15.1 Mandelstam and Quantum Mechanics

Operationalism is only one aspect of the philosophy developed by L.I. Mandelstam. This philosophy cannot be characterized without explaining his attitude to one of the most controversial problems of twentieth-century physics, namely to the problem of the interpretation of quantum mechanics. This problem was discussed by many leading physicists of the 1930s (Niels Bohr, W. Pauli, M. Born, W. Heisenberg, A. Einstein, E. Schrödinger, P. Jordan, M. Laue, H. Weyl, et al.). Mandelstam also discussed it.

Indeed, the development of quantum mechanics turned out to be in touch with the sophisticated picture of reality. Classical physical theories, including those which were the core physics of the twentieth century, described reality, nature as it exists without human beings, and regardless of a human being, i.e., described it objectively. Along with the development of quantum mechanics, its interpretation arose, the interpretation which enters into the physical theory of the generalized image of the scientist and experimenter (the image is usually called the “observer”). This interpretation was proposed by Bohr, Heisenberg, Pauli, and many other physicists who have made a decisive contribution to the creation of quantum mechanics in the 1930s. It was named the Copenhagen interpretation in honor of the Institute for Theoretical Physics in Copenhagen, headed by Niels Bohr. Since the Copenhagen interpretation is presented in basic textbooks on quantum mechanics (L.D. Landau and E.M. Lifshitz, D. Bohm, A. Messiah, etc.), it later became known as standard and orthodox.

In 1927, when the Copenhagen interpretation and quantum mechanics itself were at the stage of their early development, Albert Einstein delivered a paper at the fifth Solvay Conference [95]. He called the Copenhagen approach into question and outlined an alternative interpretation of quantum mechanics. This paper marked the beginning of Einstein’s opposition to the Copenhagen interpretation and its criticism. Along it radical criticism arose, this was the program of “hidden variables”: Introduction of “hidden variables” should make quantum mechanics closer to classical physics. In some of his critical statements, Einstein was close to the “hidden variables” program. However, the more rigid criticism was provided by the Viennese philosopher Karl Popper, who became very influential and popular in the postwar years, when he received the status of Reader at the London School of Economics and Political Science. Popper followed Einstein. However, his criticism was more philosophically systematic (see [286]).

L.I. Mandelstam belonged to the soft critics of the Copenhagen Interpretation, who did not formulate an explicit alternative, but tried to penetrate to the foundations of quantum mechanics and made skeptical comments with respect to the orthodox formulations.

In Chap. 8 the Mandelstam-Leontovich article “On the theory of Schrödinger equation” was taken under consideration. Apart from having published this article, Mandelstam did not publish anything on quantum mechanics. His collaborative with I.E. Tamm in the article about the energy–time uncertainty relation was published after L.I. Mandelstam’s death in 1945. Nevertheless, Mandelstam was in touch with quantum mechanics in his lectures and seminars. He did not formulate new problems, and he explained the foundations of quantum theory. As was mentioned in Chap. 6, L.I. Mandelstam wrote Richard von Mises about his interest in quantum mechanics. This was in 1928 when Richard von Mises published his philosophical book “Probability, statistics, and truth” (in Russian translation—“Probability and Statistics”). In this book, Richard von Mises presented his frequency conception of probability and applied this conception by discussing the foundations of physical theories, in particular the foundations of quantum mechanics.

L.I. Mandelstam systematically presented his approach to quantum mechanics in his 1939 lectures delivered at Moscow State University (these lectures had the subtitle “The theory of indirect measurement”). As is said in the Mandelstam biography, “Mandelstam came to the final clearness and clarity in his physical interpretation and understanding of the principal foundations to quantum mechanics” in these lectures [1, Vol. 1, p. 52]. However, Mandelstam went on to work on the foundations of quantum mechanics. He was planning the second part of his course, the part about the mathematical foundations of quantum mechanics. These lectures have never been delivered. We have only two small fragments. One of them, written in 1942–43, was published in [1, Vol. 3]. This fragment can be treated as a result of preparatory work under the article about the energy–time uncertainty relation. Partially in this fragment, Mandelstam’s ideas, which were present in his 1939 lectures, were specified and developed.

In his 1942–43 note “On energy in wave mechanics” (the exact data remains unknown) Mandelstam objected to L. Landau and R. Peierls’ interpretation of the energy–time uncertainty relation [199] (this interpretation was used in [197]).

The second fragment is published as an appendix to Mandelstam’s “Lectures”.

However, the “final clearness and clarity” (that was emphasized in the biography of Mandelstam) has not come with the Mandelstam–Tamm article. This article was taken under criticism by V.A. Fock in his article collaborative with his former graduate student N.S. Krylov published in 1947 [185, 186]. Louis de Broglie wrote that the energy–time uncertainty relation was deduced by Mandelstam and Tamm on the base of controversial assumptions [87, p. 160].

It is important, however, that this article came to philosophical discourse. It is cited in the writings on the history of quantum mechanics.

Not only Mandelstam’s disciples highly appreciated the Mandelstam 1939 lectures which S.M. Rytov wrote down. In A.D. Sakharov’s recollections, there is an episode: by the end of his first visit to Tamm’s office (Tamm was his supervisor) he had received two books and a manuscript. The books were W. Pauli’s books on the theory of relativity and quantum mechanics. The manuscript was Mandelstam’s lectures on quantum mechanics. A.D. Sakharov considered that these lectures are wonderful with respect to their clearness and profundity [306].

15.2 Controversies in the Interpretations of Mandelstam’s Interpretation

The Mandelstam interpretation of quantum mechanics itself was interpreted in different ways. For example, I.E. Tamm, who was close to Mandelstam, did not see any differences between the interpretation, which Mandelstam put forward, and the Copenhagen interpretation. Tamm always spoke in favor of the Copenhagen interpretation [337, 338].Footnote 1 The popular statement belongs to him: “There is not any correct interpretation of quantum mechanics that would differ from the Copenhagen interpretation” [335, p. 193, 339, p. 434].

V.L. Ginzburg, Tamm’s former graduate student, the Nobel Prize winner, too, spoke in the same spirit.

In contrast to Tamm and Ginzburg, M. Jammer, the historian of science, ascribed the Mandelstam interpretation to the type of interpretations which arose in opposition to the Copenhagen interpretation and was declared by D.I. Blokhintsev after World War II [166]. In other words, Jammer considered that the Mandelstam interpretation belongs to the statistical or ensemble interpretations. D.I. Blokhintsev himself wrote that his interpretation went back to the Mandelstam interpretation [51].Footnote 2

V.A. Fock, who was asked to participate in the preparation of Mandelstam’s lectures on quantum mechanics for publication, only corrected a couple of technical mistakes. However, in private conversations he expressed his disagreement with the conception of Mandelstam’s lectures. V.A. Fock, whose interpretation of quantum mechanics was very close to the Copenhagen interpretation (see [119]), considered that the Mandelstam interpretation provided an incorrect answer to the question what the statistical collective in quantum mechanics is (see Fock’s 1951 review of the Fifth Volume of Mandelstam’s “Complete Works” [117] and his subsequent articles on the interpretation of quantum mechanics [118, 119]. Fock highly rated Mandelstam’s book, but he objected to Mandelstam’s ensemble interpretation.

A.D. Sakharov, whose admiration for Mandelstam’s lectures on quantum mechanics was mentioned, distinguished the Mandelstam interpretation from the Copenhagen interpretation as it is presented in the Landau–Lifshitz book.

We come to the question about the logical types of the interpretations of quantum mechanics and to the question what the ensemble interpretation is.

15.3 Definition of the Ensemble Interpretation

To reach historical accuracy, let me formulate an inexact definition of the ensemble interpretations. I call an “ensemble interpretation” any interpretation which places an emphasis on the concept of a statistical collective. It is well known that the Copenhagen (orthodox) interpretation treats quantum mechanics as a theory which in its foundations tracks the behavior of a single physical system (an atom, electron, etc.). If the concept of an ensemble arises under this interpretation, it arises as a logically derivative concept: The concept of ensemble is introduced as the instrumental (empirical) interpretation of the mathematical apparatus being formulated. In addition, the concept of an ensemble appears in quantum statistical mechanics to extend the principles of quantum mechanics to the mixed states which are represented by the density matrix. In turn, the ensemble interpretation connects the essence of quantum mechanics with the concept of a statistical collective.

To make this clearer, let me distinguish between two basic interpretations of any abstract physical theory: an instrumental (empirical) interpretation and an interpretation that contributes “to our understanding of the natural world”. The former interpretation consists of a set of rules which connect the mathematical symbols with brute facts, the latter interpretation constructs a “real description of the physical world” [92, p. 2]. The instrumental interpretation of quantum mechanics is statistical: It was proposed by Born, who considered electron collisions and defined the square of the amplitude of the wave function as the probability of finding the particle at a given point in space (here the present author follows traditional terminology: as shown by L. Wessels [379], what we call Born’s interpretation was really formulated by W. Pauli, who improved Born’s formulation).

Born’s interpretation was generalized by P.A.M. Dirac and von Neumann. However, Born’s interpretation was itself interpreted in two ways. The first was provided by the ensemble treatment of probability. The probability of finding a certain value of the observable (physical magnitude) refers to the fraction of all systems in the ensemble which are characterized by the prescribed value. One can find a refinement of the ensemble probability in Richard von Mises’ definition: probability is the limit of the sequence of relative frequencies with increasing number of trials (1919) (see: Chap. 2, Sect. 2.6). Given this definition, one can test the probability that results from the wave function with the probability that follows from measurement (actually, the latter probability emerges from a finite number of trials).

The second approach to Born’s interpretation of the wave function refers to the probabilities of single events. This interpretation does not allow one to test the probability resulting from the wave function against a measurement. However, quantum mechanics itself guarantees the empirical status of this probability, since it is verified by its ample application.

By distinguishing between the ensemble interpretations of quantum mechanics and its Copenhagen interpretation, one proceeded from their reconstruction of the physical world. In explaining quantum mechanics, the Copenhagen-type interpretations refer to thought experiments with a single physical system, whereas the ensemble interpretations basically use the image of statistical collectives. As a matter of fact, both the Copenhagen and the ensemble interpretations referred to Born’s statistical interpretation of the wave function. Historically, however, the Copenhagen interpretations tended to consider probabilities of single events. Nevertheless, there are several exceptions. For example, D.I. Blokhintsev in his 1944 Copenhagen-oriented “Introduction to Quantum Mechanics” used the ensemble concept of probability. In turn, the proponents of the ensemble approach unanimously use the ensemble probability. It is not surprising: by referring to ensemble probabilities, they show the fundamental importance of ensembles for quantum theory. In this connection, Blokhintsev’ trajectory is remarkable: In his publications after World War II, Blokhintsev already combined the ensemble treatment of probability with the ensemble interpretation of quantum mechanics (he referred to Mandelstam as a predecessor). But the present chapter does not embrace the post World War II discussions.

15.4 The “Real” and Ideal (Gibbsian) Ensembles

To describe the ensemble interpretations of quantum mechanics, a narrower definition should be formulated. For his part, Popper regarded quantum mechanics as a theory of “real ensembles” (to use E.J. Post’s term; in developing Popper’s interpretation Post introduces the concept of the real ensemble as the “phase and direction randomized ensemble” [287, p. 55]). By contrast, the scientists whose ensemble interpretations are described here dealt (with a reservation concerning Mandelstam) with ideal Gibbs ensembles. Popper meant the aggregate of particles which all are in the same state—to produce such an aggregate one needs to fix the macroscopic parameters of the producing device. For example, if in a vacuum tube a hot filament emits a beam of electrons, the temperature, voltage, configuration, etc., of the filament must be specified. The leading proponents of the ensemble interpretation presupposed experimental ensembles producing identical systems in identical quantum states (or if you like, they meant experiments which repeatedly and many times placed the same system in the same quantum state).

The leading proponents of the ensemble interpretation distinguished between two kinds of experimental operations in quantum mechanics: state preparation and measurement. Popper, however, spoke of the preparation of “real ensembles”. His “preparation of state” was the production of an ensemble in a fixed state. The American physicist E. Kemble and the physicist–philosopher H. Margenau spoke of an ensemble of preparations. They meant a set of operations, each of which placed a system in a quantum state [173, 174, 218, 226].

The quotes for “real ensemble” are intentional, since these ensembles are real only in comparison with ideal Gibbsian ensembles. Actually, “real ensembles” are also a consequence of thought experiments. The “real ensemble” must have such a low density that one can treat its elements as independent from each other. Each system in such an ensemble is in its quantum state, but all these states are identical. American physicist collaborating with N. Bohr in 1924 John Slater is one of the first proponents of the ensemble approach. In 1928, he delivered a paper at a symposium on quantum mechanics held under auspices of the American Physical Society [319]. He stressed that quantum mechanics “operates with ensembles” [ibid., p. 453]. “Just as in ordinary statistical mechanics, we must here choose an ensemble,… by considering the sort of statistical distributions actually present in the repetitions of the experiment being performed” (ibidem).

In his recollections he provides the following explanation: “An ensemble… represents a collection of many repetitions of the same experiment, agreeing as concerns the large scale or macroscopic properties which we can control, but taking different values of microscopic properties which are on such a small scale that we cannot experimentally determine or control them. It does not necessarily imply a system with a great many particles in it. The essence of the ensemble is the large number of repetition of the experiment… The probability of finding certain coordinates in certain ranges… means simply the fraction of all systems in the ensemble which lie within the prescribed limits” [321, p. 44].

E.C. Kemble (he was a teacher of Slater) followed Slater but he used more refined terminology. He spoke of an ensemble of pairs “the preparation of state and measurement”. He meant the “Gibbsian assemblage of identical systems so prepared that the past histories of all its members are the same in all details that can affect the future behavior as that of original system” [173, p. 54]. Measurement consisted in a “series of observations on suitably prepared assemblage of completely independent systems each in its own separate box and laboratory” (ibid., p. 55).

The American physicist and philosopher H. Margenau first described measurement on the “real ensemble”: “Numerous observations, or a single collective observation, on a physical assemblage of many similar systems in the same state” [226, p. 352]. Although he did not use the term “Gibbsian ideal ensemble”, he subsequently specified it: “Numerous repeated observations on the same system, state in question being reprepared before each observation” [ibid.]. Margenau emphasized that namely the latter ensembles lay in the foundations of quantum mechanics.

K.V. Nikolsky is the first Soviet physicist who proposed the ensemble approach, describing it in his 1936 article [256]. In the foundations of quantum mechanics, Nikolsky stated the ensemble of “quantum processes”, that is, the “ensemble of experiments with single quantum particles that had initially been set in the certain conditions” [256, pp. 26, 27] or the “ensemble of passages of microparticles through a diffraction device” [ibid., p. 148]. Since Nikolsky, in contrast to Margenau, did not sharply distinguish between preparation of state and measurement, he spoke of two kinds of measurement: the former is measurement that formed (in Kemble terminology, prepared) an ensemble. The latter is measurement in the proper sense, measurement that sorted an ensemble according to the values of a physical magnitude, and provided the measurement of the magnitude. As Nikolsky spoke of the ensembles of quantum processes, he meant measurement in the latter sense: his quantum ensembles were ensembles of measurement operations which issued statistics.

How did Nikolsky define probability? Let α be a magnitude which is measured. As a result of measurement, the original ensemble is divided into a number of subensembles corresponding to values of the magnitude α1, α2… resulting from measurement. The probability that under measurement α = α1 equals the ratio Nα1/N, at N → ∞, where N is a number of measurement operations (elements of the ensemble) and Nα1 is a number of the measurements issuing the value α1 (elements of the subensemble).

In his 1939 Lectures on Quantum Mechanics, L.I. Mandelstam presupposed “real ensembles”. However, in his lectures on the reduction of the wave packet and on the Einstein–Podolsky–Rosen argument, ideal Gibbsian ensembles manifested themselves. Mandelstam also refers to the ideal Gibbsian ensembles in his 1942 manuscript “On energy in wave mechanics” which is historically and logically connected with his article written with I.E. Tamm “The uncertainty energy-time relation in nonrelativistic quantum mechanics” [1, vol. 2, pp. 306–315, 339, 340] (this article was published in Russian in Izvestia AN SSSR, Seria of Physics and in English in Journal of USSR Physics in 1945). In this manuscript, the concept of measurement of energy at a given moment is under discussion. “Let the wave function be ψ(x, t). In order for statistics to make sense, reiteration must be performed, that is, the experiment must be repeated many times, where t is the time elapsed from the beginning of the experiment in each of the experiments. The measurement at a “given moment of time” is the measurement in different experiments, but each time at the same time from the beginning of that experiment” [1, Vol. 3, p. 402].

15.5 One More Distinction: Hidden Variables

E. Post believes that the interpretation of quantum mechanics in terms of the ideal Gibbsian ensembles looks like the “Copenhagen-oriented text” [287, p. 11]. However, the relation of the Gibbsian ensemble interpretation to the Copenhagen approach calls for further explanation. Gibbsian ensembles were not in themselves an antidote against the “hidden variable” spirit. One more distinction should be drawn. Following D. Home and M. Whitaker [160], let us distinguish between minimal ensembles and ensembles of which the elements are characterized by preexisting initial values (the PIV ensembles). The minimal ensemble simply is ensembles of similar physical systems prepared in the same quantum states. No physical properties beyond those which the instrumental interpretation attaches to the ensembles are envisaged; that is, elements of the ensembles are characterized by the probabilities and the means (mathematical expectations) of physical magnitudes (observables) which are intended to be measured. In addition, the PIV ensembles are characterized by their premeasured objective probabilistic structures. The minimal ensembles are described with respect to actual or potential measurement, whereas in the PIV ensemble at all times all physical magnitudes have precise values.

Popper was a “believer in PIVs” [160, p. 280]. The American and Soviet proponents of the ensemble approach tended to hold the minimal ensemble approach. In view of von Neumann’s theorem, they proceeded from the fundamental completeness of quantum mechanics. It is true that this gave Nikolsky pause. In his polemics with V.A. Fock, Nikolsky spoke in favor of Einstein’s 1935 approach, which opened the way to the PIV interpretation [257, p. 558]. However, all our protagonists eventually came to the minimal ensembles. Nikolsky’s personal trajectory provides a good example. In his 1941 book, he refused to acknowledge Einstein’s approach and spoke in favor of the minimal ensembles [259, p. 147]. In this book, the quantum ensembles were treated by him in the strong connection with the process of measurement (the above citations).

In this connection, it is useful to follow how the proponents of the ensemble approach treated the Heisenberg uncertainty relations. M. Jammer distinguished between two ways of the interpretation of the uncertainty relations: a “non-statistical way”, according to which these relations provided the principle of limitations in measurement precision, and a “statistical” way which could be summarized as follows: the product of the standard deviations of two canonically conjugate variables has a lower bound given by h/4 [285, p. 81]. All the proponents of the ensemble interpretation accepted the latter approach. However, to demonstrate the significant importance of the uncertainty relations, Kemble and Mandelstam additionally referred to the thought experiments with a single system. In turn, Nikolsky and Margenau restricted themselves to the “statistical” interpretation of the uncertainty relations. “The scattering of measurements has its roots in a fact more fundamental than the destruction of states by interaction with measuring device, namely in the definition of states peculiar to quantum mechanics” [218, p. 422]. According to Nikolsky, “the formulae which express those which are called the uncertainty relations by the Copenhagen school allow us to quantitatively formulate how quantum ensembles differ from classical ones” [259, p. 65].

By contrast the “believer in PIV” considered the uncertainty relations to be just macroscopic formulas which did not rule out the possibility of exact predictions concerning single particles [285, pp. 218, 223, 229, 234–235]. Here we cite an English version.

In the preceding section, Mandelstam and Tamm’s article on the energy–time uncertainty relation was mentioned. This article also formulates the uncertainty relation in a statistical way. As was mentioned in Sect. 14.1, V.A. Fock and his former graduate student N.S. Krylov criticized Mandelstam and Tamm’s article. Fock and Krylov distinguished between two senses of the energy–time uncertainty relation [185, 186]. They credited N. Bohr with the former sense: this is concerned with a single particle and a single measurement. The latter sense (Mandelstam and Tamm) is concerned with statistics of the measurements. This is the relation between the uncertainty in energy of the ensemble of particles prepared with a given energy, on the one hand, and “the standard time”, that is, the time that it takes for some other ensemble magnitude to change its value over the value of its standard, on the other hand.

In essence, Tamm agreed with Fock and Krylov and acknowledged that his deduction of the energy–time uncertainty relation developed with Mandelstam had a lack of generality [337]. In [339, 340], the old version of this article is published. This remark on the generality is absent in it.

15.6 Soviet Ensemble Interpretations: K.V. Nikolsky

It is interesting that the ensemble interpretations of quantum mechanics were rather popular in the USSR in the 1930s. Konstantin Viacheslavovich Nikolsky, who, like Mandelstam, worked for the Physics Institute (FIAN), was the main propagandist. According to recollections and archival material,Footnote 3 his quantum endeavors were supported by the institute’s director S.I. Vavilov (who became President of the Academy of Sciences after World War II).

The FIAN archives hold the letter of Nikolsky to Mandelstam. However, nothing is known about their contact. Mandelstam’s former graduate student S.M. Rytov recalled that K.V. Nikolsky spoke “reasonable things” (his interview given to the present author).

In common with some foreign proponents, Nikolsky was a “molecular structuralist”: Nikolsky’s major book which summed up his results was “Quantum mechanics of a molecule” (1934) [255]. He also wrote a popular book Photon (1936). In 1935, Nikolsky enthusiastically wrote to his former supervisor V.A. Fock that his article on the foundations of quantum mechanics was accepted by Physics Uspekhy and was about to appear. This article led to a hard polemics [116, 257] that broke up their relationship and which was restored only in the 1960s when Nikolsky, who had just returned from the mental sanatorium, begun to send mathematical puzzles to his former supervisor and friend. Nikolsky asked Fock to help him with the publication of his puzzles.

Nikolsky’s “Quantum mechanics of a molecule” was written in the Copenhagen spirit. But it contained hints as to why he later arrived at the ensemble interpretation. Like many books on quantum mechanics of molecules, it widely used approximations which historically and logically connected with the Bohr–Sommerfeld quantum theory and the prequantum models of molecules, hence sharing in the ontology of particles. However, Nikolsky directly pointed to his preference for particles. This preference proceeded from the role of the potential energy in the quantum theory of molecules. The system of interacting particles is characterized by a complicated structured potential energy. The essence of approximations in this field consists in simplification of the potential energy by neglecting one or another of its components.

Some of the proponents of the ensemble approach held to the particle-wave symmetry in their discussion of the interpretation of quantum theory. Nikolsky spoke in favor of particles. He suggested what amounted to modeling waves by means of particles. “When we take under consideration a particle with a definite energy and momentum, to determine its future behavior, we must invite the totality of its possible motions with the following initial conditions: the definite momentum and arbitrary coordinate. This is just what we call a plane wave” [255, p. 15].

Nikolsky went further. In his 1941 book, Nikolsky treated the particle-wave duality as the duality of a single particle and an ensemble of particles (p. 28). As the wave function must represent the state of an ensemble, the particle-wave duality is resolved in favor of ensembles. “Quantum mechanics has yet not been elaborated as a theory of individual processes… An individual process is treated via the prism of the statistical method” [259, p. 28].

Nikolsky pushed materialism as early as his 1934 book.Footnote 4 He insisted that “all the physical phenomena are processes which are progressing in time” (p. 10). Later he contrasted his point of view to the Copenhagen one, identifying the contrast as one between materialism and idealism. “Heisenberg’s approach”, he wrote, “leads to giving up objective processes progressing in space and time, that is, it leads to an explicitly idealistic conclusion” [258, p. 28]. He also contrasted his point of view to the approach of what he called the “the Soviet branch of the Copenhagen school”, that is, to the conceptions pushed forward by V.A. Fock, L.D. Landau, and M.P. Bronstein [258, p. 557].

Besides Fock and Landau, Nikolsky mentioned Matvey Petrovich Bronstein (1906–1938) who worked for Leningrad Institute of Physics and Technology. In 1937, he was arrested and then executed.

By contrasting scientific objectivity to the Copenhagen school, Nikolsky, however, proceeded more than philosophical materialism. He took ensembles as vehicles of scientific objectivity. Let us follow Nikolsky’s discussion. In his terminology, the quantum particles did not exist independently from the macroscopic bodies which they composed, and they could not be cognized independently from the macroscopic bodies. The quantum of action specified the type of interconnection between the quantum particles and the macroscopic bodies, whose behavior is classical. In particular, the measuring device is a macroscopic body. When a measurement is performed, a microscopic system reacts to the measuring device, but this reaction is inevitably uncertain. “However, it is possible to avoid this uncertainty resulting from the use of classical means in the quantum realm. To do this the problem must be posed in a statistical way. A statistical treatment does not imply an elimination of uncertainty. But it is a method to describe quantum processes as objective reality in spite of the uncertainty” [256, p. 54].

As was mentioned, in his 1941 book, Nikolsky directly referred to Kemble’s “objective states”. By formulating quantum mechanics as a theory of an individual atomic system, Nikolsky claimed, we inevitably come to the conclusion that the “wave function is a notebook of an observer”. The quantum ensembles allow us to restore objectivity [259, p. 150].

In essence in his 1936 article and 1941 book, Nikolsky used Gibbsian ensembles. True, he did not refer to Gibbs. It should, however, be taken into consideration that in parallel to his work on the foundations of quantum mechanics Nikolsky worked hard to translate Gibbs’ Principles of Statistical Mechanics into Russian. Although his translation was published in 1946, judging by the date of the translator’s Preface, the translation was completed and prepared for publication in 1940.

As a materialist, Nikolsky was happy to note that statistical mechanics celebrated atomism (the atomic theory of matter). Nevertheless, in the spirit of Americans, he put emphasis on Gibbs’ idea that “ensembles possess more reality than individual events” [254, p. 8] and he stressed that Gibbs’ method is of much importance for quantum theory.

We have a brief outline of Nikolsky’s biography. He was born in 1905. He graduated from the North Caucasus State University in Nalchik in 1927. In 1927–28, he was a graduate student at this University. In 1929–30, he worked at the theoretical department of the State Optics Institute (Leningrad). In 1930–34, he prepared his Doctor Science Dissertation at the Institute of Mathematics and Mechanics of Leningrad University (his scientific supervisor was V.A. Fock). The extended text of this dissertation was published as a book (“Quantum mechanics of a molecule”). Since 1936, he had worked at the Lebedev Physics Institute of the Academy of Sciences in Moscow. In 1946, he was arrested for his anti-Soviet statements. However, the judge concluded that his statements resulted from his mental disease. He was treated in the psychiatric sanatorium and then he was under guardianship of his sister and (after her death) the psychiatrist Dr. Beniash. Nikolsky died in 1979.

15.7 The Prerequisites of the Mandelstam Interpretation

As a matter of fact, Mandelstam’s interpretation of quantum mechanics has been characterized in the previous sections. Its prerequisites  can be summarized in the following three statements. (1) Richard von Mises’ empirical frequency conception of probability, called by some of Mandelstam colleagues an “objective” conception; (2) operationalism: as was shown in Chap. 15, in his lectures on quantum theory and in some other his courses Mandelstam developed his original operationalism, which can be traced back to Mach’s philosophyFootnote 5; (3) the statistical ensemble treatment of physical experiment and measurement—Mandelstam developed the “oscillatory ideology” which presupposed a transformation of the theory of oscillations into the universal language of physics (and perhaps science) and emphasized regularly repeatable phenomena, ensembles of phenomena.

In contrast to Nikolsky, Mandelstam was strongly attracted by the wave ontology. His favorite way of explanation was to appeal to the “wave notions” (the wave packet, modulation, etc.). In fact, his article written with Tamm 1945, where the uncertainty relation between energy and time was derived with the benefit of the Schrödinger equation, repleted with the wave ontology. Mandelstam and Tamm started their section concerning “examples” by considering the wave packet for which the center of gravity, the width ΔR, and the time of travel ΔT were fixed. In their opinion, the uncertainty relation showed that the localization precision of the time of travel of the wave packet through a certain space point is directly dependent on the dispersion of the full energy and cannot be large at a small value of the latter.

V.A. Fock in his article written with N.S. Krylov criticizing Mandelstam and Tamm’s paper (see Sect. 14.4) showed that this example could be translated into the particle language. According to Krylov and Fock, the wave packet presented the statistics of the measurements on the ensemble of particles prepared with an average value of energy.

For the proponents of the ensemble approach, objectivity was associated with the description of regularly repeatable experiments and measurements. By contrast, the Copenhagen scientists, who proclaimed the reduction of the wave packet resulting from a single act of observation, attached much importance to a single experimental event.

It is natural to state that Slater, Kemble, Nikolsky, and Mandelstam adopted the classical culture of macroscopic experimentation which results in statistics (see [124]). By contrast, the Copenhagen physicists who pushed the conception of the wave packet reduction were theoreticians who presumably conducted thought experiments. Nevertheless, let me suggest a material counterpart of their thought experiments with single particles. By analyzing the experimentation culture of the first half of the twentieth century, P. Galison distinguishes between two traditions which he conventionally calls “logic” and “image” traditions [125]. In the logic tradition, the classical culture of macroscopic experimentation has been continued. This is a tradition to “sacrifice the details of one for the stability of many” [125, p. 20]. The image tradition has a presumption that “a single picture can serve as evidence for a new entity or effect” [ibid.]. This tradition is provided by the invention and usage of the devices that worked like an eye. Let me cite an explanation of the quantum measurement which runs in the style of the image tradition: “In general between the observer and the quantum object there is a so-called classical device which under the action of the measured quantum object irreversibly changes its own state in a manner that the observer can directly comprehend. Examples of such classical measuring devices are the photo emulsion of photographic plate, the supersaturated steam in a Wilson cloud chamber… There is even an example in which the device was a human eye: Pavel Cherenkov discovery of Cherenkov radiation in which he detected individual photons directly with his eyes” [58, p. 39].

15.8 B.M. Hessen and A.A. Andronov

Nikolsky did not influence Hessen as Nikolsky started to publish his articles on the foundations on quantum mechanics in 1936. Hessen was arrested and executed in 1936. Hessen’s approach to quantum mechanics was formed under the influence of Mandelstam and his graduate students.

B.M. Hessen touched upon quantum mechanics in the course of his discussion of the concept of probability [152, 154]. He wrote in favor of the ensemble interpretation by arguing in the Mandelstam school manner. In particular, he [147] referred to the principle of expedient idealization typical for Mandelstam’s disciples (see Chap. 13, Sect. 13.6). Hessen wrote that we use a causal idealization, when we treat the “macroscopic world”. We consider the space-time trajectories of macroscopic bodies then. With respect to microscopic particles, we concentrate on the ensembles. We put aside the causal behavior of particles.

In 1934, Mandelstam’s former student A.A. Andronov (see Chaps. 8–10)  delivered a course on quantum mechanics at Gorkii University, where he started to work by moving from Moscow in 1931. Andronov’s lectures characterized the “air” in which the fundamental problems of quantum mechanics were discussed in Mandelstam’s community (see also Chap. 10, Sect. 10.5). Andronov delivered a regular course dedicated to the theory of Schrödinger equation (at the present author’s disposal  there is a copy of the notes written Andronov’s graduate student A.G. Liubina). However, like Hessen, Andronov expressed his dissatisfaction with Copenhagen position. Probably, his dissatisfaction resulted  from his philosophical position. Andronov’s book written in coauthorship with his wife Andronova-Leontovich [21] shows that he sympathized with materialism as a philosophical position.

M. Jammer in his 1974 celebrated book discussed the Soviet ensemble interpretations of quantum mechanics in a special section entitled “Ideological reasons” [166].Footnote 6 In the article [277], the present author compared American and Soviet interpretations which arose before World War II. He showed that, contrary to Jammer, the Soviet physicists shared with their American colleagues not only scientific problems, theories, and ailments, but also the philosophical backgrounds, that is, philosophical problems, theories, and diseases. The relation between American and Soviet ensemble interpretations can be understood as a mutual self-elucidation: By comparing these interpretations, we understand them better.