Reply to V. A. Fock

K. V. Nikolsky, Moscow

V. A. Fock’s critical remarks on my article “Principles of Quantum Mechanics I*” give me occasion to make certain clarifications concerning the situation that has arisen. It seems to me that this will be of great benefit to Soviet theoretical physics.

In the development of modern theoretical physics, quantum mechanics is of very great importance, having achieved enormous successes in solving various special problems of atomic physics. In the Soviet Union we have theoretical physicists engaged in quantum mechanics. Quantum mechanics is already being taught in higher educational institutions as a separate discipline. But, despite the ten years that have passed since its emergence, quantum mechanics still does not have firmly established principles. On the question of its principles there exist considerable differences of opinion. Thus, it is noted that Niels Bohr and his followers, in particular W. Heisenberg, occupy

one position; Erwin Schrödinger, one of the founders of quantum mechanics, another; Albert Einstein a third, and so on. For what follows it is essential to note the point of view of N. Bohr and W. Heisenberg, who headed the so-called Copenhagen school of quantum mechanics. This is essential because in our Union there exists, as it were, a “branch” of this school, despite the fact that the conception of quantum mechanics developed by N. Bohr is completely incompatible with the progressive trend of theoretical physics, being a consistently pursued idealistic conception, namely a Machist conception. This conception is very stubbornly and consistently defended here by M. P. Bronstein (Leningrad), L. D. Landau (Kharkov), I. E. Tamm (Moscow), and V. A. Fock (Leningrad).* Since these persons do not openly defend the views of N. Bohr that they propagate, but disguise them “as materialism,” it is expedient to set forth briefly the essence of these views.

For a materialistic description of any phenomenon it is characteristic that it—the phenomenon—is objectified in space and time quite independently of any observer of this phenomenon whatsoever. Accordingly, every materialistic physical theory must satisfy this requirement. Quantum mechanics, in its Bohr interpretation, does not satisfy this requirement and therefore (contrary to the assertion of the persons mentioned) cannot be recognized as a definitive physical theory, even in the nonrelativistic domain. The conception of quantum mechanics developed by N. Bohr (and, in our country, by his followers listed above) has the following characteristic features.

In creating quantum mechanics in 1925, W. Heisenberg implemented an entirely definite program, a specific circle of problems that can be posed within the framework of quantum mechanics. Namely, quantum mechanics is knowingly constructed in such a way that in all problems there is always conceived to be present a macroscopic observer standing outside the process. It is precisely as a consequence of such a formulation of all problems of quantum mechanics that Heisenberg’s famous “uncertainty principle” appears, and with it the uncertainty of the phase of the wave function.

For this principle it is extremely essential that every quantum problem be posed in advance in such a form that the interaction between the quantum particle and some macroscopic body associated with the observer is considered. Here is how W. Heisenberg formulates this state of affairs**:

“Whereas in classical theory the mode of observation was inessential for the course of the process, in quantum theory the disturbance with which every observation of atomic phenomena is connected plays a decisive role.” And further, “the circumstance that a part of this disturbance remains fundamentally unknown comes down, according to Bohr, ultimately to the uncertainty that is introduced by the very concept of measurement. Indeed, the experimental description of any space-time processes always presupposes some fixed system (for example, a coordinate system in which the observer is at rest), relative to which all measurements are made. Assuming that this system is ‘fixed,’ we thereby abandon in advance knowledge of its momentum, since the concept ‘fixed’ precisely means that any changes of the momentum of the system in interaction with it must not take place. The fundamentally necessary uncertainty admitted at this point, spread-

* It is asserted, moreover, that there is no other point of view. Thus, for example, in setting forth the Einstein–Bohr discussion in Advances in the Physical Sciences, V. A. Fock kept silent about Einstein’s reply (a note in correction).

** See “Modern Quantum Mechanics,” Nobel speeches of Heisenberg, Schrödinger, Dirac, Heisenberg’s article (Russian translation).

is then further transmitted through the measuring apparatus to atomic processes* (pp. 29–30) (emphasis everywhere mine, K. N.). Further, the space-time localization of the object is regarded as connected with the presence of the subject studying this phenomenon. Accordingly, in every quantum problem there figures a macroscopic observer, because, as W. Heisenberg writes, “the behavior of the observer and of his measuring instrument must therefore be analyzed according to the laws of classical physics, for otherwise there would in general be no physical problem at all” (p. 30). Thus, as is proposed, if this observer never eliminates himself, then, according to Heisenberg: “classical physics comes to an end precisely at the point where it is no longer possible to refrain from taking into account the influence of observation upon the processes under investigation. Quantum mechanics, on the contrary, purchases the possibility of considering atomic processes by means of a partial renunciation of their description in space and time and of their objectification” (ibid., p. 32, emphasis mine, K. N.).

Accordingly, W. Heisenberg also says that “in quantum mechanics there is no question at all of an objective determination of space-time events” (p. 27).

For the conception of quantum mechanics developed by N. Bohr and his followers, the following is still more characteristic. According to N. Bohr, quantum mechanics deals not only with statistical problems, as, for example, A. Einstein thinks, but with individual, elementary processes. (Cf. the note at the end of V. A. Fock’s objection.) Considering quantum mechanics from this point of view, N. Bohr sees in it its distinctive “theoretical complementarity,” based on Heisenberg’s uncertainty principle. According to N. Bohr, every description of physical events, always carried out in accordance with the uncertainty principle, is formulated either by the fact that the coordinates of particles are measured, i.e. their position in space and time, or by the fact that their momenta and energy are determined. According to Heisenberg’s principle, the one excludes the other. N. Bohr sees in Heisenberg’s principle not a statistical relation worked out for a fictitious, average statistical specimen of a quantum particle, but an analysis of a separate individual quantum process of measurement. This point of view leads to very serious consequences. Namely, it turns out that position or momentum not only cannot be measured by the considered processes, but also that these concepts in the corresponding cases are simply meaningless. Namely, according to N. Bohr it follows that if a quantum system is in a state with given energy and momentum (and such will be every isolated quantum system), then the concept of space-time localization is inapplicable to it. (Let us recall that, from this point of view, “space-time localization is connected with the subject observing the phenomenon.”) N. Bohr states directly in his articles that in these cases we must consider that, if our object exists independently of us, then it exists outside space and time. (See, for example, in W. Heisenberg’s book The Physical Principles of Quantum Theory* the instructive table in which there is a column characterizing the essence of the quantum mechanics of an isolated system as a “mathematical scheme outside space and time.”)

With these questions, i.e. with the recognition of things outside space and time, there is directly connected the idea that quantum mechanics is a completed (in the nonrelativistic domain) discipline and that physics is forever bound by the necessity of using classical notions, and that we thereby arrive at a limit to the rational description of nature.

Thus, for example, P. A. M. Dirac says this in the second edition of his course on quantum mechanics (Leningrad, translation edited by M. P. Bronstein). N. Bohr speaks in this connection of the “irrationality” inherent in quantum mechanics. And indeed, the “irrationality” of such a point of view is so great that not a single physicist-experimenter defending the point of view of the Copenhagen school has met me. However

* See the end of V. A. Fock’s objection.

the authority of your “branch” is so great that one is forced to reckon precisely with this point of view and only with it, which is extremely hindering the development of our science. I note, for example, that the teaching of quantum mechanics in Moscow, Leningrad, and Kharkov is determined precisely by these persons, as is likewise admission to press for works on quantum mechanics. The situation, only just outlined, is further complicated by the fact that N. Bohr’s views are presented in a highly disguised form (see in particular the articles by M. P. Bronstein and, for example, L. D. Landau’s article published in the summer of this year in Izvestiia), which leads to extraordinary confusion and the utter disorientation of experimental physicists. A group playing a leading role of theoretical physicists, instead of heading Soviet experimental physics and, by critically reviewing existing theories, creating new, genuinely progressive ones, is engaged in slavish, pitiful copying of views utterly alien to us. Moreover, recently we have seen a whole series of active speeches popularizing and defending this position (especially V. A. Fock’s speech). The sooner Soviet physicists reveal the true meaning of their position, its reactionary character for the present stage of the development of physics, the greater will be the benefit to the cause of creating our materialist theoretical physics. This is the affair of the entire physical community. It is a matter of exchanging opinions. One may boldly assert that we shall have real theorists who will create a genuine materialist theory of atomic phenomena. For, if such a theory can now be created anywhere, then it is only here in the Union. I note, finally, that the situation, only just outlined, already began to be clarified at the March session of the Academy of Sciences of the USSR devoted to problems of physics. However, the authority and position of the “Copenhagen branch” were not shaken, since the argumentation of a number of speakers proceeded from circles not competent in questions of quantum mechanics.

In the article of mine mentioned at the beginning I tried clearly to formulate the contemporary conception of quantum mechanics, indicating, at the same time, the possibility of another, deeper understanding of its foundations. It is not surprising that this attempt can meet only with a sharply negative attitude on the part of the above-mentioned persons, as representatives of a completely consistently pursued conception. V. A. Fock’s remarks have as their aim to make impossible the further development of the statistical conception of quantum mechanics, clearing the way at the same time for his (i.e. Bohr’s) views.

Let us consider the remarks made by V. A. Fock.

1. V. A. Fock asserts that my task is “to construct quantum mechanics wholly on the basis of statistics and to derive the quantum-mechanical apparatus, considering exclusively probabilities and means.” This assertion distorts the content of my article. Such a task is not of interest. In my article I assert the following. Quantum mechanics is a fundamentally statistical theory, which has nothing to do directly with individual quantum phenomena. Otherwise, we would have to adopt N. Bohr’s completely unacceptable point of view, expressed by the “principle of complementarity.” In the quantum principle of superposition of states one usually sees the expression of the possibility of the non-spatio-temporal being of a quantum particle (or, as they say, “a state in which the particle has no definite coordinate or any other quantity”). I consider this incorrect, seeing in the quantum principle of superposition only a judgment about the properties of an average fictitious quantum ensemble of particles. The matter stands, from my point of view, as follows. An analysis is made of the statistical ensemble from a definite process of interaction of quantum particles and a macroscopic body, which is put each time, when the reaction is repeated, into conditions identical only “to within a quantum of action.” On the basis of a series of such individual processes a notion is established of a fictitious average process, and it is precisely such an average fictitious process that is meant in the uncertainty principle, which has no relation to a real, indi-

individual process, if only because the fundamental formula \(\varepsilon=h\nu\) is realized physically only statistically and is meaningless for an individual act, just like any other formula that contains Planck’s constant \(h\). Quantum mechanics is a theory of the properties of such a “mean, fictitious representative” of quantum particles, and moreover only of those of their properties which manifest themselves in interaction with the participation of macroscopic bodies. The theory of an individual process, as well as the problem of “how quantum particles behave when a macroscopic observer is not looking at them,” remain unresolved problems to this day. This is the basic problem of all contemporary theoretical physics, and it is a very great mistake to imagine the matter as though this problem has no bearing on existence, and quantum mechanics is already a completed discipline. Quantum mechanics has not even posed such a problem. It is no more than a very artificial, roundabout path for the solution of certain problems of atomic physics.

The point of view of quantum mechanics as a statistical discipline, incompletely describing atomic processes, developed in my article, is, however, incorrectly attributed to the views of A. Einstein, expressed by him in his recent discussion with N. Bohr on physical reality.

In accordance with what has been said, I single out as a part not depending on individual phenomena the entire part of quantum mechanics contained in § 3, namely the quantum principle of superposition (the formula on p. 846, line 6 from the bottom) and the operator method. However, at the same time it is emphasized that the quantum statistical method is a special quantum method, pursuing the analysis of the atomism of action. This is already expressed by the title of the 3rd paragraph. Its peculiarity is that all problems are posed in an entirely distinctive way, namely every quantum process is necessarily considered in relation to a macroscopic observer. Mathematically this methodological peculiarity is expressed by the indeterminacy of phase and by the quantum principle of superposition. (Since the type of operators is not defined, we are dealing only with a method.) It is from precisely here that all the difficulties arise which occur in the Bohr conception of quantum mechanics. Of course, these difficulties, noted above, are seen only by the physicist-materialist. The idealist Machian school will maintain that everything is “complete” and in the best possible order*.

Thus, having ascribed to me the assertion mentioned, V. A. Fock poses also the natural problem of phases, and then considers the entire work from this, obviously inappropriate, point of view.

1. The question of introducing wave functions. I do not at all assert that the formula \(w_{1*}(k_s)=|C_{1s}(k_s)|^2\) defines wave functions. This is stated further on. It is most expedient to introduce the wave function into the discussion in this way, and only then to formulate their properties by the usual quantum principle of superposition, which I do on p. 546, regarding the latter as a methodological device of quantum mechanics. V. A. Fock notes all this himself, “if one understands the quoted words of K. V. Nikolsky in the sense that he wants to express parametrically by means of auxiliary (wave) functions, then before him there arises the problem of deriving and investigating the properties of these auxiliary functions.” But this problem is equally—

* It is curious to note the defense of N. Bohr’s theory “against materialism,” expressed by M. P. Bronstein in one of his articles in Priroda. The author argues as follows: the uncertainty principle says that we shall never be able to measure the coordinate of an electron, and therefore it is the “worst form of idealism” to recognize the possibility of the existence of an electron in space independently of measurement. The author “does not notice” that the whole point is that the problem is from the very beginning posed in a Machian way: one always speaks only of electron + macroscopic observer. The problem electron + electron, for example, independently of a macroscopic observer carrying out measurement within the framework of the existing method of quantum mechanics, is conceivable only as a statistical problem, i.e. as an incomplete solution of the problem.

is strong due to the independent postulation of Hilbert space and of the entire apparatus of quantum mechanics, i.e., precisely of what he wants to avoid*. It would be strange, in my opinion, if my aim were “to avoid obtaining the apparatus of quantum mechanics,” when this is precisely what is at issue. My task was to understand the meaning of this apparatus. The point is precisely to single out that part of quantum mechanics which does not depend on the special choice of operators, and to make clear that this part is nothing other than a special statistical method, developed in accordance with the distinctive formulation of all problems of quantum mechanics. I consider the “postulation” of Hilbert space quite reasonable, i.e., as the expression of quite definite statistical assertions. Thus, for example, I have shown that the condition of the “Eigenwertsproblem”:

\[ R\psi_\mu = r_\mu \psi_\mu \]

does not “fall from the sky” and is not obtained by “translation into physical language with the help of a special dictionary of purely mathematical relations,” but is simply an expression of the known statistical requirement of zero scattering \(\Delta R = 0\) for a quantity \(R\) having a given value. Thus, step by step, one can trace the statistical meaning of every assertion contained in the mathematical apparatus of quantum mechanics.

After the quotation given above, V. A. Fock nevertheless returns to a well-known misunderstanding and speaks of phases and then of averages, attributing to me the view that from the mathematical expectation \(k\) I derive the whole operator \(K\). I do not do this because there is no need to do it. It is necessary to show that the condition \(R\psi = r\psi\) is equivalent to the condition \(\Delta R = 0\) and can be obtained from it and from the connection between the probabilities formulated earlier. I note that V. A. Fock writes that \(c(k_s,l^*) = \sum_\mu c(k_s,n)\,\varphi(r_\mu,l^*)\) are determined only by the fact that they have a given modulus. This is incorrect, since this formula expresses the connection between the probabilities of different quantities, which follows from comparison of it with the statistical principle of reciprocity:

\[ w_{ks}(l^*) = w_{l^*}(k_s) \]

valid for any \(k\) and \(l\).

Further, the closedness of systems of functions expresses the known statistical requirements of the presence of all possible values of the quantity under consideration. Finally, V. A. Fock notes that “all arguments relating to the calculation of mean square deviations are devoid of any basis.” However, the same function \(\varphi\) enters into the mean, and the proof of the formula

\[ (\Delta A)(\Delta B) \geq \frac{1}{2}\left|\int \varphi\,(AB-BA)\,\varphi\,dt\right| \]

is widely known (see, for example, M. Born, Lectures in Ann. de l’Inst. H. Poincaré, p. 222, or Yu. B. Rumer, Introduction to Wave Mechanics, part I, p. 141). V. A. Fock writes that, if we take as basic propositions the formulae contained in § 3, then they will already contain all the basic hypotheses of quantum mechanics. This is incorrect, because these formulae, of course, contain everything, but only what does not depend on a special type of operators. For me, precisely the analysis of the foundations of quantum mechanics from this point of view was of interest, since it makes it possible to give a rational exposition of the physical content of quantum mechanics—an exposition not based on reasoning by analogy, which only the authors themselves can use and which cannot be recognized as scientific (see, for example, P. A. M. Dirac’s course on quantum mechanics and V. A. Fock’s course on quantum mechanics, which copies it). In fact, what is new and most essential in the whole article is not at all § 3, which contains the known material, but the introduction of canonically conjugate variables, given on pp. 553–554. This, in my opinion, is the most direct and natural way of introducing the concept of a particle into the quantum domain, because the invariance of actions makes it possible to substantiate all quantum mechanics quite independently of classical theory, which also accom-

is explained further. It is essential that, in such a presentation, it becomes perfectly clear that the difference between classical mechanics and quantum mechanics is due only to the atomism of action. At the same time, the presentation shows that notions of the “wave nature” of quantum particles are devoid of any physical content and are extremely harmful, since they only obscure the essence of the matter, disguising by their peculiarity the permanent task of quantum mechanics. They lead to the conception of quantum mechanics as an ordinary physical theory of the classical type, which in fact it is not.

V. A. Fock’s objection contains no other concrete remarks. However, I believe that my work contains a very large number of very serious defects from the mathematical point of view. Moreover, I do not undertake to assert that all the necessary statistical propositions required to substantiate a new understanding of quantum mechanics have been identified in it. Thus, for example, it is necessary to formulate more clearly the special role of the principle of reversibility of the quantum process, i.e., the condition

\[ W_{ls}^{*}(k_s)=W_{ks}(l_s^{*}). \]

This work is no more than the beginning of a program that must be carried out. Despite all its defects, it seemed expedient to me to undertake this work, first, in order to bring clarity into the existing state of affairs and, second, to begin the work of creating a genuinely physical understanding of atomic phenomena, possible only on a materialist basis.

Note added in proof. This article was submitted in the autumn of last year, whereas V. A. Fock’s “Principles of Quantum Mechanics I,” which is under discussion, was submitted more than a year ago. At present it is very outdated, since during the past year I have carried out a detailed analysis of the principles of the statistical conception of quantum mechanics. Along with this, we see also a definitive crystallization of views in the Copenhagen school on the principles of positivism. See the exposition of this point of view in the book by the German physicist P. Jordan, Anschauliche Quantentheorie, Springer, 1936.

It is quite possible that the views of the persons mentioned in the article have also changed.