q is it not possible to make the corresponding measurement; therefore
a statement like P and Q is meaningless. Similarly, other combinations are
considered meaningless, such as P or Q. The logic of quantum mechanics is
constructed, according to Strauss, so that not all statements are connectable;
there are nonconnectable statements.
It may be regarded as an advantage that this interpretation constructs the
language of physics in such a way that it contains only meaningful elements.
A determination concerning meaningless ｅxpressions is formulated only in
terms of the rules for the connection of statements. on the other hand, the
physical law of complementarity is once more expressed as a semantic rule,
not as & statement of the object language.
Leaving it open whether or not the latter fact is to be considered as a disad-
vantage, we now must point out a serious difficulty with this interpretation
resulting from definition 6. If we consider U as a function of t, it is indeed
always possible to measure the entity u, and therefore U is always meaningful.
It is different when we consider U as a function of the general physical situa-
tion s characterized by a general function \f/. Is it possible to measure the entity
u in a general situation s? Since we know that the measurement of u destroys
the situation s, this is not possible. Thus, it follows that in a general situation s
even the individual statement U is meaningless. The given interpretation
therefore is reduced to the interpretation based on definition 5. on the other
hand, if the meaning of definition 6 is so construed that it is possible to measure
u even in a general situation s, the obtained value u must mean the value of
the entity in the situation s and therefore before the measurement. The inter-
pretation is thereby shown to use definition 1, 25. But if the latter definition
is used, it is possible to make statements about simultaneous values between
two measurements, such as explained in 25; this means that the rules con-
cerning nonconnectable statements break down.
If we are right, therefore, in considering Strauss's interpretation as given
through definition 4 and definition 6, we come to the result that this interpre-
tation leads back to the interpretation of definition 4 and definition 5. 6
30. Interpretation through a Three-valued Logic
The considerations of the preceding section have shown that, if we regard
statements about values of unobserved entities as meaningless, we must in-
clude meaningless statements of this kind in the language of physics. If we
* Mr. Strauss* informs me that he is planning to publish a new and somewhat modified
presentation of his conceptions.
30. THREE-VALUED LOGIC 145
wish to avoid this consequence, we must use an interpretation which excludes
such statements, not from the domain of meaning, but from the domain of
assertability. We thus are led to a three-valued logic, which has a special cate-
gory for this kind of statements.
Ordinary logic is two-valued; it is constructed in terms of the truth values
truth and falsehood. It is possible to introduce an intermediate truth value
which may be called indeterminacy, and to coordinate this truth value to the
group of statements which in the Bohr-Heisenberg interpretation are called
meaningless. Several reasons can be adduced for such an interpretation. If an
entity which can be measured under certain conditions cannot be measured
under other conditions, it appears natural to consider its value under the latter
conditions as indeterminate. It is not necessary to cross out statements about
this entity from the domain of meaningful statements; all we need is a direc-
tion that such statements can be dealt with neither as true nor as false state-
ments. This is achieved with the introduction of a third truth value of
indeterminacy. The meaning of the term "indeterminate" must be carefully
distinguished from the meaning of the term "unknown' '. The latter term
applies even to two-valued statements, since the truth value of a statement of
ordinary logic can be unknown; we then know, however, that the statement
is either true or false. The principle of the tertium non datur, or of the
excluded middle, expressed in this assertion, is one of the pillars of traditional
logic. If, on the other hand, we have a third truth value of indeterminacy, the
tertium non datur is no longer a valid formula; there is a tertium, a middle
value, represented by the logical status indeterminate.
The quantum mechanical significance of the truth-value indeterminacy is
made clear by the following consideration. Imagine a general physical situa-
tion s, in which we make a measurement of the entity q; in doing so we have
once and forever renounced knowing what would have resulted if we had made
a measurement of the entity p. It is useless to make a measurement of p in
the new situation, since we know that the measurement of q has changed the
situation. It is equally useless to construct another system with the same
situation s as before, and to make a measurement of p in this system. Since the
result of a measurement of p is determined only with a certain probability,
this repetition of the measurement may produce a value different from that
which we would have obtained in the first case. The probability character of
quantum mechanical predictions entails an absolutism of the individual case;
it makes the individual occurrence unrepeatable, irretrievable. We express this
fact by regarding the unobserved value as indeterminate, this word being
taken in the sense of a third truth value.
The case considered is different, with respect to logical structure, from a case
of macrocosmic probability relations. Let us assume that John says, "If I cast
the die in the next throw, I shall get 'six* ". Peter states, "If I cast the die,
instead, I shall get 'five' ". Let John throw the die, and let 'four' be his result;
146 PART III. INTERPRETATIONS
we then know that John's statement was false. As to Peter's statement, how-
ever, we are left without a decision. The situation resembles the quantum me-
chanical case so far as we have no means of determining the truth of Peter's
statement by having Peter cast the die after John's throw; since his throw
then starts in a new situation, its result cannot inform us about what would
have happened if Peter had cast the die in John's place. Since casting a die,
however, is a macrocosmic affair, we have in principle other means of testing
Peter's statement after John has thrown the die, or even before either has
thrown the die. We should have to measure exactly the position of the die, the
status of Peter's muscles, etc., and then could foretell the result of Peter's
throw with as high a probability as we wish; or let us better say, since we
cannot do it, Laplace's superman could. For us the truth value of John's
TABLE 2
Observation
al language
u
Quantum
mechanical
language
U
T
T
T
T
F
F
F
T
I
F
F
I
statement will always remain unknown; but it is not indeterminate, since it is
possible in principle to determine it, and only lack of technical abilities pre-
vents us from so doing. It is different with the quantum mechanical example
considered. After a measurement of q has been made in the general situation s,
even Laplace's superman could not find out what would have happened if we
had measured p. We express this fact by giving to a statement about p the
logical truth-value indeterminate.
The introduction of the truth-value indeterminate in quantum mechanical
language can be formally represented by table 2, which determines the truth
values of quantum mechanical statements as a function of the truth values of
statements of the observational language. We denote truth by T, falsehood
by F y indeterminacy by 7. The meaning of the symbols m u , u, and U is the
same as explained on page 141. *
Let us add some remarks concerning the logical position of the quantum
mechanical language so constructed. When we divide the exhaustive interpre-
tations into a corpuscle language and a wave language, the language introduced
by table 2 may be considered as a neutral language, since it does not determine
one of these interpretations. It is true that we speak of the measured entity
1 As to the use of the functor, "the value of the entity," cf . fn. 4, p. 159.
30. THREE-VALUED LOGIC 147
sometimes as the path of a particle, sometimes as the path of needle radiation;
or sometimes as the energy of a particle, sometimes as the frequency of a wave.
This terminology, however, is only a remainder deriving from the corpuscle
or wave language. Since the values of unobserved entities are not determined,
the language of table 2 leaves it open whether the measured entities belong to
waves or corpuscles; we shall therefore use a neutral term and say that the
measured entities represent parameters of quantum mechanical objects. The
difference between calling such a parameter an energy or a frequency then is
only a difference with respect to a factor h in the numerical value of the param-
eter. This ambiguity in the interpretation of unobserved entities is made pos-
sible through the use of the category indeterminate. Since it is indeterminate
whether the unmeasured entity has the value u\ 9 or u 2 , or etc., it is also inde-
terminate whether it has the values u\, and u<i, and etc., at the same time; i.e.,
it is indeterminate whether the quantum mechanical object is a particle or a
wave (cf. p. 130).
The name neutral language, however, cannot be applied to the language of
table 1, page 142. This language does not include statements about unobserved
entities, since it calls them meaningless; it is therefore not equivalent to the
exhaustive languages, but only to a part of them. The language of table 2, on
the contrary, is equivalent to these languages to their full extent ; to statements
about unmeasured entities of these languages it coordinates indeterminate
statements.
Constructions of multivalued logics were first given, independently, by E. L.
Post 2 and by J. Lucasiewicz and A. Tarski. 3 Since that time, such logics have
been much discussed, and fields of applications have been sought for; the origi-
nal publications left the question of application open and the writers restricted
themselves to the formal construction of a calculus. The construction of a logic
of probability, in which a continuous scale of truth values is introduced, has
been given by the author. 4 This logic corresponds more to classical physics
than to quantum mechanics. Since, in it, every proposition has a determinate
probability, it has no room for a truth value of indeterminacy; a probability
of i is not what is meant by the category indeterminate of quantum mechan-
ical statements. Probability logic is a generalization of two-valued logic for the
case of a kind of truth possessing a continuous gradation. Quantum mechanics
is interested in such a logic only so far as a generalization of its categories true
and/aZse is intended, which is necessary in this domain in the same sense as in
classical physics; the use of the "sharp" categories true and/afee must be con-
sidered in both cases as an idealization applicable only in the sense of an
2 E. L. Post, ' Introduction to a General Theory of Elementary Propositions/' Am.
Journ. of Math., XLIII (1921), p. 163.
8 J. Lucasiewicz, Comptes rendus Soc. d. Sciences Varsovie, XXIII (1930), Cl. Ill, p.
51; J. Lucasiewicz and A. Tarski, op. cit., p. 1. The first publication by Lucasiewicz of his
ideas was made in the Polish journal Ruch Filozoficzny, V (Lwow, 1920), pp. 169-170.
4 H. Reichenbach, "Wahrscheinlichkeitslogik," Ber. d. Preuss. Akad., Phys,-Math. Kl.
(Berlin, 1932).
148 PART III. INTERPRETATIONS
approximation. The quantum mechanical truth-value indeterminate, however,
represents a topologically different category. The application of a three-valued
logic to quantum mechanics has been frequently envisaged; thus, Paulette
F^vrier 8 has published the outlines of such a logic. The construction which we
shall present here is different, and is determined by the epistemological con-
siderations presented in the preceding sections.
31. The Rules of Two- valued Logic
Before we turn to the presentation of the system of three-valued logic, let us
give a short exposition of the rules of two-valued logic. We shall use the
logistic, or symbolic, form of logic, since only this form is sufficiently precise to
make possible an extension to a three-valued system.
Classical, or two-valued, logic is represented, with respect to its structure,
by truth tables which determine the truth values resulting for propositional
operations as functions of those of the elementary propositions. The most
important of these operations are the following :
a non-a, negation
a V & a or &, or both, disjunction
a . 6 a and &, conjunction
a D 6 a implies 6
a 55 b a equivalent to b
The two-valued truth tables are set forth in tables 3A and 3B.
These tables can be read in two directions : from the elementary propositions
to the propositional combination, or from the combination to the elementary
propositions. For the "or", for instance, the tables tell us by the first direction:
"If a is true and b is true, a V b is true". By the second direction they tell us:
"If a V b is true, a is true and 6 is true, or a is true and b is false, or a is false
and 6 is true." The more T's the column of an operation contains, the weaker
is the operation, since it tells less. Thus the "or" is weaker than the "and"; it
informs us to a less extent about the truth values of a and b than the "and".
A weaker operation is easier to verify than a stronger one, since any of the
T-cases, if observed, will verify it. For the implication of table 3B, which
corresponds only to a certain extent to the implication of conversational lan-
guage, Russell has introduced the name of material implication.
8 Paulette FeVrier, "Les relations d'incertitude de Heisenberg et la logique," Comptes
rendus de VAcad. d. Sciences, T. 204 (Paris, 1937), pp. 481, 958. Cf. also the report given
by L. Rougier, "Les nouvelles logiques de la m^canique quantique," Journ. of Unified
Science, Erkenntnis, Vol. 9 (1939), p. 208. In P. Feeder's paper the third truth value is not
considered as indeterminacy, but as "reinforced falsehood" or absurdity; accordingly,
her truth tables are different from ours. She makes a distinction between "propositions
conjugue*es" and "propositions non conjugues" which is similar to the distinction intro-
duced by M. Strauss. Our objections against the latter conception (p. 144) therefore
apply also to this conception.
31- RULES OF TWO-VALUED LOGIC
149
A logical formula is a combination of propositions which, for every truth
value of the elementary propositions, has the truth value T. Such a formula is
called a tautology. It is necessarily true, since it is true whatever be the truth
values of the elementary propositions. on the other hand, a tautology is empty,
or tells nothing, since it does not inform us at all about the truth values of the
elementary propositions. This property does not make tautologies valueless;
on the contrary, their value consists in their being necessary and empty. Such
formulae can always be added to physical statements, since no empirical
content is added by them; and we must add them if we want to derive conse-
TABLE 3A
a
Negation
a
T
F
F
T
TABLE 3B
Disjunction
Conjunction
Implication
Equivalence
a
b
aV6
a.b
O6
a^b
T
T
T
T
T
T
T
F
T
F
F
F
F
T
T
F
T
F
F
F
F
F
T
T
quences from physical statements. The construction of elaborate tautologies
therefore presents the physicist with a powerful instrument of derivation; the
whole of mathematics must be regarded as an instrument of this kind.
Examples of simple tautologies are given by the formulae :
a ss a rule of identity
a ss a rule of double negation
a V a tertium non datur
a . a rule of contradiction
a . b ss a V 5\ , rT-k TV/T
^V6-a.6J rules of De Morgan
a . (b V c) = a. 6 V a.c 1st distributive rule
a V b . c = (a V b) . (a V c) 2nd distributive rule
(a 3 b) = bDd) rule of contraposition
(0=6) = (a D 6) . (b D a) dissolution of equivalence
a D b ss a V 6 dissolution of implication
(a D &) D d reductio ad absurdum
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
150 PART III. INTERPRETATIONS
All these formulae can easily be verified by case analysis, i.e., by assuming
for a and b successivley all truth values and proving on the basis of the truth
tables that the truth value of the formula is always T. To simplify our nota-
tion we have used the following rule of binding force for the symbols :
Strongest binding force . V 3 = weakest binding force
This rule saves parentheses.
A formula which has sometimes T, sometimes F, in its column is called
synthetic. It states an empirical truth. All physical statements, whether they
be physical laws or statements about physical conditions at a given time, are
synthetic. A formula which has only F's in its column is called a contradiction;
it is always false.
Starting with the given tautologies, we can easily construct rules which allow
us to manipulate logical formulae in a way similar to the methods of mathe-
matics. We shall not enter into a description of this procedure here; it is ex-
plained in textbooks of symbolic logic.
32. The Rules of Three-valued Logic
The method of constructing a three-valued logic is determined by the idea
that the metalanguage of the language considered can be conceived as belong-
ing to a two-valued logic. We thus consider statements of the form "A has the
truth value T" as two-valued statements. The truth tables of three- valued
logic then can be constructed in a way analogous to the construction of the
tables of two- valued logic. The only difference is that in the vertical columns
to the left of the double line we must assume all possible combinations of the
three values T, 7, F.
The number of definable operations is much greater in three-valued tables
than in two-valued ones. The operations defined can be considered as generali-
zations of the operations of two-valued logic; we then, however, shall have
various generalizations of each operation of two-valued logic. We thus shall
obtain various forms of negations, implications, etc. We confine ourselves to
the definition of the operations presented in truth tables 4 A and 4B. 1 As before,
three-valued propositions will be written with capital letters.
The negation is an operation which applies to one proposition; therefore
only one negation exists in two-valued logic. In three-valued logic several
operations applying to one proposition can be constructed. We call all of them
negations because they change the truth value of a proposition. It is expedient
to consider the truth values, in the order T y /, F, as running from the highest
1 Most of these operations have been defined by Post, with the exception of the com-
plete negation, the alternative implication, the quasi implication, and the alternative
equivalence, which we introduce here for quantum mechanical purposes. Post defines
sqme further implications which we do not use. Our standard implication is Post's impli-
cation D with m * 3 and M = 1, i.e., for a three-valued logic and M = t\ =* truth.
32. RULES OF THREE -VALUED LOGIC
value T to the lowest value F. Using this terminology, we may say that the
cyclical negation shifts a truth value to the next lower one, except for the case
of the lowest, which is shifted to the highest value. We therefore read the ex-
pression ~ A in the form next- A. The diametrical negation reverses T and F,
but leaves / unchanged. This corresponds to the function of the arithmetical
TABLE 4A
Cyclical
negation
Diametrical
negation
Complete
negation
A
~A
-A
A
T
I
F
I
I
F
I
T
F
T
T
T
TABLE 4B
A
B
Disjunc-
tion
AVB
Conjunc-
tion
A. B
Standard
implica-
tion
A>B
Alternative
implica-
tion
A-+B
Quasi
implica-
tion
A^-B
Standard
equiva-
lence
A** B
Alternative
equiva-
lence
AmB
T
T
T
T
T
T
T
T
T
T
I
T
I
I
F
I
I
F
T
F
T
F
F
F
F
F
F
I
T.
T
I
T
T
I
I
F
I
I
I
I
T
T
I
T
T
I
F
I
F
I
T
I
I
F
F
T
T
F
T
T
I
F
F
F
I
I
F
T
T
I
I
F
F
F
F
F
T
T
I
T
T
minus sign when the value I is interpreted as the number 0; and we therefore
call the ｅxpression A the negative of A, reading it as minus- A. The complete
negation shifts a truth value to the higher one of the other two. We read A
as non-A. The use of this negation will become clear presently.
Disjunction and conjunction correspond to the homonymous operations of
two-valued logic. The truth value of the disjunction is given by the higher one
of the truth values of the elementary propositions; that of the conjunction, by
the lower one.
There are many ways of constructing implications. We shall use only the
152 PART III. INTERPRETATIONS
three implications defined in table 4B. Our first implication is a three-three
operation, i.e., it leads from three truth values of the elementary propositions
to three truth values of the operation. We call it standard implication. Our
second implication is a three-two operation, since it has only the values T
and F in its column; we therefore call it alternative implication. Our third im-
plication is called quasi implication because it does not satisfy all the require-
ments which are usually made for implications.
What we demand in the first place of an implication is that it makes possible
the procedure of inference, which is represented by the rule: If A is true, and
A implies B is true, then B is true. In symbols :
A
AlB (1)
B
All our three implications satisfy this rule; so will every operation which has a
T in the first line and no T in the second and third line of its truth table. In the
second place, we shall demand that if A is true and B is false, the implication
is falsified; this requires an F in the third line & condition also satisfied by
our implications. These two conditions are equally satisfied by the "and",
and we can indeed replace the implication in (1) by the conjunction. If we do
not consider the "and" as an implication, this is owing to the fact that the
"and" says too much. If the second line of (1) is A.B, the first line can be
dropped, and the inference remains valid. We thus demand that the implica-
tion be so defined that without the first line in (1) the inference does not hold;
this requires that there are some T's in the lines below the third line. This
requirement is satisfied by the first and second implication, though not by
the quasi implication. A further condition for an implication is that a implies a
is always true. Whereas the first and second implication satisfy this condition,
the quasi implication does not. The reason for considering this operation, in
spite of these discrepancies, as some kind of implication will appear later
(cf. 34).
It is usually required that A implies B does not necessarily entail B implies
A, i.e., that the implication is nonsymmetrical. Our three implications fulfill
this requirement. The latter condition distinguishes an implication from an
equivalence (and is also a further distinction from the "and") . The equivalence
is an operation which states equality of truth values of A and jB; it therefore
must have a T 7 in the first, the middle, and the last line. Furthermore, it is
required to be symmetrical in A and B, such that with A equivalent B we also
have B equivalent A. These conditions are satisfied by our two equivalences.
Since these conditions leave the definition of equivalence open within a certain
frame, further equivalences could be defined; we need, however, only the two
given in the tables.
3*. RULES OF THREE-VALUED LOGIC 153
To simplify our notation we use the following rule of binding force for our
" strongest binding force
complete negation
cyclical negation \ lf ~
-,. , . , ,. > equal force
diametrical negation ) ^
conjunction
disjunction V
quasi implication -s-
standard implication D
alternative implication ->
standard equivalence =
alternative equivalence =
weakest binding force
If several negations of the diametrical or cyclical form precede a letter A, we
convene that the one immediately preceding A has the strongest connection
with Aj and so on in the same order. The line of the complete negation ex-
tended over compound ｅxpressions will be used like parentheses.
Our truth values are so defined that only a statement having the truth value
T can be asserted. When we wish to state that a statement has a truth value
other than T, this can be done by means of the negations. Thus the assertion
~~A (2)
states that A is indeterminate. Similarly, either one of the assertions
~A -A (3)
states that A is false.
This use of the negations enables us to eliminate statements in the meta-
language about truth values. Thus the statement of the object language
next-next-A takes the place of the semantical statement "A is indeterminate".
Similarly, the statement of the metalanguage "A is false" is translated into
one of the statements (3) of the object language, and then is pronounced, re-
spectively, "next-A", or "minus-A". We thus can carry through the principle
that what we wish to say is said in a true statement of the object language.
As in two-valued logic, a formula is called tautological if it has only T's in its
column; contradictory, if it has only F's; and synthetic, if it has at least one T
in its column, but also at least one other truth value. Whereas the statements
of two-valued logic divide into these three classes, we have a more compli-
cated division in three-valued logic. The three classes mentioned exist also in
the three-valued logic, but between synthetic and contradictory statements
we have a class of statements which are never true, but not contradictory; they
have only I's and F's in their column, or even only /'s, and may be called
asynthetic statements. The class of synthetic statements subdivides into three
categories. The first consists of statements which can have all three truth
154 PART III. INTERPRETATIONS
values; we shall call them, fully synthetic statements. The second contains state-
ments which can be only true or false; they may be called true-false statements,
or plain-synthetic statements. They are synthetic in the simple sense of two-
valued logic. The use of these statements in quantum mechanics will be indi-
cated on p. 159. The third category contains statements which can be only
true or indeterminate. Of the two properties of the synthetic statements of
two-valued logic, the properties of being sometimes true and sometimes false,
these statements possess only the first; they will therefore be called semi-
synthetic statements.
The cyclical or the diametrical negation of a contradiction is a tautology;
similarly, the complete negation of an asynthetic statement is a tautology. A
synthetic statement cannot be made a tautology simply by the addition of a
negation.
All quantum mechanical statements are synthetic in the sense defined. They
assert something about the physical world. Conversely, if a statement is to
be asserted, it must have at least one value T in its column determined by the
truth tables. Asserting a statement means stating that one of its T-cases
holds. Contradictory and asynthetic statements are therefore unassertable.
On the other hand, tautologies and semisynthetic statements are indisprovable;
they cannot be false. But whereas tautologies must be true, the same does not
follow for semisynthetic statements. When a semisynthetic statement is
asserted, this assertion has therefore a content , i.e., is not empty as in the case
of a tautology. For this reason we include semisynthetic statements in the
synthetic statements; all synthetic statements, and only these, have a content.
The unique position of the truth value T confers to tautologies of the three-
valued logic the same rank which is held by these formulae in two-valued
logic. Such formulae are always true, since they have the value T for every
combination of the truth values of the elementary propositions. As before, the
proof of tautological character can be given by case analysis on the base of
the truth tables; this analysis will include combinations in which the elemen-
tafy propositions have the truth value /. We now shall present some of the
more important tautologies of three-valued logic, following the order used in
the presentation of the two-valued tautologies (1)-(12), 31.
The rule of identity holds, of course:
A ^ A (4)
The rule of double negation holds for the diametrical negation:
A = --A (5)
For the cyclical negation we have a rule of triple negation:
A m A (6)
32. RULES OF THREE-VALUED LOGIC 155
For the complete negation the rule of double negation holds in the form
3 = 2 (7)
It should be noticed that from (7) the formula A = A cannot be deduced,
since it is not permissible to substitute A for A ; and this formula, in fact, is
not a tautology. We shall therefore say that the rule of double negation does
not hold directly. A permissible substitution is given by substituting A for A
in this Way we can increase the number of negation signs in (7) correspond-
ingly oh both sides. This peculiarity of the complete negation is explained by
the fact that a statement above which the line of this negation is drawn is
thus reduced to a semisynthetic statement; further addition of such lines will
make the truth value alternate only between truth and indeterminacy.
Between the cyclical and the complete negation the following relation holds :
A 55 ~A V ~~A (8)
The tertium non datur does not hold for the diametrical negation, since
A V A is synthetic. For the cyclical negation we have a quartum non datur:
A V ~A V ~~A (9)
The last two terms of this formula can be replaced by A, according to (8) ;
we therefore have for the complete negation a formula which we call a pseudo
tertium non datur: A V A (1 ftt
This formula justifies the name "complete negation" and, at the same time,
reveals the reason why we introduce this kind of negation; the relation (8),
which makes (10) possible, may be considered as the definition of the complete
negation. The name which we give to this formula is chosen in order to indicate
that the formula (10) does not have the properties of the tertium non datur of
two-valued logic. The reason is that the complete negation does not have the
properties of an ordinary negation: It does not enable us to infer the_truth
value of A if we know that A is true. This is clear from (8) ; if we know A, we
know only that A is either false or indeterminate. This ambiguity finds a
further ｅxpression in the fact that for the complete negation no converse opera-
tion can be defined, i.e., no operation leading from A to A. Such an operation
is impossible, because its truth tables would coordinate to the value T of A,
sometimes the value / of A y and sometimes the value F of A.
The rule of contradiction holds in the following forms :
ZI (11)
(12)
(13)
156 PART III. INTERPRETATIONS
The rules ofDe Morgan hold only for the diametrical negation :
-4V r (14)
-A.-B (15)
The two distributive rules hold in the same form as in two-valued logic :
A.(BVC)^A.BVA.C (16)
AVB.C= (AVS).(AVC) ' (17)
The rule of contraposition holds in two forms
-AlB = -BlA (18)
A-*B = B-+A (19)
Since for the diametrical negation the rule of double negation (5) holds, (18)
can also be written in the form
AlB=-Bl-A (20)
This follows by substituting -A for A in (18). For (19), however, a similar
form does not exist, since for the complete negation the rule of double negation
does not hold directly.
The dissolution of equivalence holds in its usual form only for the standard
implication in combination with the standard equivalence :
(A sB) s (ADB).(J?DA) (21)
The corresponding relation between alternative implication and alternative
equivalence is of a more complicated kind:
(AmB)s(A^tB). (-A^-B) (22)
By the double arrow implication we mean implications in both directions. This
double implication does not have the character of an equivalence, since it has
values T in its column aside from the first, the middle, and the last line. By
the addition of the second term these T's are eliminated, and the column of the
alternative equivalence is reached. For a double standard implication, and
similarly for two-valued implication, a second term of the form occurring on
the right hand side of (22) is dispensable, because such a term follows from the
first by means of the rule of contraposition (20). For the double alternative
implication this is not the case. This relation states only that B is true if A is
true, and that A is true if B is true; but it states nothing about what happens
when A and B have one of the other truth values. A corresponding addition
is given by the second term on the right hand side of (22).
The dissolution of implication holds for the alternative implication in the
form A^B^ -- (IVB) (23)
3*. RULES OF THREE -VALUED LOGIC 157
The reductio ad absurdum holds in the two forms :
(A DZ) 3 A (24)
(A-*Z)-*Z (25)
Next to the tautologies, those formulae offer a special interest which can
only have two truth values. Among these the true-false statements, or plain-
synthetic statements, are of particular importance. An example is given by
the formula . . .. ( .
f^s y'^'./jL V ^^"^ l ^*~ J?j* J \^ii\j j
which assumes only the truth values T and F when A runs through all three
truth values. The existence of such statements shows that the statements of
three-valued logic contain a subclass of statements which have the two-
valued character of ordinary logic. For the formulae of this subclass the tertium
non datur holds with the diametrical negation. Thus, if D is a true-false
formula, for instance the formula (26), the formula
. . . . DV-Z) (27)
is a tautology. v '
The other two-valued formulae can easily be transformed into true-false
formulae by the following device. An asynthetic formula A, which has the two
values 7 and F in its truth table, can be transformed into the true-false formula
~A. A semisynthetic formula A, which has the two values I and 3T, can be
transformed into the true-false formula ~~A.
We now turn to the formulation of complementarity. We call two statements
complementary^ if they satisfy the relation
AV~A-~~B (28)
The left hand side is true when A is true and when A is false; in both these
cases, therefore, the right hand side must be true. This is the case only if B is
indeterminate. When A is indeterminate the left hand side is indeterminate;
then we have no restriction for the right hand side, according to the definition
of the alternative implication. Therefore (28) can be read: If A is true or false,
B is indeterminate.
Substituting in (8) ~~A for A, and using (6), we derive
~~A = AV~A (29)
We therefore can write (28) also in the form
~~4- B (30)
Applying (19) we see that (30) is tautologically equivalent to
^B + A (31)
Substituting B for A in (29) we can transform (31) into
'B-+~~A (32)
1 58 PART III. INTERPRETATIONS
It follows that (32) is tautologically equivalent to (28) . 2 The condition of
complementarity is therefore symmetrical in A and B\ if A is complementary
to B y then B is complementary to A .
The relation of complementarity, opposing the truth value of indeterminacy
to the two values of truth and falsehood, is a unique feature of three-valued
logic, which has no analogue in two-valued logic. Since this relation deter-
mines a column in the truth tables of A and J5, it can be considered as estab-
lishing a logical operation of complementarity between A and J3, for which
we could introduce a special sign. It appears, however, to be more convenient
to dispense with such a special sign and to express the operation in terms of
other operations, in accordance with the corresponding procedure used for
certain operations of two-valued logic.
The rule of complementarity of quantum mechanics can now be stated as
follows: If u and v are noncommutative entities, then
C/V~t/-> ~~F (33)
Here U is an abbreviation for the statement, "The first entity has the value
u"\ and V for, "The second entity has the value v". Because of the symmetry
of the complementarity relation, (33) can also be written
FV~F-* ~~17 (34)
Furthermore, the two forms (30) and (31) can be used.
With (33) and (34) we have succeeded in formulating the rule of comple-
mentarity in the object language. This rule is therefore stated as a physical
law having the same form as all other physical laws. To show this let us con-
sider as an example the law: If a physical system is closed (statement a), its
energy does not change (statement 6). This law, which belongs to two-valued
logic, is written symbolically : s
a D b (35)
This is a statement of the same type as (33) or (34). It is therefore not neces-
sary to read (33) in the semantical form: "If U is true or false, V is indetermi-
nate." Instead, we can read (33) in the object language: " J7 or next-[7 implies
next-next- V".
The law (33) of complementarity can be extended to prepositional functions.
Our statement U can be written in the functional form
Vl(ei,t) = u (36)
meaning: "The value of the entity e\ at the time t is u". The symbol "
2 This result could not be derived if we were to use the standard implication, instead
of the alternative implication, in (28) and (32).
8 We simplify this example. A complete notation would require the use of propositional
functions.
32. RULES OF THREE-VALUED LOGIC 159
used here is a functor, meaning "the value of ... ", 4 and will be similarly
applied to other entities. Then the law of complementarity can be expressed
in the form:
(u) (v) (t) {[Vl( ei ,t) = u] V ~[Vl(e,fi = u] - [Vl(e^ = v]} (37)
The symbols (u), (v), (f), represent all-operators and are read, as in two-valued
logic : "for all u" . . . "for all F.
The relation of complementarity is not restricted to two entities; it may hold
between three or more entities. Thus the three components of the angular
momentum are noncommutative, i.e., each is complementary to each of the
two others (cf. p. 78). In order to express this relation for the three entities
u,v,w, we add to (33) the two relations:
V v ~ V -* ~~ W W V ~W -+ ~~C7 (38)
Each of these relations can be reversed, as has been shown above. The three
relations (33) and (38) then state that, if one of the three statements is true or
false, the other two are indeterminate.
Since the alternative implication is the major operation in (33) and (34), 5 it
follows that these formulae can only be true or false, but not indeterminate.
The rule of complementarity, although it concerns all three truth values, is
therefore, in itself, a true-false formula. Since the rule is maintained by
quantum mechanics as true, it has the truth of a two-valued synthetic state-
ment. We see that this interpretation of the rule of complementarity, which is
implicitly contained in the usual conception of quantum mechanics, appears
as a logical consequence of our three-valued interpretation. 6
This result shows that the introduction of a third truth value does not make
all statements of quantum mechanics three-valued. As pointed out above, the
frame of three-valued logic is wide enough to include a class of true-false
formulae. When we wish to incorporate all quantum mechanical statements
4 The use of functors in three-valued logic differs from that in two-valued logic in that
the existence of a determinate value designated by the functor can be asserted only when
the statement (36) is true or false, whereas the indeterminacy of (36) includes the inde-
terminacy of a statement about the existence of a value. We shall not give here the
formalization of this rule.
5 I.e., the operation which divides these formulae into two major parts.
6 It can be shown that the true-false character of (28) and (32) is not bound to the par-
ticular form which we gave to the arrow implication, but ensues if the following postulates
are introduced: one, the relation of complementarity is symmetrical in A and B, i.e., (28)
is equivalent to (32); two, if A in (28) is indeterminate, B can have any one of the three
truth values; three, the implication used in (28) is verified if both implicans and implicate
are true, and is falsified if the implicans is true and the implicate is false. We shall only
indicate the proof here. Postulate two requires that the arrow implication have a T in
the three cases where A is indeterminate; postulate three requires a T in the first value
of the column of the arrow implication, and an F in the third. It turns out that with this
result, seven of the nine cases of (28) are determined, and contain only T's and F's. The
missing case T, F of (28) then must be equal to the case F, T, according to the first postu-
late, and is thus determined as F. It then can be shown that in order to furnish this result,
the arrow implication must have an F in the second value from the top. With this, the last
case of (28), the F,F-case, is determined as F. The arrow implication is not fully deter-
mined by the given postulates; its last three values can be arbitrarily chosen.
l6o PART III. INTERPRETATIONS
into three-valued logic, it will be the leading idea to put into the true-false
class those statements which we call quantum mechanical laws. Furthermore,
statements about the form of the ^-function, and therefore about the proba-
bilities of observable numerical values, will appear in this class. only state-
ments about these numerical values themselves have a three-valued character,
determined by table 2.
33. Suppression of Causal Anomalies through
a Three-valued Logic
In the given formulae we have outlined the interpretation of quantum me-
chanics through a three-valued logic. We see that this interpretation satisfies
the desires that can be justifiably expressed with respect to the logical form of
a scientific theory, and at the same time remains within the limitations of
knowledge drawn by the Bohr-Heisenberg interpretation. The term "meaning-
less statement" of the latter interpretation is replaced, in our interpretation,
by the term "indeterminate statement". This has the advantage that such
statements can be incorporated into the object language of physics, and that
they can be combined with other statements by logical operations. Such com-
binations are "without danger", because they cannot be used for the derivation
of undesired consequences.
Thus, the and-combination of two complementary statements can never be
true. This follows in our interpretation, because the formula
[A V ~A -*> ~~B] -> A.B (1)
is a tautology. This is not an equivalence; therefore the condition of comple-
mentarity cannot be replaced by the condition A.B. But the implication (1)
guarantees that two complementary statements cannot be both true. Such a
combination can be false; but only if the statement about the measured entity
is false. Now if a measurement of q has resulted in the value q it it will certainly
be permissible to say that the statement "the value of q is q 2 , and the value of
p is pi", is false. Similarly, it is without danger when we consider the or-
combination, "the value of q is qi or the value of p is pi", as true after a
measurement of q has furnished the value qi. With such a statement nothing is
said about the value of p.
Furthermore, the reductio ad absurdum (24), 32, or (25), 32, cannot
be used for the construction of indirect proofs. If we have proved by means
of the reductio ad absurdum that A is true, we cannot infer that A is false ;
A can also be indeterminate. Similarly, we cannot construct a disjunctive
derivation of a statement C by showing that C is true both when B is true
and when B is false; we then have proved only the relation
BDC (2)
33. SUPPRESSION OF ANOMALIES 1 6 1
Since the implicans need not be true, we cannot generally infer that C must
be true.
This shows clearly the difference between two-valued and three-valued logic.
In two-valued logic a statement c is proved when the relation
&V6Dc ' (3)
has been demonstrated, since here the implicans is a tautology. The analogue
of (3) iji three-valued logic is the relation
BVBlC (4)
which according to (8), 32, is the same as
BV~BV~~B D C (5)
If (5) is demonstrated, C is proved, since here the implicans is a tautology.
But this means that in order to prove C we must prove that C is true in the
three cases that B is true, false, or indeterminate. An analysis of quantum
mechanics shows that such a proof cannot be given if C formulates a causal
anomaly; we then can prove, not (5), but only the relation (2), or a generaliza-
tion of the latter relation which we shall study presently.
For this purpose we must inquire into some properties of disjunctions. Let us
introduce the following notation, which applies both to the two-valued and the
three-valued case.
A disjunction of n terms is called dosed if, in case n 1 terms are false,
the n-th term must be true.
A disjunction is called exclusive if, in case one term is true, all the others
must be false.
A disjunction is called complete if one of its terms must be true; or what is
the same, if the disjunction is true.
For the two-valued case the first two properties are expressed by the follow-
ing relations: _ _
bi = b 2 6s . . . b n
& 2 = ,. &,...&. (6)
b n S5 bl . 62 . . En-1
That a disjunction for which these relations hold, is closed, follows when we
read the equivalence in (6) as an implication from right to left; that it is ex-
clusive follows when we read the equivalence as an implication from left to
right. Now it can easily be shown that if relations (6) hold, the disjunction
6iV6 2 V...V6 (7)
must be true. This result can even be derived if we consider in (6) only the
implications running from right to left. In other words: A two-valued dis-
PART III. INTERPRETATIONS
junction which is closed is also complete, and vice versa. This is the reason
that in two-valued logic the terms "closed" and "complete" need not be dis-
tinguished. Furthermore, it can be shown that one of the relations (6) can be
dispensed with, since it is a consequence of the others.
For the three-valued case a closed and exclusive disjunction is given by the
following relations :
BI+ BZ . BZ . . . "B n
B t ^-B i .-B 3 ...-B n (8)
As before, the closed character of the disjunction follows when we use the
implications from right to left; and the exclusive character follows when we
use the implications from left to right.
We now meet with an important difference from the two-valued case. From
the relations (8) we cannot derive the consequence that the disjunction
B l VB 2 V...VB n (9)
must be true. This disjunction can be indeterminate. This will be the case if
some of the B are indeterminate and the others are false. All that follows from
(8) is that not all J? t - can be false simultaneously; the disjunction (9) therefore
cannot be false. But since it can be indeterminate, we cannot derive that the
disjunction is complete. In three-valued logic we must therefore distinguish
between the two properties closed and complete] a closed disjunction need not
be complete. A further difference from the two-valued case is given by the
fact that the conditions (8) are independent of each other, i.e., that none is
dispensable. This is clear, because, if the last line in (8) is omitted, the remain-
ing conditions would be satisfied if the BI . . . B n -\ are false and B n is inde-
terminate, a solution excluded by the last line of (8).
The disjunction B V B is a special case of a closed and exclusive disjunc-
tion. Corresponding to (2), the proof of the relation
B l V J5 2 V . . . V B n D C (10)
does not represent a proof of C if the B\ . . . B n constitute a closed and exclusive
disjunction, since the implicans can be indeterminate. only a complete dis-
junction in the implicans would lead to a proof of C.
We now shall illustrate, by an example, that the distinction of closed and
complete disjunctions enables us to eliminate certain causal anomalies in
quantum mechanics.
Let us consider the interference experiment of 7 in a generalized form in
which n slits BI . . . B n are used. Let Bi be the statement: "The particle passes
through slit B". After a particle has been observed on the screen we know that
the disjunction BI V -82 V B n is closed and exclusive; namely, we know that
33- SUPPRESSION OF ANOMALIES l6g
if the particle did not go through n 1 of the slits, it went through the n-th
slit, and that if it went through one of the slits, it did not go through the others.
In other words, the observation of a particle on the screen implies that rela-
tions (8) hold. But since from these relations the disjunction (9) is not deriv-
able, we cannot maintain that this disjunction is complete, or true; all we can
say is that it is not false. It can be indeterminate. This will be the case if no
observation of the particle at one of the slits has been made.
The disjunction will also be indeterminate if an observation is made at the
n-th slit with the result that the particle did not go through this slit. Such
observations will, of course, disturb the interference pattern on the screen. But
we are concerned so far only with the question of the truth character of the
disjunction. The fact that, if observations with negative result are made in
less than n 1 slits, the remaining slits will still produce a common inter-
ference pattern finds its ｅxpression in the logical fact that in such a case the
remaining disjunction is indeterminate. only when an observation with nega-
tive result is made at n 1 slits, do we know that the particle went through
the n-th slit; then the disjunction will be true. on the other hand, if the particle
has been observed at one slit, we know that the particle did not go through the
others; the disjunction then is also true.
Although the disjunction (9) may be indeterminate, our knowledge about
the relations holding between the statements BI . . . B n will not be indetermi-
nate, but true or false. This follows because relations (8), whose major opera-
tion is the alternative implication, represent true-false formulae. 1
For the case n = 2, relations (8) are simplified. We then have
Using relations (22), 32, and (5), 32, we can write this in the form:
B, m -B 2 (12)
This means that BI is equivalent to the negative, or the diametrical negation
of -B 2 . This relation, which we shall call a diametrical disjunction, can be con-
sidered as a three-valued generalization of the exclusive "or" of two-valued
logic. In the case of a diametrical disjunction we know that BI is true if B% is
false, that BI is false if B* is true, and that BI is indeterminate if B 2 is indeter-
minate. This expresses precisely the physical situation of such a case. If an
observation at one slit is made, the statement about the passage of the particle
1 As before, the true-false character of these formulae is not introduced by us deliber-
ately, but results from other reasons. If we were to use in (8) a double standard implica-
tion, this would represent a standard equivalence according to (21), 32; then the only
case in which the disjunction (9) is indeterminate is the case that all B are indetermi-
nate. But we need also cases in which some # are false and the others indeterminate, for
the reasons explained above.
164 PART III. INTERPRETATIONS
at the other slit is no longer indeterminate, whether the result of the observa-
tion at the first slit is positive or negative. We see that this particular form of
the disjunction results automatically from the general conditions (8) if we
put n = 2. 2
Now let us regard the bearing of this result on probability relations. Using
the notation introduced in 7, we can determine the probability that a particle
leaving the source A of radiation and passing through slit BI or slit J? 2 or . . . or
slit B n will arrive at C by the formula 3
t ) P(A.B i ,C)
(13)
t-1
This formula is the mathematical ｅxpression of the principle of corpuscular
superposition; it states that the statistical pattern occurring on the screen,
when all slits are open simultaneously, is a superposition of the individual
patterns resulting when only one slit is open. Formula (13), however, can only
be applied when the two statements A and B\ V B 2 V . . . V B n are true. Now A
is true, since it states that the particle came from the source A of radiation.
But we saw that J3i V B 2 V . . . V B n cannot be proved as true. Therefore (13)
is not applicable for the case that all slits are open. The probability holding
for this case must be calculated otherwise, and is not determined by the prin-
ciple of corpuscular superposition. We see that no causal anomaly is derivable.
We can interpret this elimination of the anomaly as given by the impossi-
bility of an inference based on the implication (10) when we regard the C of this
relation as meaning: The probability holding for the particle has the value (13).
The inference then breaks down because our knowledge, formulated by (8),
does not prove the implicans of (10) to be true.
In the example of the grating we applied the relations (8) to a case where the
localization of the particle is given in terms of discrete positions. The same
relations can also be applied to the case of a continuous sequence of possible
positions, such as will result when we make an unprecise determination of
position. We then usually say: The particle is localized inside the interval Ag,
but it is unknown at which point of this interval it is. The latter addition repre-
2 Dr. A. Tarski, to whom I communicated these results, has drawn my attention to
the fact that it is possible also to define a similar generalization of the inclusive "or".
We then replace in the column of the disjunction in table 4B the "I" of the middle row
by a "T", leaving all other cases unchanged. This "almost or", as it may be called, thus
means that at least one of two propositions is true or both are indeterminate. This opera-
tion can be shown to be commutative and associative, while it is not distributive or
reflexive (the latter term meaning that "almost A or A" is not the same as "A"). Applied
to more than two propositions, the "almost or" means: "At least one proposition is true
or at least two are indeterminate". A disjunction in terms of the "almost or", therefore,
represents what we called above a closed disjunction. It can be shown that if relations (8)
hold, the BI . . . B n will constitute a disjunction of this kind. The latter statement, of
course, is not equivalent to the relations (8), but merely a consequence of the latter.
Cf. the author's Wahrscheinlichkeitslehre (Leiden, 1935), (4), 22.
33. SUPPRESSION OF ANOMALIES 1 65
sents the use of an exhaustive interpretation. Within a restrictive interpreta-
tion we shall also say that the particle is localized inside the interval A<j. But
we shall not use the additional statement because we cannot say that the par-
ticle is at a specific point of the interval. Rather, we shall define: The phrase
''inside the interval A#" means that when we divide this interval into n small
intervals dqi . . . dq n adjacent to each other, the relations (8) will hold for the
statements t made in the form "the particle is situated in bq". We see that the
statement "the position of the particle is measured to the exactness Ag" is thus
given an interpretation in terms of three-valued logic. The statement itself is
true or false, since the relations (8) are so. But since we can infer from (8) only
that the disjunction is closed, although it need not be complete, the statement
is not translatable into the assertion "the particle is at one and only one point
of the interval Aq". The fact that the latter consequence cannot be derived
makes it impossible to assert causal anomalies.
In a similar way other anomalies are ruled out. Let us consider, as a further
example, the anomaly connected with potential barriers. A potential barrier
is a potential field so oriented that particles running in a given direction are
slowed down, as the electrons emitted from the filament of a radio tube are
slowed down by a negative potential of the grid. In classical physics a particle
cannot pass a potential barrier unless its kinetic energy is at least equal to the
additional potential energy H which the particle would have acquired in
running up to the maximum of the barrier. In quantum mechanics it can be
shown that particles which, if measured inside the barrier, possess a kinetic
energy smaller than H can later be found with a certain probability outside
the barrier. This result is not only a consequence of the mathematics of quan-
tum mechanics, derivable even in so simple a case as a linear oscillator, but its
validity is, according to Gamow, proved by the rules of radioactive disintegra-
tion. It is important to realize that the paradox cannot be eliminated by a suit-
able assumption about a disturbance through the measurement. Let us consider
a swarm of particles having the same energy H, with H <H . That each of these
particles has this energy can be shown by an energy measurement applied to
each, or by taking fair samples out of a swarm originating in sufficiently
homogeneous conditions. According to the considerations given on page 140,
we must consider the measured value H as the value of the energy after the
measurement. After passing* through the zone of measurement the particles
enter the field of the potential barrier. Beyond the barrier, even at a great dis-
tance from it, measurements of position are made which localize particles at
that place. Because of the distance, the latter measurements cannot have
introduced an additional energy into the particle before it reached the barrier;
this means we cannot assume that the measurement of position pushes the
particle across the barrier, since such an assumption would itself represent a
causal anomaly, an action at a distance. We must rather say that the paradox
constitutes an intrinsic difficulty of the corpuscle interpretation; it is one of
l66 PART III. INTERPRETATIONS
the cases in which the corpuscle interpretation cannot be carried through
without anomalies. The anomaly in this case represents a violation of the
principle of the conservation of energy, since we cannot say that in passing
the barrier the particle possesses a negative kinetic energy. In view of the fact
that the kinetic energy is determined by the square of the velocity, such an
assumption would lead to an imaginary velocity of the particle, a consequence
incompatible with the spatio-temporal nature of particles.
If, however, we use the restrictive interpretation by a three-valued logic,
this causal anomaly cannot be stated. The principle requiring that the sum of
kinetic and potential energy be constant connects simultaneous values of
momentum and position. If one of the two is measured, a statement about the
other entity must be indeterminate, and therefore a statement about the sum
of the two values will also be indeterminate. It follows that the principle of
conservation of energy is eliminated, by the restrictive interpretation, from
the domain of true statements, without being transformed into a false state-
ment; it is an indeterminate statement.
What makes the paradox appear strange is this: It seems that we need not
make a measurement of velocity in order to know that in passing the barrier
the particle violates the principle of energy. If only we know that the velocity
at this point is any real number, zero included, it follows that the principle of
energy is violated. The mistake in this inference originates from the assump-
tion, discarded by the restrictive interpretation, that an unmeasured velocity
must at least have one determinate real number as its value. It is true that we
know the velocity cannot be an imaginary number; but from this we can only
infer that the statement, "the velocity has one real number as its value", is not
false. The statement, however, will be indeterminate if the velocity is not
measured. This is clear when we consider the statement as given by the closed
and exclusive disjunction, "the value of the velocity is vi or v 2 or . . .", 4 which
is indeterminate for the reasons explained in the preceding example.
We see that a three- valued logic is the adequate form of a system of quan-
tum mechanics in which no causal anomalies can be derived.
34. Indeterminacy in the Object Language
We said above that the observational language of quantum mechanics is two-
valued. Although this is valid on the whole, it needs some correction. We shall
see this when we consider a question concerning a test of predictions based on
probabilities, such as raised in 30. For such questions the relation of comple-
mentarity introduces an indeterminacy even into the observational language.
Let us consider the two statements of observational language : "If a measure-
4 It would be more correct to speak here of an existential statement instead of a dis-
junction of an infinite number of terms. It is clear, however, that the considerations given
can be equally carried through for existential statements.
34- INDETERMINACY IN THE OBJECT LANGUAGE 167
ment m q is made, the indicator will show the value qi", and "if a measurement
m p is made, the indicator will show the value p\". We know that it is not pos-
sible to verify both these statements for simultaneous values. The case is
different from the example given in 30 concerning a throw of the die by
Peter or by John; we said that in the latter case the statement about a throw
of Peter's can be verified in principle even if Peter does not throw the die, by
means of physical observations of another kind. For the combination of the
two observational statements about measurements, however, a verification is
not even possible in principle. We must therefore admit that we have in the
observational language complementary statements.
Now the complementary statements of the observational language are not
given by the two statements, "the indicator will show the value qi", and "the
indicator will show the value pi". These statements are both verifiable, since
the indicator will, or will not, show the said value even if the measurement is
not made. It is rather the implications m q implies qi and m p implies pi which
are complementary. We therefore have here in the observational language an
implication which is three-valued and can have the truth-value indeterminate.
What is the nature of this implication? It is certainly not the material
implication of the two-valued truth table (table 3B, p. 149), since this implica-
tion is true when the implicans is false. Thus, the statement
WpDpi (1)
conceived as written in terms of the material implication, will be true if a
measurement m q is made, since, then, the statement m p is false. This difficulty
cannot be eliminated by the attempt to interpret the implication (1) as a
tautological implication or as a nomological implication, i.e., the implication of
physical laws. 1 Although such an interpretation has turned out satisfactory
for other cases in which the material implication appears unreasonable, it
cannot be used with respect to (1), since the implication of this formula carries
no necessity with it.
Now if we try to interpret the implication of (1) by the standard or the alter-
native implication of three-valued logic, the same difficulties as in the case of
the two-valued material implication obtain. Since both m p and pi are two-
valued statements, we can use in the three-valued truth table (table 4B, p. 151)
only those lines which do not contain the value / in the first two columns; but
for these lines both the first and second implication coincide with the material
implication of the two-valued truth table (table 3B). There remains therefore
only the quasi implication, and we find that we must write instead of (1) the
relation f<> .
m p -> pi (2)
This implication has the desired properties, since by canceling all lines con-
1 A complete definition of nomological implication will be given in a later publication
by the author.
i68
PART III. INTERPRETATIONS
taining an 7 in the first two columns of the table, we obtain the implication
noted in table 5. Therefore, (2) corresponds to what we want to say, since
we consider the statement m p implies pi as verified or falsified only if m p is
true, whereas we consider it as indeterminate if m p is false.
This shows that the observational language of quantum mechanics is not
two-valued throughout. Although the elementary statements are two-valued,
this language contains combinations of such statements which are three-
valued, namely, the combinations established by the quasi implication. The
truth table 3B of two-valued logic (p. 149) must therefore be complemented
by the three-valued truth table (table 5) of quasi implication. 2
TABLE 5
a
b
Quasi implication
a-3-6
T
T
T
T
F
F
F
T
I
F
F
I
We see that the three-valued logical structure of quantum mechanics pene-
trates to a small extent even into the observational language. Although the
observational language of quantum mechanics is statistically complete, it is
incomplete with respect to strict determinations. It contains a three-valued
implication. If there were no relation of indeterminacy in the microcosm, this
three-valued implication could be eliminated; the implication in "m q implies
qi" then could be interpreted as a nomological implication which, in principle,
could be verified or falsified. In observational relations of the kind considered,
however, the uncertainty of the microcosm penetrates into the macrocosm. The
same holds for all other arrangements in which an atomic occurrence releases
macrocosmic processes. Such arrangements need not be measurements; they
may consist as well in the lighting of lamps, or the throwing of bombs. The
fact that no strict predictions can be made in microcosmic dimensions thus
leads to a revision of the logical structure of the macrocosm.
2 The quasi implication of the latter table is identical with an operation which has
been introduced by the author, by the use of the same symbol, in the frame of probability
logic (cf. Wahrscheinlichkeitslehre (Leiden, 1935), p. 381, table lie). It can be considered
as the limiting case of a probability implication resulting when only the probabilities 1
and can be assumed. It can also be considered as the individual operation coordinated
to probability implication as a general operation; the probability then is determined by
counting only the T-cases and F-cases of the quasi implication, the /-cases being omitteu.
In this sense the quasi implication has been used, under the name of comma-operation,
or operation of selection, in my paper "Ueber die semantische und die Objectauffassung
von Wahrscheinlichkeitsausdrucken," Journ. of Unified Science, Erkenntnis,VIII (1939),
pp. 61-62.
35' LIMITATION OF MEASURABILITY 169
35. The Limitation of Measurability
Our considerations concerning exhaustive and restrictive interpretations lead
to a revision of the formulation given to the principle of indeterminacy. We
stated this principle as a limitation holding for the measurement of simul-
taneous values of parameters; this corresponds to the form in which the prin-
ciple has been stated by Heisenberg. We now must discuss the question
whether the principle, in this formulation, is based on an exhaustive or a re-
strictive interpretation.
If we start from a general situation s, and consider the two probability distri-
butions d(q) and d(p) belonging to s, these distributions refer to the results of
measurements made in systems of the type s. Therefore, if we apply definition
4, 29, i.e., if we regard the measured values as holding only after the measure-
ment, the obtained values do not mean values existing in the situation s, and
thus the two distributions are not referred to numerical values belonging to
the same situation. The values q and p y to which these distributions refer,
rather pertain, respectively, to the two different situations m q and m p . We
therefore cannot say that the inverse correlation holding for the distributions
d(q) and d(p) states a limitation of simultaneous values; instead, the relation
of uncertainty then must be formulated as a limitation holding for values ob-
tainable in two different situations. It follows that the usual interpretation of
Heisenberg's principle as a limitation holding for the measurement of simul-
taneous values presupposes, not definition 4, 29, but definition 1, 25, since
only the use of this definition enables us to interpret the results of measure-
ments m q and m p as holding before the operation of measurement and thus as
holding for the situation s. Therefore, if Heisenberg's inequality (2), 3, is to
be regarded as a cross-section law limiting the measurability of simultaneous
values of parameters, this interpretation is based on the exhaustive interpre-
tation expressed in definition 1, 25.
Now we saw in 25 that if the latter definition is assumed, we can speak of
an exact ascertainment of simultaneous values when we consider a situation
between two measurements and add the restriction that the situation to which
the obtained combination of values belongs no longer exists at the moment
when the values are known. For such values, therefore, Heisenberg's principle
does not hold. It follows that the principle of uncertainty must be formulated
with a qualification: The principle states a limitation holding for the measura-
bility of simultaneous values existing at the time when we have knowledge of
them. There is no limitation for the measurement of past values; only present
values cannot be measured exactly, but are bound to Heisenberg's inequality
(2), 3. The limitation so qualified is, of course, sufficient to limit the predicta-
bility of future states; for the knowledge of past values cannot be used for
predictions.
170 PART III. INTERPRETATIONS
These considerations show that the conception of Heisenberg's principle as a
limitation of measurability must be incorporated into an exhaustive interpre-
tation. Within a restrictive interpretation we cannot speak of a limitation of
exactness, since then the standard deviations Ag and Ap, used in the inequality
(2), 3, are not referred to the same situation. Within such an interpretation
we must say that, if we regard a situation for which q is known to a smaller or
greater degree of exactness Ag, p is completely unknown for this situation and
cannot even be said to be at least within the interval Ap coordinated to A# by
the Heisenberg inequality. The corresponding statement holds for the reversed
case. By completely unknown we mean here, either the category indeterminate
of our three-valued interpretation, or the category meaningless of the Bohr-
Heisenberg interpretation. We see that for a restrictive interpretation Heisen-
berg's principle, in its usual meaning, must be abandoned.
36. Correlated Systems
In an interesting paper A. Einstein, B. Podolsky, and N. Rosen 1 have attempted
to show that if some apparently plausible assumptions concerning the meaning
of the term "physical reality" are made, complementary entities must have
reality at the same time, although not both their values can be known. This
paper has raised a stimulating controversy about the philosophical interpre-
tation of quantum mechanics. N. Bohr 2 has presented his views on the subject
on the basis of his principle of complementarity with the intention of showing
that the argument of the paper is not conclusive. E. Schrodinger 3 has been
induced to present his own rather skeptical views on the interpretation of the
formalism of quantum mechanics. Other authors have made further contribu-
tions to the discussion.
In the present section we intend to show that the issues of this controversy
can be clearly stated without any metaphysical assumptions, when we use the
conceptions developed in this inquiry; it then is easy to give an answer to the
questions raised.
In their paper Einstein, Podolsky, and Rosen construct a special kind of
physical systems which may be called correlated systems. These are given by
systems which for some time have been in physical interaction, but are sepa-
rated later. They then remain correlated in such a way that the measurement
of an entity u in one system determines the value of an entity v in the other
system, although the latter system is not physically influenced by the act of
measurement. The authors believe that this fact proves an independent reality
1 A. Einstein, B. Podolsky, N. Rosen, "Can Quantum Mechanical Description of
Physical Reality Be Considered Complete?" Phys. Rev. 47 (1935), p. 777.
2 N. Bohr, "Can Quantum Mechanical Description of Physical Reality Be Considered
Complete?" Phys. Rev. 48 (1935), p. 696.
8 E. Schrodinger, "Die gegenwartige Situation in der Quantenmechanik," Natur-
wissenschaften 23 (1935), pp. 807, 823, 844. We shall use the term "correlated systems"
as a translation of Schrodmger's "verschrankte Systeme".
36. CORRELATED SYSTEMS 171
of the entity u considered. This result appears even more plausible by a proof,
given in the paper, stating that an equal correlation holds for entities other
than u in the same systems, including noncommutative entities.
Translating these statements into our terminology, we can interpret the
thesis of the paper as meaning that by means of correlated systems a proof can
be given for the necessity of definition 1, 25, which states that the measured
value holds before and after the measurement. If we refuse to consider the
measured value as holding before the measurement, such as does the Bohr-
Heisenberg interpretation with definition 4, 29, we are led to causal anom-
alies, since, then, a measurement in one system would physically produce the
value of an entity in another system which is physically not affected by the
measuring operations. This is what is claimed to be proved in the paper.
In order to analyze this argument, let us first consider the; mathematical
form in which it is presented. Let us assume two particles which for some time
enter into an interaction; their ^-function will then be a function \l/(q\ . . . g e )
of six coordinates, which include the 3-position coordinates of each particle.
When, after the interaction, the particles separate, the ^-function will be given
by a product of ^-functions of the individual particles (cf. (12), 27). Let
us expand the individual ^-functions in eigen-f unctions <?i of the same entity u]
we then have
t(qi 8e) =
Here the <r ik determine the probability d(u,-,u*) that the value Ui is measured in
the first system and the value Uk is measured in the second system:
d(ui,u k ) =*\<rik\* (2)
We have, of course,
Now let us assume that a measurement of u is made in the first system, and
furnishes the value u\. The subscript 1 is not meant here to denote the first
or "lowest" eigen-value, but the value obtained in the measurement. We then
shall have a new ^-function such that
Since the second system is not involved in the measurement, its part in (1)
remains unchanged; therefore the new ^-function results from (1) simply by
PART III. INTERPRETATIONS
canceling all terms possessing a 0-^ with i ^ 1. Thus, the new ^-function will
have the form
, 2, ^3) ' 2 ff ^<pk(q^ ft, tfe) (5)
with
Z I ** I 2 = Z
A; fc
Let us put
Ti2X2(g4, ft, 2e) = ^ffikpkfa, % 2e) (7)
k
Then (5) assumes the form
iKffi ge) = iWiCgi, ft, gs)x2(g4, , ffe) (8)
This introduction of a new ^-function is sometimes called the reduction of the
wave packet. Now the physical conditions of the systems, and of the measure-
ment of u in the first system, can be so chosen that X2(</4, % <?e) represents
an eigen-function of an entity v. Then (8) represents a situation which is de-
termined both in u and v. This means that the situation depicted by (8) cor-
responds to a situation which would result from measurements of both u and v,
although only a measurement of u has been made. We therefore know: If we
were to measure v in the second system, we would obtain the value u.
We can simplify the consideration by choosing the entity v identical with
the entity u. It can be proved that this is physically possible. This means that
it is possible to construct physical conditions such that, after a measurement
of u in the first system, a ^-function results for which all coefficients (r^ of (1)
vanish except for the value ai 2 . We then have
2 =
(9)
Here the measurement of u in the first system, resulting in ui, has made the
second system definite in u, for the value u^\ i.e., if we were to measure u
in the second system we would obtain the value UQ.
We now see the way the conclusion of Einstein, Podolsky, and Rosen is
introduced: We must assume that the value u^ exists in the second system
before a measurement of u is made in this system; otherwise we are led to the
consequence that a measurement of u in the first system produces, not only the
value u\ in the first system, but also the value t^ in the second system. This
would represent a causal anomaly, an action at a distance, since the measure-
ment in the first system does not physically affect the second system.
With this inference the main thesis of the paper is derived. It then goes on
to show that similar results can be obtained for an entity w which is comple-
36. CORRELATED SYSTEMS 173
mentary to u. Let o> be the eigen-f unctions of w; then it is possible to make,
instead of a measurement of w, a measurement of w in the first system which
results in the production of an eigen-function
We therefore have a free choice, either to measure u in the first system and
thus to make the second system definite in u, or to measure w in the first
system and to make the second system definite in w. This is considered as
further evidence for the assumption that the value of an entity must exist
before the measurement.
In his reply to the Einstein-Podolsky-Rosen paper, Nils Bohr sets forth the
opinion that an assumption of this kind is illegitimate. Within this exposition
he gives a physical interpretation of the formalism developed for correlated
systems, i.e., of the above formulae (1)-(10). Let us consider this illustration
before we turn to a logical analysis of the problem.
Bohr assumes that the systems considered consist of two particles, each of
which passes through a slit in the same diaphragm. If we include measure-
ments concerning the diaphragm, he continues, it is possible to determine
momentum or position of the second particle by measurements of the first
particle after the particles have passed through the slits. For the determina-
tion of the momentum of the second particle, we would have to measure:
1) the momentum of each particle before the particles hit the
diaphragm
2) the momentum of the diaphragm before the particles hit
3) the momentum of the diaphragm after the particles hit
4) the momentum of the first particle after the particle hit the
diaphragm
The momentum of the second particle is then determined by subtracting the
change in the momentum of the first particle from the change in the momen-
tum of the diaphragm, and adding this result to the initial momentum of the
second particle.
In order to determine the position of the second particle, we would measure :
1) the distance between the slits in the diaphragm
2) the position of the first particle immediately after passing
the slit
From the second result we here would infer the position of the diaphragm,
which is determined because the position of the particle tells us the position of
the slit through which it went (we consider here the position of the plane of the
diaphragm as known, and allow only a shifting of the diaphragm within its
plane, caused by the impact of the particles). Since the distance between the
slits is known, the position of the second slit and with this the position of the
second particle in passing, or immediately after passing the slit, is determined.
174 PART III. INTERPRETATIONS
It is interesting to see that Nils Bohr in these derivations uses the very
definition which Einstein, Podolsky, and Rosen want to prove as necessary,
namely, our definition 1, 25. This definition, although not assumed for the
measurements 1 and 2 of our first list, is assumed for 3 and 4, and likewise for
the measurement 2 of our second list. Otherwise, for instance, the difference
between the measurements 2 and 3 of the momentum of the diaphragm could
not be interpreted as equal to the amount of momentum which the diaphragm
has received through the impacts of both particles. If the measurement 3
changes the momentum of the diaphragm, the latter inference could not be
made. Bohr does not mention his use of a definition which considers the meas-
ured value as holding before the measurement. 4 Fortunately, the use of this
definition does not make Bohr's argument contradictory, as can be seen when
we incorporate his answer into an analysis given in terms of our conception.
Using this conception, we shall answer the criticism of Einstein, Podolsky,
and Rosen in a way different from Bohr's consideration. We shall not maintain
that the use of definition 1, 25, is impermissible] we shall say, instead, that
this definition is not necessary. It may be used; thereby the corpuscle interpre-
tation is introduced, and in the case of correlated systems of the kind consid-
ered it is this interpretation which is free from causal anomalies. Thus, even
Bohr uses this exhaustive interpretation which makes his inferences plausible;
he follows here the well-established habit of the physicist of switching over to
the interpretation which is free from anomalies. What can be derived in such
an interpretation must hold for all interpretations; this is the principle at the
base of Bohr's inferences.
It would be wrong, however, to infer that because definition 1, 25, fur-
nishes here an interpretation free from anomalies, this definition must be
chosen. We should be glad if we could identify this view of the problem,
resulting from our inquiries, with Bohr's opinions. The latter do not seem to
us to be stated sufficiently clearly to admit of an unambiguous interpretation;
in particular, we should prefer to disregard Bohr's ideas about the arbitrari-
ness of the separation into subject and object, which do not appear to us
relevant for the logical problems of quantum mechanics. Let us therefore
continue the analysis by the use of our own notation, applying the three-
valued logic developed in 32.
What is proved by the existence of correlated systems is that it is not per-
missible to say that the value of an entity before the measurement is different
from the value resulting in the measurement. Such a statement would lead to
4 once this definition is chosen, the correlated systems can even be used for a measure-
ment of simultaneous values of noncommutative entities. We then measure u in the first
system, and w in the second; then the obtained values m and Wk represent simultaneous
values in the second system. But the possibility of measuring such simultaneous values
has been pointed put already in 25 as being a consequence of definition 1, 25. If only
definition 4, 29, is used, the two measurements on correlated systems would not furnish
simultaneous values.
36. CORRELATED SYSTEMS 175
causal anomalies, since it would involve the consequence that a measurement
on one system produces the value of an entity in a system which is in no phys-
ical interaction with the measuring operation. In the paper of Einstein,
Podolsky, and Rosen, the inference is now made that we must say that the
entity before the measurement has the same value as that found in the
measurement. It is this inference which is invalid.
The inference under consideration would hold only in a two-valued logic;
in a three-valued logic, however, it cannot be made. Let us denote by A the
statement "the value of the entity before the measurement is different from
the value resulting in the measurement' ' ; then what the existence of correlated
systems proves is that, if causal anomalies are to be avoided, the statement
A must hold. This statement A, which states that A is not true, does not
mean, however, that A is false; A can also be indeterminate. The state-
ment, "the value of the entity before the measurement is equal to the
value resulting in the measurement", is to be denoted by A, i.e., by means
of the diametrical negation, since this statement is true when A is false. If we
could infer from the existence of correlated systems the statement -A, the
restrictive interpretation would indeed be shown to be contradictory. But this
is not the case; all that can be inferred is A, and such a statement is compatible
with the restrictive interpretation, since it leaves open the possibility that A
is indeterminate.
We see that the paper of Einstein, Podolsky, and Rosen leads to an impor-
tant clarification of the nature of restrictive interpretations. It is not permis-
sible to understand the restrictive definition 4, 29, as meaning that the value
of the entity before the measurement is different from the results of the
measurement; such a statement leads to the same difficulties as a statement
about this value being equal to the result of the measurement. Every statement
determining the value of the entity before the measurement will lead to causal
anomalies, though these anomalies will appear in different places according
as the determination of the value before the measurement is given. The anom-
alies appearing if equality of the values is stated are described in the inter-
ference experiment of 7; 6 the anomalies resulting if difference of the values
is stated are given in the case of correlated systems.
The given considerations constitute an instructive example for the nature
of interpretations. They show the working of an exhaustive interpretation,
and make it clear that restrictive interpretations are introduced for the pur-
pose of avoiding causal anomalies; they prove, on the other hand, that the
6 This is to be understood as follows. When we put a Geiger counter at the place of each
of the two slits #1 and B 2 we shall always locate the particle either in one or in the other
of these two counters. (This measurement, of course, disturbs the interference pattern on
the screen.) The assumption that the particle was at that place before it hits the counter
leads to the consequence that the particle would have been there also when no measure-
ment was made. This result implies that, when no observation at the slits is made, the
particle will go either through one or the other slit. We showed in 7 that this assump-
tion leads to causal anomalies.
176 PART III. INTERPRETATIONS
restrictive interpretation is consistent if all statements involved are dealt with
by the rules of a three-valued logic.
Within the frame of the restrictive interpretation it is even possible to
express the condition of correlation holding between the two systems after
their interaction has been terminated. Using the functor Vl() introduced on
page 159 and indicating the system I or II within the parentheses of this
symbol we can write :
(w) { [ Vlfal) = u]m[ Vl(e lf II) = f(u) ] } ; (11)
where the function / is known. Similarly we can write for a noncommutative
entity v: (|>) { ( ^^ = y] _ { ^^ = g(v} } } (12)
where the function g is known. Both relations hold so long as no measurement
is made in one of the systems. After a measurement has been made in one of
them only that relation continues to hold which concerns the measured entity.
For instance, if u has been measured in system 7, only (11) continues to hold.
It would be a mistake to infer from (11) or (12) that there exists a determi-
nate value of u, or v, in one of the systems so long as no measurement is made.
This would mean that the ｅxpressions in the brackets must be true or false;
but the equivalence will hold also if these ｅxpressions are indeterminate. In
(11) and (12) we thus have a means of expressing the correlations of the sys-
tems without stating that determinate values of the respective entities exist. 8
This represents an advantage of the interpretation by a three-valued logic over
the Bohr-Heisenberg interpretation. For the latter interpretation the state-
ments (11)-(12) would be meaningless. only the three-valued logic gives us
the means to state the correlation of the systems as a condition holding even
before a measurement is made, and thus to eliminate all causal anomalies.
We need not say that the measurement of u in system / produces the value of
u in the distant system //. The predictability of the value Uz in the system II,
after u has been measured in the system /, appears as a consequence of condi-
tion (11), which in its turn is a consequence of the common history of the sys-
tems.
37. Conclusion
Combining the general considerations of Part I with the mathematical and
logical analysis of Parts II and III, we can summarize the results of our inquiry
as follows. The relation of indeterminacy is a fundamental physical law; it holds
for all possible physical situations and therefore involves a disturbance of the
object by the measurement. Since the relation of indeterminacy makes it
impossible to verify statements about the simultaneous values of comple-
mentary entities, such statements can be introduced only by means of defi-
nitions. The physical world therefore subdivides into the world of phenomena,
6 Cf. fn. 4, p. 159.
37* CONCLUSION 177
which are inferable from observations in a rather simple way and therefore can
be called observable in a wider sense; and the world of interphenomena, which
can be introduced only by an interpolation based on definitions. It turns out
that such a supplementation of the world of phenomena cannot be constructed
free from anomalies. This result is not a consequence of the principle of inde-
terminacy; it must be considered as a second fundamental law of the physical
world, which we call the principle of anomaly. Both these principles are deriv-
able from the basic principles of quantum mechanics.
Instead of speaking of the structure of the physical world, we may consider
the structure of the languages in which this world can be described; such
analysis expresses the structure of the world indirectly, but in a more precise
way. We then distinguish between observational language and quantum me-
chanical language. The first shows practically no anomalies with the exception
of unverifiable implications occurring in some places. Quantum mechanical
language can be formulated in different versions; we use in particular three
versions: the corpuscle language, the wave language, and a neutral language.
All three of these languages concern phenomena and interphenomena, but
each of them shows a characteristic deficiency. Both the corpuscle language
and the wave language show a deficiency so far as they include statements of
causal anomalies, which occur in places not corresponding to each other and
therefore can be transformed away, for every physical problem, by choosing
the suitable one of the two languages. The neutral language is neither a cor-
puscle language nor a wave language, and thus does not include statements
expressing causal anomalies. The deficiency reappears here, however, through
the fact that the neutral language is three-valued; statements about interphe-
nomena obtain the truth-value indeterminate.
The stated deficiencies are not due to an inappropriate choice of these
languages; on the contrary, these three languages represent optima with re-
spect to the class of all possible languages of quantum mechanics. The de-
ficiencies must rather be regarded as the linguistic ｅxpression of the structure
of the atomic world, which thus is recognized as intrinsically different from
the macro-world, and likewise from the atomic world which classical physics
had imagined.
INDEX
(The numbers refer to pages)
action by contact, 29
active interpretation of a transformation,
56
adjoint matrix, 58
anomaly, causal, 26, 28, 43, 160
anomaly, principle of, 33, 44, 117, 129
assumption r, 127
asynthetic statements, 153
Bargmann, V., 92
basic functions of an expansion, 45
basic principles of quantum mechanical
method, 109, 110
binding force, rule of, in two-valued logic,
150; in three-valued logic, 153
Bohr, N., 22, 40, 170, 173
Bohr-Heisenberg interpretation, 22, 139
Boltzmann, L., 1
Born, M., 8, 22, 80, 82
Born's statistical interpretation of the
waves, 36
Broglie, L. de, 6, 8, 21, 32, 66, 80
canonically conjugated parameters, 77, 100
Carnap, R., 136, 141
case analysis, 150
causal anomaly, 26, 28, 43, 160; eliminabil-
ity of, 34
causal chain, 117, 122
chain structure, 123
closed disjunction, 161
commutation rule, 77
commutative operators, 76
commutator, 76
complementarity, principle of, 22
complementary parameters, 77
complementary statements, 157
complete disjunction, 161
complete set of basic functions, 45
complex functions, 45
complex vectors, 54
concentrated distribution, 96
configuration space, 65
conservation of energy, 89, 165
context, physical, 81
context of discovery or of justification, 67
continuous case of an expansion, 49
contradiction, 150, 153, 155
contraposition, rule of, 156
converse of a transformation, 58
corpuscle interpretation, 21, 25, 118
corpuscle language, 146
correlated systems, 170
correlation, inverse, 10
cross-section law, 4, 16
data, observable, 5
Davisson, C. J., 21
definite, 97
definition 1, 118; definition 2, 120; defini-
tion 3, 130; definition 4, 140; definition
5, 141; definition 6, 144
De Morgan, rules of, 156
density function, 51
derivative relations of classical physics,
111; of classical-statistical physics, 112;
of quantum mechanics, 115
descriptions, equivalent, 19
descriptive simplicity, 20
determinism, 1
diametrical disjunction, 163
Dirac, P. A.M., 80
Dirac function, 51, 79
discovery, context of, 67
disjunction, closed, 161; complete, 161; dia-
metrical, 163; exclusive, 161
dispersion, law of, 69
dissolution of equivalence, 156
dissolution of implication, 156
distribution, concentrated, 96; probability,
5,91,111
distributive rule, 156
disturbance by the means of observation,
15, 99, 103, 104
dualism of waves and corpuscles, 68
duality of interpretations, 33, 71
dynamic parameters, 11
eigen-functions, 48, 73
eigen-f unctions (practical), 97
eigen-values, 73
eigen-values, rule of, 82
Einstein, A., 8, 14, 15, 20, 21, 26, 67, 170
eliminability of causal anomalies, 34
D79H
i8o
INDEX
elimination, rule of, 28, 101
enumeration, induction by, 105
equivalence, dissolution of, 156
equivalent descriptions, 19
excluded middle, principle of, 145
exclusive disjunction,.161
exhaustive interpretations, 33, 139
Exner, F., 1
expansion, Fourier, 7, 48
expansion of a function, 45
expectation formula, 85
Feenberg, E., 92, 131
Fourier expansion, 7, 48
Fourier functions, 51
fully synthetic statements, 154
function space, 57
functions, basic, 45; complex, 45
functor, 159
fundamental tone, 7
Gauss function, 6
Germer, L. H., 21
Heisenberg, W., 3, 12, 40, 66, 80
Hermitean operators, 74
heterogeneous case (of an expansion), 49
Hilbert space, 55
holistic transformation, 65
homogeneous case (of an expansion), 49
Huygens, C.,6,21
Huygens, principle of, 26
identity, rule of, 154
implication, dissolution of, 156
implication, material, 2, 148
implication, material, in three-valued
logic, 152, 159
implication, nomological, 167; probability,
2
indeterminacy, principle of, 3, 9, 14, 35, 44,
98; (truth value), 42, 145, 146
induction by enumeration, 105
inductive simplicity, 20
inference, rule of, 152; statistical, 105
inner product, 52
integrable, quadra tically, 46
integral transformation, 65
interference experiment, 24, 162
interference of probabilities, 105
interphenomena, 21
interpretation, active or passive, of a trans-
formation, 56; Bohr-Heisenberg, 22, 139;
corpuscle, 21, 25, 118; duality of, 33, 71;
exhaustive, 33, 139; restrictive, 33, 40,
139; wave, 26, 129
interpretation by a restricted meaning, 141
inverse correlation, 10, 11
Jordan, P., 80
justification, context of, 67
Kemble, E. C., 92
kinematic parameters, 11
Klein, F., 56
Kramers, H. A. ,78, 91
Land6, A.,23, 25
Laue, M. von, 21
law, cross-section, 4, 16
laws, causal or statistical, 1
linear operators, 74
Liouville, theorem of, 113
logic, multivalued, 147; three-valued, 42,
147, 150; two-valued, 42, 148
Lucasiewicz, J., 147
macrocosmic analogy, 39
major operation, 159
material implication, 148
matrix, 48; adjoint, 58; operator, 78
matrix mechanics, 79
matrix multiplication, 59
maximal measurement, 100
meaning, probability, 121; restricted, 42,
141; truth, 121; verifiability theory of, 29
measurability, limitation of, 4, 13, 169
measurement, general definition of, 95;
quantum mechanical definition of, 97;
disturbance by the, 99, 103, 104; maxi-
mal, 100
metalanguage, 141
minimum system, 33
mixture (statistical assemblages), 106
momentum, 5
monochromatic waves, 7
Morris, C. W., 141
multivalued logic, 147
needle radiation, 21
negation, rule of double, 154, 155; rule of
triple, 154
INDEX
Neumann, J. von, 14
neutral language, 146
Newton, I., 21
nomological implication, 167
noncommutative operators, 76
nonconnectable statements, 144
normal system, 19, 20; in the wider sense, 24
normalized functions, 46, 81
object language, 141
observable data, 5
observation, disturbance by the, 15, 104
observational language, 136
operator, 56, 72
operator matrix, 78
operators, commutative, 76; linear and
Hermitean, 74
orthogonal coordinates on the sphere, 34
orthogonal straight-line coordinates, 20
orthogonality of basic functions, 45; of
transformations, 52, 53
overtones, 7
passive interpretation, 56
Pauli,W.,23,100
phenomena, 21
physical context, 81
physical situation, 81, 108
pilot waves, 32
plain-synthetic statements, 154
Planck, M., 8, 21, 67
Planck relation, 12
Podolsky, B., 170
point transformation, 65
Post, E. L., 147, 150
potential barriers, 165
pragmatics, 141
predictability, limitation of, 3, 13
principle of superposition, 87, 130
principles (basic) of quantum mechanical
method, 109, 110
probability, a posteriori determination of,
105; a priori determination of, 106; func-
tion (relative), 122; interference of, 105;
statistically inferred, 105; theoretically
introduced, 106
probability amplitudes, 84
probability chains, 122
probability distributions, 5, 81, 111
probability implication, 2
probability meaning, 29, 121
probability waves, 22
pseudo tertium non datur, 155
^-function, observational determination of
the, 92; rule of the squared, 7, 84
pure case (statistical assemblages), 106
quadrangle of transformations, 62
quadratically integrable, 46, 81
quanta, Planck's theory of, 21
quantum mechanical language, 136
quantum mechanics, derivative relations
of, 115
reductio ad absurdum, 156
reduction of the wave packet, 172
reference class, 95, 107
reflexive (set of basic functions), 47
relative probability function, 122
relativity, theory of, 15, 20
restricted meaning, 42; interpretation by,
141
restrictive interpretation, 33, 40, 139
reversed transformation, 58
Rosen, N., 170
scalar product, 52
Schrodinger, E., 1, 6, 21, 36, 66, 80, 95, 170
Schrodinger's first equation (time-inde-
pendent), 7 1,73
Schrodinger 's second equation (time-de-
pendent), 31, 70, 85
semantics, 141
semisynthetic statements, 154
simultaneity, 29
simultaneous values, 3, 119
singularities in a system of coordinates, 34
situation, physical, 81, 108
spectral decomposition, rule of, 8, 82
spectrum, 7
standard deviation, 12
stationary case, 87
statistical inference, 105
statistical laws, 1
statistically complete, 138
Strauss, M., 143
superposition, principle of, 87, 130
superposition of stationary states, 88
swarm of particles, 133
syntax, 141
synthetic statements, 150, 153; a priori, 24;
fully, 154; plain, 154; semi-, 154
INDEX
Tarski, A., 147, 164
tautology, 149, 153
tertiumnondatur, 142, 145, 149
tertium non datur, pseudo, 155
theory of relativity, 15, 20
three-valued logic, 42, 147, 150
time, direction of, 119
transforming away of anomalies, 34
triangle of transformations, 60
truth meaning, 121
truth tables, two-valued logic, 148; three-
valued logic, 151
truth value, 42, 144, 148
two-canal element, 30
two-valued logic, 42, 148
unique, 46
unitary space, 55
unitary transformation, 55
unobserved objects, 17, 118
verifiability theory of meaning, 29
wave interpretation, 21, 26, 129
wave language, 146
waves, monochromatic, 7
wave number, 67
wave packet, 10; reduction of, 172
Weierstrass symbol, 46
Zilsel, E., 17