CRITIQUE OF BOHR’S VIEWS ON QUANTUM MECHANICS

V. A. Fock

In 1948, in the pages of the Swiss journal Dialectica, a discussion took place on fundamental questions of quantum mechanics. Many outstanding foreign physicists took part in this discussion: Bohr, Heisenberg, Einstein, Pauli, and others, whose articles were printed in nos. 7–8 of the aforementioned journal for 1948. The leading article is Bohr’s “On the Notions of Causality and Complementarity.” This article may be regarded not only as an expression of the views of the Copenhagen school proper, but also as an expression of those views on quantum mechanics that prevail among foreign physicists.

These views were developed on the basis of an idealist philosophy—the so-called positivism, incompatible with the Marxist-Leninist philosophy of dialectical materialism. Incorrect philosophical premises, when applied to some concrete field of science, almost inevitably lead to errors within that field itself. It is therefore especially important to analyze whether, in the views of the representatives of the Copenhagen school on quantum mechanics, besides unacceptable general philosophical pronouncements whose incorrectness is obvious to the Soviet reader, there are also specific errors leading to an incorrect interpretation of quantum mechanics itself. Since Bohr’s article mentioned above represents a résumé of these views, formulated by the recognized head of the Copenhagen school, it provides convenient material for such an analysis.

To be precise, we shall have to set forth in detail the content of the aforementioned article by Bohr and to quote from it at sufficient length.

In the introductory part of his article Bohr speaks of the causal mode of describing phenomena in classical physics, and writes: “the theory of relativity, which gave classical physics an unprecedented unity and breadth, made it possible to formulate the principle of causality in its most general form.”

V. A. FOK

On this subject the following may be observed. There is no doubt that, in the field of physics, the theory of relativity has significantly clarified the concept of causality. But the precise formulation of the principle of causality in its most general form is a question of general philosophy, and the solution of this question belongs to the classics of Marxism.

Further, Bohr writes the following:

“A completely new state of affairs in physical science was created, however, by the discovery of the universal quantum of action. This discovery revealed a primary feature of the divisibility of atomic processes, going far beyond the old doctrine of the limited divisibility of matter, which had originally been introduced for the causal explanation of the properties of substance and served as the basis for this explanation. This new feature is not only entirely foreign to the classical theories of mechanics and electromagnetism, but is even incompatible with the very idea of causality.”

Thus Bohr asserts that the existence of the quantum of action is incompatible with the very idea of causality. How does he substantiate this? Let us continue the quotation.

“In fact, if the state of a physical system is given, this, obviously, still does not determine which precisely of the various elementary processes of transition to other states will take place. In taking quantum effects into account, one must essentially operate with the concept of the probability of the various possible transition processes.”

What is said here is correct, but it can in no way serve as an argument in favor of nonobservance of the principle of causality. First of all, causality is not reducible to unambiguous determinacy, but includes the more general concept of regularity. Moreover, in quantum phenomena it is necessary to distinguish between the development in time of the state of a system that is under strictly definite physical conditions, and that abrupt change of state which has to be introduced when there is a sudden and not exactly controllable change in the physical conditions. (Such a sudden change in physical conditions is necessary for the measurement of one or another physical quantity.) It is quite natural that in quantum mechanics the wave equation, which describes with one or another degree of accuracy the state of a system under definite physical conditions, ceases to be applicable when these conditions change abruptly and become not quite definite. When the physical conditions are given, the change of state proceeds according to a definite law, in full accordance with the principle of causality. But when these conditions change abruptly and in an indeterminate manner, it is in no way possible to demand that the change of state be predetermined in advance. Yet this change of state is by no means unsystematic, but is subject to definite probabilistic laws, which also corresponds to the principle of cau-

...certainty. The state of an object at the moment of time immediately preceding the external intervention determines the probability of one or another behavior of the object under this intervention. One may even say that the state itself is characterized by the probability of one or another behavior of the object under all possible potential external actions. We shall return to this question below.

Bohr further points to the presence of corpuscular and wave properties of particles and says that “any determination of Planck’s constant is based on the comparison of two aspects of the phenomenon, to which correspond pictures incompatible within the framework of classical theories.” This latter remark seems to us rather trivial, since from experiments in which quantum effects are not manifested, it is of course impossible to determine Planck’s constant. Bohr, however, after this says:

“Under such circumstances we are faced with the necessity of a radical revision of the very foundations for describing and explaining physical phenomena,” prefacing with these words an exposition of the basic ideas of quantum mechanics as he understands it.

Passing to the exposition of quantum mechanics, Bohr first of all emphasizes “that the description of the experimental arrangement and the registration of observations must always be expressed in ordinary language, using the terminology of classical physics.” In itself this assertion is undoubtedly correct. However, it by no means follows from it that the conclusions Bohr draws follow, namely, that everything else, apart from the description of the experimental arrangement and the registration of observations, has only a symbolic character. We shall return to this point of Bohr’s view below.

Then Bohr says:

“The fact that quantum phenomena cannot be analyzed in the spirit of classical physics entails the impossibility of separating the behavior of atomic objects from their interaction with measuring instruments, which fix the conditions under which the phenomena occur.

In particular, the indivisibility of typical quantum effects is expressed in the fact that any attempt to subdivide the phenomenon requires a change in the experimental arrangement, and this change introduces new sources of uncontrollable interaction between the objects and the measuring instruments.”

In the first sentence it is correctly pointed out that the state of atomic objects and their “behavior,” i.e. the change of state, depend on external conditions, and that therefore it is essentially necessary to fix these external conditions. The thought expressed in the second sentence is also generally correct, although the words about uncontrollable interaction are, strictly speaking, devoid of exact meaning. In fact, what is involved here is interaction in the very act of measur—

tion. But what does “to control” mean? In essence, it means “to measure.” Thus, “uncontrollability” means, in our opinion, simply the impossibility of introducing an additional measurement without violating the conditions of the given measurement. Apparently, Bohr’s words about an uncontrollable interaction represent an attempt to explain, in the language of classical physics, the state of affairs that arises from Heisenberg’s relations as applied to the interaction between object and apparatus. Whatever Bohr may have meant by these words, from our point of view they in any case have no deeper meaning and, still less, should be understood in the sense of some assertion about the unknowability of the process of interaction, and so forth.

Bohr’s assertion that it is impossible to separate the behavior of atomic objects from their interaction with measuring instruments can likewise be accepted only with reservations. As will be seen from what follows, Bohr understands this assertion not in the sense of an interconnection between the object under study and the field surrounding it and external objects, in particular measuring instruments, but in the sense that the atomic object is, as it were, dissolved in the measuring instrument and becomes less real than the measuring instrument.

Moreover, Bohr does not indicate what exactly he means by a measuring instrument: whether he considers that the measuring instrument includes that part in which the object is under fixed external conditions and the phenomenon develops without disturbance, or whether by the instrument he means the device that is brought into action only at the last stage of the experiment and serves to register the result of the measurement.

For example, in the phenomenon of electron diffraction we have fixed external conditions inside the crystal lattice through which the electron passes (the regular arrangement of atoms, the elastic character of the collisions of the electrons with them). To observe the diffraction pattern we may place behind the crystal a photographic plate, a row of counters, or some other instrument capable of measuring the coordinate of the electron. The details of its construction are immaterial; in the case of a photographic plate or a counter we are dealing with an amplifying device in which avalanche-type processes develop. The presence of the crystal, however, is essential for the diffraction phenomenon: if there is no crystal, there will be no diffraction, and if it is present, diffraction will occur, independently of how the diffraction pattern is registered and whether it will be registered at all. For brevity we shall call the part of the instrument in which the phenomenon itself takes place the working part of the instrument, and the part in which it is registered the registering part. In our example the working part is the crystal, and the registering part is the photographic plate or the coun—

CRITIQUE OF BOHR’S VIEWS ON QUANTUM MECHANICS

Besides the working and registering parts, one may speak of a preparatory part: this is the part that fixes the state of the object before it enters the working part. In our example, the preparatory part is the source of the monochromatic electron beam, as well as the diaphragms and other devices placed before the crystal. It must be noted that, since the phenomenon can also occur under natural conditions, the preparatory part, as well as the working part, may exist by themselves, and only the registering part, generally speaking, requires a special device.

The above-mentioned inaccuracy of Bohr consists in the fact that he does not draw a distinction between the various parts of the measuring apparatus and leaves open whether by the measuring apparatus he means only its registering part, or whether he includes in this concept the working part as well. The words about an “uncontrollable interaction” obviously refer to the registering part, whereas the assertion that measuring instruments fix the conditions under which phenomena occur, apparently, can refer only to the working part.

As a result of this inaccuracy (possibly deliberate), the essentially important distinction between the behavior of an object under fixed external conditions and its behavior under conditions that are not fully determined is obscured. The behavior of an object under fixed external conditions is fully determined by the initial state and by the properties of the object itself, and can be described quite precisely in the language of quantum mechanics. The result of the final stage of measurement, connected with the interaction of the object with the registering part of the measuring apparatus, must already be described in the language of classical physics, while the probability of one or another result is calculated by the formulas of quantum mechanics.

From what he has said Bohr draws the following conclusion:

“In such a situation, any attribution to atomic objects of ordinary, conventionally accepted, physical attributes (conventional physical attributes) is connected with ambiguity.”

This somewhat obscure phrase about conventional physical attributes requires deciphering. If what is meant here is the impossibility of ascribing to, for example, an electron under all conditions a definite coordinate or a definite quantity of motion, then one can agree with this, since, according to quantum mechanics, whatever classical models of atomic objects there may be have only limited applicability*. But the phrase is formulated in such a general form that by physical attributes one may understand

* An interpretation of Heisenberg’s uncertainty relations as the principle of the limited applicability of classical models is given in our work (Bulletin of Leningrad University, No. 4, p. 31, 1949).

and such properties of the electron as charge, mass, spin, properties connected with the form of the energy operator and determining the interaction of the electron with other particles and with external fields. All these properties are entirely objective and indubitable, and there can be no question of any conditionality of them. Therefore, if Bohr here had in view the disputing of the presence in atomic objects of objective properties like those enumerated, then one cannot in any way agree with him in this.

Further, Bohr introduces his term “complementarity” (at first in application to observations made with different experimental arrangements) and proceeds to characterize quantum mechanics. Extremely curious is the role that Bohr assigns to quantum mechanics in the description and explanation of atomic phenomena. Instead of directly acknowledging that the subject and purpose of quantum mechanics is a deeper study of the properties of atomic objects, Bohr from the very beginning says: “The proper means for an additional mode of description is the apparatus of quantum mechanics.” Thus the role of quantum mechanics is reduced in advance to the compilation of formal recipes for calculating the probabilities of various results of observations, without any penetration into the essence of the phenomenon. In characterizing the apparatus of quantum mechanics, Bohr in every way emphasizes its supposedly exclusively symbolic character. Bohr says that in this apparatus “the canonical equations of classical mechanics are preserved, but the physical variables are replaced by symbolic operators subject to noncommutative algebra.” It is not clear why the mathematical apparatus of quantum mechanics is more symbolic than the apparatus of any other physical theory. Mathematics always operates with symbols, but since in a physical theory these symbols admit a definite physical interpretation, they cease to be abstract symbols and reflect reality. This is so regardless of the complexity of the mathematical apparatus employed. The Schrödinger equation of quantum mechanics is neither more nor less symbolic than the Hamilton–Jacobi equation of classical mechanics. Bohr, however, emphasizes that classical mechanics deals with physical variables, whereas quantum mechanics deals as if only with symbolic operators. It is difficult to refrain from the thought that Bohr needs such a contrast in order to retain for himself the right to cast doubt on the reality of the existence and properties of the atomic objects with which quantum mechanics operates. The striving to eliminate a direct physical interpretation of the apparatus of quantum mechanics, leaving to it only a symbolic meaning, leads Bohr to absurdity. Thus, he says: “These very symbols do not admit of a visualizable interpretation, as is indicated by the very use of imaginary numbers.” As though imaginary nu-

CRITIQUE OF BOHR’S VIEWS ON QUANTUM MECHANICS

numbers contain something mystical and precludes the possibility of a visualizable interpretation! Yet it is constantly used by classical physics as well, especially by electrical engineering.

Bohr expresses directly his idea that the mathematical apparatus of quantum mechanics is allegedly not connected directly with the properties of atomic objects (the reality of which Bohr nowhere mentions at all), but plays a purely auxiliary role in coordinating the results of measurements. He writes:

“The whole apparatus should be regarded as a means for deriving predictions, of an unambiguous or statistical character, concerning results that can be obtained under specified experimental conditions.”

This idea—that not simply knowledge of the objective properties of real atomic objects, but something purely symbolic, is the basis for predictions—was undoubtedly suggested to Bohr by his idealistic philosophical position. Bohr is not an entirely consistent idealist, since he apparently does nevertheless acknowledge the reality of those phenomena that can be described in the language of classical physics; but his point of view on quantum mechanics is unquestionably idealistic.

Proceeding then to consider Heisenberg’s relations, Bohr points to “mutually exclusive conditions under which one can, without contradiction, use localization in space and time, on the one hand, and the conservation laws of mechanics, on the other.” “Indeed,” says Bohr, “any attempt to localize an atomic object in space and time requires such an experimental arrangement in which there occurs an exchange, in principle uncontrollable, of momentum and energy between the object and the scales and clocks defining the frame of reference. Conversely, no arrangement suitable for controlling the balance of momentum and energy permits an exact description of the phenomenon as a chain of events localized in space and time.”

All this is perfectly true provided that the analysis of the behavior of the atomic object is carried out on the basis of classical mechanics (as Bohr himself says in his next sentence). But in fact only the last stage of measurement (usually connected with the action of some amplifying device) requires such a classical analysis. The preceding behavior of the object, when it was still under fixed external conditions, can be analyzed much more accurately on the basis of quantum-mechanical conceptions. Therefore one cannot agree with Bohr’s assertion that “the uncertainty relations manifestly indicate the limitation of causal consideration.” The uncertainty relations impose limitations on the appli-

ness of classical models, but the principle of causality does not constitute a monopoly of classical physics, and the rejection of classical models does not mean the rejection of causality.

Proceeding to enumerate the positive achievements of quantum mechanics, Bohr seems to forget that on the preceding pages he had assigned to its mathematical apparatus only a purely symbolic significance and had connected it not with the properties of atomic objects, but with the coordination of the results of measurements.

Bohr writes the following:

“Quantum mechanics represents a generalization of classical mechanics, making it possible to take into account the existence of the quantum of action. Its framework is sufficiently broad to include within itself an explanation of the regularities observed in experiment that do not yield to description in the classical manner. We shall point here to the characteristic stability of atoms, the existence of which gave the first impetus to the development of quantum mechanics. Moreover, one may point to the peculiar regularities observed in systems composed of identical particles, for example photons or electrons. These regularities play a decisive role in equilibrium radiation and in the properties of matter. As is known, they are adequately represented by the symmetry properties of wave functions representing the states of complex systems.”

In the quotation cited, Bohr seems to pass over to the materialist point of view—to the point of view of the natural scientist. He speaks of the objective properties of atoms (stability), of objectively existing regularities, and of their representation by the mathematical apparatus of quantum mechanics.

With these statements one may fully agree. At the same time, one cannot fail to see their contradiction with what Bohr said several pages earlier.

But Bohr does not remain long at this materialist point of view. After analyzing the thought experiment that at one time (1933) served as the subject of discussion between Bohr and Einstein, Bohr returns to the question of the subject matter of quantum mechanics and says that in the quantum-mechanical description “we are dealing with a mathematically consistent scheme, which is applicable (within the limits of its domain of applicability) to every measurement process.” Thus, here Bohr again assigns to quantum mechanics the significance of a scheme applied to the coordination of the results of measurement processes. There is no longer any mention of the reflection by quantum mechanics of the objective properties of real atomic objects.

Bohr’s idealist philosophical orientation is manifested especially clearly in the interpretation he recommends for the concept of “phenomenon.” Bohr writes:

“We would strongly insist that the word ‘phenomenon’ be used only in a restricted sense. The word ‘phenomenon’ must refer only to observations made under precisely defined conditions, including an account of the whole experiment as a whole.”

In view of the importance of this definition for characterizing Bohr’s philosophical position, we shall give the corresponding quotation (to which Pauli approvingly refers in his editorial article) also in the original: “As a more appropriate way of expression, one may strongly advocate limitation of the use of the word phenomenon to refer exclusively to observations obtained under specified circumstances, including an account of the whole experiment” (Dialectica, 7/8, 1948, p. 317).

Let us analyze this definition. First, it contains no indication whatsoever of the object that in fact produces the phenomenon. The reference to the object is replaced by a reference to the observations produced. Second, it does not take into account that measurement itself constitutes only the final stage in the setting up of an experiment, while this final stage must be preceded by the preparation of the phenomenon itself, if possible in a pure form, i.e. under precisely defined conditions; the final stage, however, knowingly violates these conditions, as Bohr himself says many times in other places when he speaks of “uncontrollable interaction.” In the last words of Bohr’s definition it is emphasized that the experiment must be considered as a whole, i.e. without subdivision into stages; evidently Bohr connects this with what he said earlier about the indivisibility of atomic processes. Meanwhile, if one refuses to subdivide the experiment into stages, a contradiction will result even in Bohr’s own words: on the one hand, he speaks of precisely defined conditions, while on the other, he repeatedly emphasizes that measurement involves an “uncontrollable interaction” between the object and the instrument.

It is clear that “precisely defined conditions” and “uncontrollable interaction” can relate only to different stages of the experiment.

The necessity of subdividing the experiment into the undisturbed phenomenon and the act of measurement itself is self-evident. For an undisturbed phenomenon may also occur under natural conditions, which we cannot even influence (for example, the radiation of atoms on some star).

Thus, the definition proposed by Bohr does not meet the most basic scientific requirements and is erroneous both from the standpoint of philosophy and from the standpoint of physics. Bohr’s philosophical error is obvious. To please positivist idealist philosophy, he strives to avoid even mentioning

knowledge about the object subject to study in the given experiment; he needs this in order to regard this object only as an auxiliary logical construction necessary for coordinating the readings of instruments. If Bohr does not say this directly, he clears the way for such an interpretation, and this path is then followed by others, more overt adherents of idealistic philosophy.

But here we should like especially to emphasize Bohr’s physical error, which may not be so obvious, but is no less important. For it is precisely by means of an incorrect interpretation of physical facts that foreign idealist physicists try to substantiate their philosophy.

The fundamental physical error of Bohr, which is reflected not only in the definition of “phenomenon” he gives, but also in his understanding of quantum mechanics, and which runs through his whole article, we see in the following.

Bohr ignores the fact that in a physical experiment, besides the initial and final stages (the preparation of the object and the act of measurement proper), there also exists an intermediate stage, when the object prepared in a definite way is in fixed external conditions.

The division of an experiment into stages is not speculative, but quite real. It corresponds (in those cases when the whole experiment takes place under laboratory conditions) to the division of the experimental apparatus into preparing, working, and recording parts. In ignoring the intermediate stage of the experiment, Bohr as it were wishes to divert our attention from the very essence of the phenomenon, from that part of it which can be most deeply studied by the methods of quantum mechanics. For it is precisely to this intermediate stage that the wave equation of quantum mechanics applies, whereas the preparation of the object and the result of the measurement must be described in the language of classical physics. The state of the object in the intermediate stage is described by the wave function. This state is characterized, as we have already said, by the probability of one or another potential behavior of the object under each of the possible external actions. The state gives, as it were, a tuning of the object, which is something quite real, despite the fact that in order to determine this tuning it must be disturbed. The circumstance that the wave function characterizes the behavior of the object not in the classical sense, not in itself, but only in relation to a concrete external action, makes it possible to speak of the non-absolute character of the wave function, in contrast to the absolute character of the fields of classical physics. The wave function represents a new (as compared with classical physics) form of description of the state of an object.

Ignoring the middle stage of the experiment, Bohr wants to reduce it to preparation and measurement, which are described in the language of classical physics with account taken only of the uncertainty relation. Thereby he diminishes the role of quantum mechanics (leaving to it only a certain symbolic meaning) and exaggerates the significance of the uncertainty relation. In reality this relation limits the applicability of classical mechanics, but says nothing about the essence of quantum mechanics.

A substantial part of quantum mechanics, making it possible to penetrate more deeply into the study of the properties of atomic objects, is formulated not by means of the uncertainty relation, but by means of the mathematical apparatus that includes operators in Hilbert space, wave equations, wave functions, and so on. The uncertainty relation for coordinates and momentum is a consequence of the apparatus of quantum mechanics, and only the relation for energy and time, pertaining to the act of measurement, is introduced separately; moreover, here too it must be shown that it does not contradict the Schrödinger equation.

The overestimation of the uncertainty relation leads Bohr to a far-reaching overestimation of his principle of complementarity. Initially, complementarity meant that state of affairs which followed directly from the uncertainty relation: complementarity referred to the uncertainties in coordinate and in momentum (in the sense of their inverse proportionality), and the term “principle of complementarity” was understood as a synonym for Heisenberg’s relations*). Very soon, however, Bohr began to see in his principle of complementarity a certain universal principle limiting not only the possibilities of description in the spirit of classical physics, but also the possibilities of any scientific description, and applicable not only in physics, but also in biology, psychology, sociology, and all the sciences. Such a point of view is also pursued in Bohr’s article under consideration (on the last pages).

In view of the obvious groundlessness of such a point of view, we shall not analyze it here. However, since the term “principle of complementarity” has lost its original meaning and has come to be used as a designation for nonexistent limitations of knowledge and for other incorrect concepts, it is best to abandon it altogether.

In the concluding words of his article Bohr returns to his thought that the idea of complementarity is intended to replace

*) In particular, in our article in the journal Pod znamenem marksizma, No. 1, 1938, p. 149, the term “principle of complementarity” is used exclusively in the sense of Heisenberg’s relations.

with it the former idea of causality, and that quantum mechanics carries out this replacement in the realm of physics.

It seems to us, however, that quantum mechanics proves precisely the opposite. In it have been found new forms of expression of the principle of causality, which, along with the other principles of the philosophy of dialectical materialism, will remain a reliable guide in our striving toward the knowledge of nature.