The early statistical interpretations of quantum mechanics: H.Margenau, K.Popper, E.C.Kemble, K.V.Nikol’skii, and L.I.Mandelstam.



This article is about the first attempts (before War II) to develop a consistent alternative to the Copenhagen interpretation of quantum mechanics. It reconstructs the statistical (ensemble) interpretations of quantum mechanics proposed by the Viennese philosophers K.Popper, the American scientists H.Margenau and E.C.Kemble, and the Soviet physicists K.V.Nikol’skii and L.I.Mandelstam. The author subsequently considers their statistical treatment of the main interpretative problems and he emphasizes a remarkable similarity between the statements which arose in different scientific, philosophical, and even political contexts. He extends his comparative analysis to scientific and philosophical traditions which laid behind the statistical interpretation and its versions.


This article is about the first attempts (before War II) to develop a consistent interpretation of quantum mechanics, the interpretation which would be an alternative to the Copenhagen interpretation elaborated by the founders of this theory (N.Bohr, W.Heisenberg, M.Born, W.Pauli., P.A.M.Dirac) and supported by some prominent physicists of different generations (A.Sommerfeld, W.Heitler, L.D.Landau, R.Peirls, That interpretation used to be called the statistical interpretation or, following Einstein, the pure statistical one, emphasizing that it differed from the Copenhagen interpretation which was statistical, too, but deviated from the statistical point of view in the way of introducing the concept of state (according to the Copenhagen faction, the fundamental quantum state is a state of a single system). In recent times the name "ensemble interpretation" (or "statistical ensemble interpretation") has become more popular, since it clearer expresses the position of the statistical interpretation in the modern network of the interpretation of quantum mechanics. A number of interpretations which preserve a statistical point of view arose, and using the term "ensemble interpretation" one emphasizes that he does not go to the many worlds and many minds metaphysics, does not speculate about propensities, and his "metaphysics" (like those who proposed the statistical interpretation before War II) is restricted to the common physical concept of an ensemble.

The statistical interpretations (since now I shall speak on the statistical interpretation in the plural) arose in the thirties years in different regions of the World. In Wien the young philosopher and physics school teacher Karl Popper, in the USA the young philosopher and physicist (who spent his postdoc fellowship at Sommerfeld Institution) Henry Margenau, the young and prominent physicist contributed to quantum theory John Slater at Harvard, and his older colleague E.C.Kemble formulated their versions of the statistical interpretation. The statistical interpretation was rather popular in the USSR: The young physicist K.V.Nikol’skii rose with such an interpretation against his supervisor V.A.Fock, whose interpretation was very close to the Copenhagen one, the prominent specialist in radiophysics and optics L.I.Mandelstam presented a statistical interpretation in his authoritative lectures delivered at Moscow State University. The Marxist philosopher, Mandelstam’s former graduate student B.M.Hessen also came out with the statistical treatment of quantum mechanics in his philosophical essays.

There was a common source of the statistical interpretations. This was A.Einstein’s address to 15th Solvay Conference (1927) and his subsequent papers on the foundations of physics. Some of partisans of the statistical interpretation were inspired by J.von Neuman’s "Mathematishe Grundlagen" (1932), by the "great classic" which was written in the "tenor of statistical ensemble interpretation", although von Neumann "seems never to have committed himself verbis expressis to the statistical view" (Jammer, 1974, p.443). However, this would be only a partial explanation of a historical phenomenon which this article tends to reconstruct. A struggle for the statistical interpretation arose in different countries, in different scientific, philosophical, and even political contexts. Correspondingly a diversity of the versions arose, and some of the versions used concepts which were hostile to both Einstein and von Neumann. Nevertheless, there was a striking similarity between the statements to which different partisans came (probably) independently from each other!

An outline of the statistical interpretations is given in Max Jammer’s celebrated book (Jammer, 1974). Jammer regards the statistical ensemble interpretations as a special area in the philosophy of quantum mechanics, he emphasizing the role which Einstein and von Neumann played in the development of the area (I have just quoted Jammer’s characteristic of von Neumann’s book). Jammer, however, does not systematize the ensemble treatments of the fundamental quantum paradoxes and he does not go far to scientific-philosophical traditions that laid behind the statistical interpretations. His outline of the Soviet statistical interpretation is very short. He only mentions Nikol’skii and Mandelstam and concentrates on Blokhintsev’s interpretation which arose after War II on the wave of the struggle against idealism in physics (before War II Blokhintsev belonged to the Copenhagen faction). Moreover, he places Nikol’skii’s and Mandelstam’s interpretations in the special section entitled by "ideological reasons" and hence separated these interpretations from the international stream of anti-Copenhagen ideas.

In a sense, Dipankar Home and Matthew A.B Whitaker’s extensive review (1992) supplements Jammer’s book. Home and Whittaker, however, are basically concerned with modern problems which the statistical interpretations face and they leave their cultural background in shadow. The Soviet statistical interpretations are represented only by Blokhintsev’s books which have been translated into English.

Arthur Fine’s book (1986) is credited with a careful and detailed presentation of Einstein’s approach which inspired the author’s own version of the statistical interpretation.

To assess different types of the statistical interpretation I provide a classification of the interpretations of quantum mechanics (sections 2 and 3). The main interpretative problems (the uncertainty relations, the reduction of the wave packet, the Einstein 1927 argument on action-at-a distance, the 1935 Einstein-Podolsky-Rosen argument – subsequently referred as the EPR argument) subsequently follow (sections 4, 5, 7, 8, 9, 10). Section 6 reconstructs Mandelstam’s theory of indirect measurement.

Section 11 provides the conclusion of sections 2-10.

Section 12 reconstructs what laid behind the statistical interpretations of quantum mechanics (either all the statistical interpretations or some of them): operationalism, an empirical trend in the interpretation of probability, the culture of macroscopic experimentation with its emphasis on the statistical (collective) measurement, and struggle for scientific objectivity.

As an appendix I shall outline who Nikol’skii and Mandelstam were for Soviet physics.

2. The "horizontal" and "vertical" classification of interpretations of quantum mechanics.

We now pose the question, What kind interpretations of quantum mechanics should be called the statistical ones? This question requires a suitable classification of the interpretations of quantum mechanics.

In fact, we require two such classifications that could arbitrary be called "horizontal" and "vertical". The "horizontal" classification of interpretations of quantum mechanics is effected by the descriptions of physical reality given in these interpretations. In the "vertical" classifications the principle of division is according to the account of scientific experiment given in these interpretations. The "vertical" classifications are classifications according to the "layers" of experiment (brute facts and thought experiments).

The "horizontal" classifications have been suggested in many writings. We are first interested in discussing a classification whose principle of division may be formulated as the following question: Does a theory fundamentally describe the behavior of an individual physical system (e.g. electron) or an ensemble of similar (identically prepared) physical systems? In the case of quantum mechanics, the Copenhagen interpretation elaborated by the founders of this theory is usually taken as a paradigmatic example of the former type of interpretation. In the Copenhagen interpretation quantum mechanics is regarded as providing "the most complete description of an individual physical system" (Jammer, 1974, p.440). Some of "hidden variables" interpretations (for example, the D.Bohm interpretation) of quantum mechanics also tend to present it as a theory of an individual physical system.

The interpretations of the latter type were basically elaborated by Einstein, Popper, Margenau, Nikol’skii and other physicists and philosophers in the course of criticism of the Copenhagen interpretation and were called statistical or ensemble interpretations. Mandelstam also permanently emphasized that quantum mechanics was a statistical theory and, as such, was a theory of an ensemble. At the same time he did not limit himself to this interpretation. In discussing the "rules relating the mathematical formalism of quantum mechanics to measurement", he endorsed the Copenhagen thought experiments made on a single physical system. To some extent he was close to von Neumann, whose "endorsement of the Bohr and Heisenberg thought experiments in their original interpretation conflicts with the general tenor of statistical ensemble interpretation in which his great classic on the mathematical foundations was written" (Jammer, 1974, p.443).

Two years earlier E.C.Kemble also combined the Copenhagen approach with the statistical one. Unlike Mandelstam he directly declared this combination. Indeed, he wrote that "the y function correlated with a system is not completely independent of the observer" and hence y function represents a "subjective state" of the system. He also introduced the concept of an "objective state": if the y function correlated with a Gibbsian assemblage of identical systems, it represents an "objective state". The assemblage’s reaction to future experiments "will depend only on the nature of experiments and not on the observer" (Kemble, 1937, pp. 52, 54).

Kemble’s and Mandelstam’s interpretations were inspired by J.von Neumann‘s Mathematische Grundlagen. Following von Neumann Mandelstam paid much attention to the problem of measurement. He planned to dedicate the second part of his lectures to von Neumann’s theory of measurement, but these were never delivered. Kemble himself wrote in the Preface that his book must bridge the gap between the exacting technique of von Neumann and the usual less rigorous formulations of quantum mechanics. Unlike Mandelstam Kemble directly communicated with J.von Neumann.

In his 1940 book Nikol’skii, who proposed his statistical interpretation in 1936, explicitly joined Kemble’s "objective states" (p.150). He, however, kept a strict anti-Copenhagen line. "The standpoint of Heisenberg, who treats quantum theory as a theory of an individual physical process, abandons a representation of objectivated in space and time independent from an observer atomic processes, that is, to an explicitly idealistic conclusion, and, at least on this reason, can not be regarded as valid" (Nikol’skii, 1940, p.28).

Let us turn to the classifications which were called "vertical". In respect to such interpretations there are only minor differences in literature.

Experiment in the crudest sense can be regarded as a sequence of "brute facts" brought about by observation and measurement. With this kind of approach the problem of interpretation of a theory is to propose the "minimal instrumentalist interpretation" (Michael Redhead's term), a set of rules which relate the mathematical formalism of a theory to these facts. Yet scientific experiment can also be regarded as a complex of thought experiments which aim to explain what physical reality is. Accordingly we have an interpretation of the mathematical formalism which "contributes to our understanding of the natural world" (Redhead, 1987, p.45).

We have just referred to M.Redhead's classification of interpretations. However, similar classifications have been proposed by other authors, for example, by Bernard d’Espagnat, who writes: "We must first state precisely a set of rules that lead to experimentally verified predictions for observed phenomena, later we will assume that it is interesting to try to construct from these rules a real description of the physical world" (d’Espagnat, 1976, p.2).

D’Espagnat is inclined to take the first step as independent or at least separated from the second.

However, at least two "minimal instrumentalist interpretations" have been constructed. The first relates the mathematical formalism of quantum mechanics to the probabilities of the results of a single experiment on a single system, the second relates it to the probabilities (relative frequencies) of the results of an ensemble of identical experiments on an ensemble of identically prepared systems. "For historical reasons" Jammer uses the name the "Born hypothesis" for the thesis that quantum mechanics predicts the result of a measurement performed on a single system, in contrast to the thesis that quantum mechanics predicts probability which is showed by a collection of measurements performed on identically prepared systems called the "Einstein hypothesis".

Historically the Copenhagen interpretation was bound with the first "minimal instrumental interpretation". However, H.Weil in his 1931 Gruppentheorie and Quantummechanik supplied his Copenhagen presentation with the second "minimal instrumental interpretation". M.Readhead believed that this "minimal instrumental interpretation" is a common point of departure for three main interpretations of quantum mechanics (Copenhagen, propensity, and "hidden variables").

The partisans of the statistical interpretation started exclusively from the second "minimal instrumentalist interpretation". Where they really differ from the Copenhagen faction is in this interpretation that contributes to "our understanding of natural world". As a rule, they relate the wave function not only to an ensemble of measurements on identically prepared systems but also directly to an ensemble of such systems. A.Einstein (in his note addressed to the Fifth Solvay Conference) and K. Popper (in his Logik der Forschung, 1934) had clearly expressed this view.

H.Margenau, whose interpretation of quantum mechanics is rightly subsumed by M.Jammer under the heading "Statistical Interpretations" (Jammer, 1974, p.228), attempted to meet the quantum mechanical difficulties within the framework of the "minimal instrumental interpretation" and did not go beyond it in his interpretative assumptions. Having started from the abstract postulates of quantum mechanics, in the core of which he emphasized the concept of a state of a physical system, he formulated the "second minimal instrumental interpretation" to relate these postulates to results of measurement. He did not invite the image of ensemble of physical particles (electrons, photons, etc) to explain the meaning of the concept of a state.

In his 1937 article Margenau explained that in quantum mechanics probability becomes meaningful only with respect to "a collection of observations on an ideal ensemble consisting of an infinite repetition of a set of physical operations" [Margenau, 1937, p.353]. Thus in this theory the statistical ensemble is a "numerous repeated observations on the same system, the state in question being reprepared before each observation" (ibidem).

By emphasizing that the y -function represents the state of a "Gibbsian ensemble" Nikol’skii and Kemble indicated that they, like Popper, are closer to Einstein here. "We can identify the state of a physical system", Kemble wrote, "with that of Gibbsian assemblage of identical systems so prepared that the past history of all its members are the same in all details that can affect future behavior …" (Kemble, 1937, p. 54). Nikol’skii, in turn, described a "quantum ensemble" as an ensemble of repetitive interactions of "quantum particles" with a "macroscopic body whose parameters were each time fixed at the same values (with the precision of the quantum of action)" (Nikol’skii, 1937, p.555). "Quantum theory studies statistical collectives which were formed from independent from each other quantum entities put into homogeneous uniform experimental conditions" (Nikol’skii, 1940, p.29).

Although Mandelstam sometimes express his attitude in Margemau’s stile, he was more close to Einstein, Popper, Nikol’skii, and Kemble here. Mandelstam said the following in his lectures (Mandelstam, 1950a, p.356):

Wave mechanics states that the micro-mechanical ensemble to which the wave function refers could be determined by specifying macroscopic parameters.

If, for example, in a vacuum tube a hot filament emits a beam of electrons subjected to an accelerating voltage, then their behavior obeys wave mechanics. An element of the collective in question is the behavior of an individual electron... The definite y -function refers to these elements; that is, the statistics of the emitted electrons are specified by this function. To determine the collective of electrons, the temperature, voltage, configuration, etc. of the filament must be specified.

To asses the difference of Popper’s interpretation, on the one hand, and the interpretations proposed by Nikol’skii, Kemble and Mandelstam, on the other hand, we need a finer classification of the interpretations of quantum mechanics, namely we need distinguish between the statistical interpretations.
  3. Three statistical interpretations. Let us now go further into the "horizontal" classification of interpretations of quantum mechanics. There are three types of the statistical (ensemble) interpretations. Margenau’s interpretation provides a good example of the first type (the closest to the Copenhagen interpretation). This is an interpretation by means of the "minimal instrumentulist interpretation". The square of the module of the y function is interpreted by referring to the ensemble of measurement results. The function itself is considered to be a mathematical construct.

As indicated in a recent review of the statistical (ensemble) interpretations of quantum mechanics there are two other types of such interpretations (Home & Whitaker, 1992). According to the interpretations called in this review the pre-assigned initial values (PIV) ensemble interpretations, at all times all observables have values in quantum mechanics, values which are available to be discovered in measurement. In other words, a measurement always reveals a preexisting property of the system.

The Einstein interpretation of quantum mechanics is usually considered to be an example of the PIV interpretation. However, Arthur Fine (see:(Fine, 1986)) attacks the widespread attitude toward Einstein as a supporter of the PIV interpretation. Nevertheless the authors of the above review insist that Einstein, in his 1927 note addressed to the Fifth Solvay conference, came out namely in favor of the PIV version of the ensemble interpretation. Indeed, by discussing a simple thought experiment Einstein compared two interpretations of the wave function. The Copenhagen interpretation (Einstein’s viewpoint 2) stated that the wave function represented a state of a single particle whose position appeared as it was detected in measurement. Einstein was inclined to accept the interpretation of the wave function as representing an ensemble of particles, each of which had its preexisting position which could be discovered in measurement.

As Einstein in his following writings did not explain clearly his version of the statistical interpretation, Popper’s interpretation provides a more characteristic example of the PIV version.

The other type of interpretation is called the minimal ensemble interpretation in Home & Whitaker (1992). This simply means that a wave function represents an ensemble of similar physical systems prepared in the same quantum state. No physical properties beyond those which the "minimal instrumentalist interpretation" attaches to the ensemble are envisaged. Elements of the ensemble are only characterized by the probabilities and the means (mathematical expectations) of physical magnitudes.

The authors of the above review point to D.I.Blokhintsev as a characteristic adherent of the "minimal ensembles". Blokhintsev’s view will not be discussed in the present article. However, as Blokhintsev himself pointed out, he was influenced by Mandelstam and Nikol’skii. Mandelstam had a remedy against the PIV’s commitments, for he was retaining in his interpretation elements of the Copenhagen interpretation.

Blokhintsev did not follow Mandelstam in his turn to the Copenhagen "measurement disturbances". He was closer to Nikol’skii who opposed the Copenhagen interpretation. Nikol’skii’s remedy against PIV commitments was the quantum of action which did not allow the classical description of ensembles. The "quantum of action", the term which M.Planck introduced for his constant in 1906 (he also used the term "element of action"), acquired the role of an interpretative notion in Nikol’skii’s writings. Nikol’skii emphasized that quantum theory originated in atomic theory. The "quantum particles" did not exist independently from the "macroscopic bodies" which they compose, and they can not be cognized independently from the "macroscopic bodies". The quantum of action specified the type of interconnection of the "quantum particles" and "macroscopic bodies". It led to the "unavoidable quantum scattering which is characteristic for the micro-world". "Quantum theory", Nikol’skii wrote, "is a theory of an ‘average process’ which is established on the base of a sequence of individual processes" precise description of which is restricted to quantum action (1937, p.557). "An individual process is treated via the prism of the statistical method" (1940, p.28).

True, Nikol’skii’s philosophical comments went further. He speculated about the problem of the description of individual processes. However, since he stated that quantum mechanics did not set this problem, these speculations did not affect his interpretation which held the level of ‘minimal ensembles".

Kemble also held "minimal ensemble interpretation". Although he wrote about Gibbsian essamblages, he emphasised that in these essamblages "we have a single quantum mechanical state", rather than "infinitely many sharply defined classical states" (Kemble, 1937, p.55) and "we can not know more about an individual electron that the fact that it belongs to a suitable chosen potential assemblage" (Kemble, 1935, p.974).

Home & Whitaker point out that the dilemma of "the <<minimal ensemble>> interpretation vs. the <<PIV interpretation>> is similar to the dilemma of <<the Copenhagen interpretation>> vs. the <<hidden variables>> interpretation, the dilemma that arises as quantum mechanics is considered to be a theory of a single (individual) system. The PIV version has turned out to be extremely effective in solving so called quantum paradoxes. However, this success has come at a cost.

So, with respect to the Copenhagen interpretation the above three statistical interpretations are set out as follows: the Margenau type interpretation, the minimal ensemble interpretation, and the PIV interpretation.

4. The interpretation of the uncertainty relations. As M.Jammer warded it, the founders of the Copenhagen school interpreted Heisenberg's uncertainty relations in a "non-statistical way": they saw in these relations the principle of limitations in measurement precision. According to their interpretation "it is impossible, in principle, to specify precisely the simultaneous values of canonically conjugate variables that describe the behavior of a single (individual) physical system" (Jammer, 1974, p.81). Popper, Margenau, Kemble, Nikol’skii, and Mandelstam subsequently put forward the statistical interpretation of Heisenberg's uncertainty relations, the interpretation which can be summarized as follows: the product of the standard deviations of two canonically conjugate variables has a lower bound given by h/4.

Nevertheless, one can distinguish three approaches to the uncertainty relations within the statistical interpretations.

Kemble and Mandelstam retain the Copenhagen interpretation of the uncertainty relations, too. In fact, following Bohr and Heisenberg they admit limitations in the definability and measurability of the canonically conjugate dynamical variables and explains the Heisenberg’s formulae by referring to the fact of the interaction between a measuring device and a micro-system.

In contrast to Kemble and Mandelstam, Popper ("a believer to PIV", as he is named by Home and Whitaker) believed that Heisenberg's formulae did not rule out of the possibility of performing measurement in individual cases with arbitrary accuracy. These formulae ("scattering relations", as he called them) were for him similar to the phenomenological correlations behind which more strict dynamics laid.

Like Popper, Margenau and Nikol’skii criticized the Copenhagen interpretation of the uncertainty relations. They, however, stressed the fundamental status of the relations. Thus, in Lindsay and Margenau’s book one reads that "the formalism of quantum mechanics is such as to produce unavoidable related uncertainties in the position and momentum of particles, permitting only statistical knowledge and prediction regarding their properties (1936, p.520). Lindsay and Margenau attacked the Copenhagen approach by pointing out that it restricted to classical language which presupposed that the state is defined by the position coordinate and the momentum. In quantum mechanics state is defined by ?-function, and "experimental indeterminacy as to the outcome of a given measurement is inherent in the fundamental manner of describing states" (Lindsay, Margenau, 1936, p.420).

In the course of criticism of the Copenhagen approach Lindsay and Margenau proposed to introduce the concepts of "generalized position" and "generalized momentum". One comes across such a proposal in Mandelstam’s lectures, too.

According to Nikol’skii, the formulae which express those which are called the uncertainty relations by the Copenhagen school "allow to quantitatively formulate how quantum ensembles differ from classical ones" (1940, p.65). He preferred to use the terms "generalized position" and "generalized momentum" or "canonical variables which correspond to coordinate and momentum in the classical limit". Following Nikol’skii Blokhintsev included the uncertainty relations in the very definition of the quantum ensembles (1968, 1987).

According to Nikol’skii, Heisenberg’s interpretation of his formulae proceeded from "an erroneous transportation of the characteristics of a statistical collective to an individual process with a single electron" (1940, p.79) The celebrated thought experiments demonstrating the uncertainty of coordinate at a given momentum and vise versa is interpreted by Nikol'skii as follows. This experiments illustrate how the initial conditions for the formation of a quantum ensemble are fixed. If we form an ensemble with a definite coordinate, our ensemble has D p® „ . In other words, these experiments are relevant to what Margenau called "the state preparation", rather than to what he called "measurement".

In 1945 Mandelstam’s and Tamm’s article on the energy-time uncertainty relation was published by Tamm (Mandelstam died in 1944, but his note "On energy in quantum mechanics" published in vol.3 of his Complete Work (1950) represented a piece of his work on the problem). This article was written in the pure statistical stile (compare with the deduction and interpretation of the energy-time uncertainty relation in Landau&Lifshits’s book (1948)). In Fock’s and his graduate student Krylov’s 1947 article the Mandelstam-Tamm interpretation of the energy-time uncertainty relation had been taken under criticism. Krylov and Fock claimed that this interpretation did not capture the "statistic of errors of individual experiments performed on individual particles" (Krylov, Fock, 1947, p.115). In his 1948 note dedicated to Mandelstam’s biography I.E.Tamm in essence agreed with this criticism (Mandelstam, 1948, p. 43).

In the fifth section we shall return to the statistical interpretation of the uncertainty relations in Mandelstam’s lectures. This has a specific feature analysis of which required a previous excursion into Mandelstam’s theory of indirect measurement.

5. The reduction of the wave packet. Popper, Margenau, Kemble, Nikol’skii, and Mandelstam shared an opposition to the Copenhagen (more precisely: Heisenberg-Pauli-Dirac-von Neumann) notion of the reduction of the wave packet as an "acasual jump" resulting from a single measurement.

Margenau, who developed his ensemble interpretation on the level of the minimal instrumentalist interpretation and abandoned the ensemble of particles as a theoretical concept (see sections 2 and 3), rejected the very notion of the reduction of the wave packet. Since only a collective measurement (a set of measurements) made sense within the framework of the second minimal instrumentalist interpretation, he had no room for a reduction which presupposed a single (individual) measurement. Moreover Margenau sharply distinguished between "state preparations" such as the injection of electrons through a magnetic field, which endowed them with a new spin-state, and "measurements", which yielded numerical results. He denied what the reduction of the wave packet presupposed, namely, that a state preparation may be a measurement at the same time. According to him, all the examples of the reduction resulted from a confusing treatment of the state preparation experiments as both state preparation and measurement experiments.

In his 1934 book Popper did not discuss the problem of the reduction in its general form. He only discussed a thought experiment which was interpreted with the aid of the idea of the reduction by Heisenberg. However, what he wrote about the reduction in this book could be subsumed under his more recent definition: the "reduction of the wave packet" is not an "effect of characteristic of quantum theory, it is an effect of probability theory in general" (Popper, 1982, p. 74). Indeed, Popper reduced the reduction to a change in the of probability of an event, namely, to the fact that "the probability of an event before its occurrence is different from the probability of the same event after it has occurred" (Popper, 1959, p.157). This difference proceeded from the difference in the referent classes with respect of which the probability is estimated. To estimate the probability of an event before its occurrence we take a totality of trials which give not only this event but also different outcomes. The probability of the event after it has occurred is equal to 1 since this probability is estimated with respect to the referent class which is a subclass of the initial one. This subclass consists of only the events in question..

Popper’s interpretation is fit for very simple cases. Nevertheless, Popper came close to the interpretation of the reduction as a subensemble selection, the interpretation which was distinctly conducted by Kemble and Mandelstam both of whom were apparently influenced by J. von Neumann. Indeed, in his 1932 book von Neumann wrote (Neumann, 1955, p.300):

Even if two or more quantities, R, S in a single system are not simultaneously measurable, their probability distributions in a given ensemble [S1,…SN].can be obtained with arbitrary accuracy if N is sufficiently large.

Indeed, with an ensemble of N elements it suffices to carry out the statistical inspections, relative to the distribution of values of the quantity R, not on all N elements [S1,…SN], but on any subset of M (£ N) elements, say [S1,…SM] – provided that M, N are both large and that M is very small compared to N. Then only the M/N –th part of the ensemble is affected by changes which result from the measurement.

Von Neumann’s concern was to stress that quantum measurement can disturb only a small portion of systems constituting an ensemble. In turn, Kemble was interested in explaining what measurement in quantum mechanics was. He treated the reduction of the wave packet as a "selective process in which the systems of the original assemblage are divided into two or more subassemblages each associated with a value, or range of values, of the dynamical variable measured. The discontinuous change in the wave function of a system which accompanies the measurement of a variable a is the reflection of a mental process in which we transfer the system under consideration from an initial assemblage A to a subassemblage Aa 1, consisting of those members of A which have given the measured value a 1 of a " (Kemble, 1937, p.328)

By treating the "reduction of the wave packet" as a selection Kemble, however, introduced an inconsistency into his minimal ensemble interpretation. Indeed, he tended to regard this selection as a hidden physical process. "When a atomic system is "observed", it interacts with an observing mechanism", Kemble wrote. "If a certain type of observation is carried out for each member of an assemblage of similar systems, the assemblage will be divided, in general, into two or more subassemblages according to the outcome of the individual measurements" (Kemble, 1937, p. 75).

Although technically Kemble’s formulations differed from Popper’s, his interpretation of the reduction of the wave packet was also committed with the PIV approach. Home and Whitaker, in their review cited in section 3, point to the PIV commitment of every interpretation of the reduction as the subensemble selection. "Without the PIV interpretation, one has only a distribution of postmeasured values, before a measurement the wave function cannot be considered as a distribution… Only with PIVs is the situation so simple that the mental process of subensemble selection is all that is required" (Home, Whitaker, 1992, p.28).

In turn, Mandelstam did not attach much importance to the "reduction of the wave packet". Here his position was close to that of N.Bohr and subsequently to Landau&Lifshits (1948), where N.Bohr’s position was technically formulated. Nevertheless Mandelstam kept the concept of the reduction as an alternative way to explain measurement in quantum mechanics. Like Popper, Mandelstam was close to admitting the probability theory account of the reduction of the wave packet. The reader will come below (sec. 9) to Popper’s characteristic phrase in a passage borrowed from Mandelstam’s Lectures. "After all is probability a thing which can be physically acted upon ?" However, Mandelstam was closer to Kemble, who did not reduce the reduction to the theory of probability and treated it as a quantum effect. In some of his passages he did not escape from the PIV’s terminology either. I shall return to his interpretation in the 8 section; like his interpretation of the uncertainty relations, this interpretation will be presented after an excursion to his theory of indirect measurement.

Nikol’skii did not pay much attention to the reduction of the wave packet either. However, he agreed with its interpretation as a subensemble selection and he emphasized that this subensemble selection really was the formation of a new ensemble (Nikol’skii, 1940, pp. 153-154).

6. Mandelstam’s theory of indirect measurement.

Mandelstam’s Lectures were subtitled "The theory of indirect measurement". This subtitle expressed the peculiarity of Mandelstam’s approach to quantum mechanics in these lectures. Following von Neumann he paid much attention to the problem of measurement. However, following to some extent L.D.Landau’s and R.Peierls’ 1931 article, he developed the theory of indirect measurement while von Neumann was mainly concerned with the principles of interaction of a microparticle with a macroscopic measuring device and observation. As was above mentioned, Mandelstam planned to dedicate the second part of his lectures to von Neumann’s theory of measurement, but these were never delivered.

By discussing indirect measurement Mandelstam achieved an extended physical framework for formulating the fundamental concepts and principles of quantum mechanics. "Thus, by elucidating the question how in quantum mechanics the mathematical symbols are related to the real objects", Mandelstam concluded his first lecture, "we came to the problem of developing a theory of indirect measurement. This is necessary, insofar as we are certain that direct measurements are an exception, and this is not accidental. This exceptionality of direct measurement is of fundamental importance" (Mandelstam, 1950a, p. 360).

I.E.Tamm wrote in this connection (Tamm, 1991, p.275):

As far as I know, Leonid Isaakovich was the first to include in lectures the very important distinction between direct and indirect measurements in quantum systems. The last stage in any measurement of a quantum system necessarily has a macroscopic character. L.I. calls measurement direct when the first measurement step is macroscopic. Example: An electron incident on a photographic film leaves a blackened spot. The macroscopic coordinate of the spot, by definition, is the coordinate of the electron upon its impact on the film. It is important to note that the direct measurements are possible only for free or nearly free particles in weak fields. For example, it is impossible to determine the coordinate of an electron in a hydrogen atom by placing a photographic film inside the atom.

In addition to direct measurements, indirect ones are also possible. In these we force the quantum system on which we want to make measurements to interact with another micro-system on which direct measurements are possible. The data of these direct measurements we use for theoretical calculations of the values of the quantities relevant to the first system. Example: By measuring the angular distribution of electrons scattered by an atom, we can find the distribution of bound electrons in this atom.

To gain access to Mandelstam’s theory of indirect measurement let us introduce his terminology. Let us assume that in order to measure the quantity R attached to system 1, one needs to measure the quantity S attached to system 2 which interacts with system 1. To simplify the problem Mandelstam took first the one-dimensional case (x is the position coordinate of system 1 and y is the position coordinate of system 2). Mandelstam indicates the initial state of the system 1 as y 0(x), the eigenfunctions of R as y 1(x), y 2(x),..., the corresponding eigenvalues as l 1, l 2..., the initial state of system 2 as j 0(y), and the state of the pair after the interaction as Y (x,y). Mandelstam wrote (Mandelstam, 1950a, p. 367): After the interaction we have, as a rule, a mixture rather than a production (for the whole system we have a pure case, but for each particle we have a mixture). However, Y (x,y) can be expanded in the series of eigenstates of the operator S referring to system 2.

Y (x,y)=å ui (x)j i(y). (1)

Let y 0(x) is an eigenfunction of our operator R, say, y 0(x) =y i(x), that is, before the interaction l was equal to l i. Generally speaking Y (x,y) is a mixture but it may happen that at y 0(x)=y i(x) we have a mere product of two functions, where the function, which refers to system 2, corresponds to the eigenvalue µ= µi of the operator S connected in a one-to-one way with the operator R eigenvalue l i , which system 1 possessed before the interaction:

Y (x,y) = ui(x) j i(y) (2)

Then, having observed that µ=µi after the interaction for the system 2 we conclude that l =l i for the system 1 before the interaction. If such an experimental device for which (2) is valid for all i has been found we can say that we found a device for measurement of R.

Mandelstam concluded this passage by stating that quantum mechanics postulated that such an experimental device existed for any quantum observable (dynamical variable) and therefore one can construct for any observable the chain of indirect measurements completed by a direct measurement.

Since "direct measurements are an exception", what the theory of indirect measurement implies (the perturbation of the state of system 1 and the formation of mixture) attends the fundamental significance for Mandelstam’s interpretation of quantum mechanics.

7. The interpretation of the uncertainty relations again.

As was mentioned, Mandelstam and Kemble accepted both statistical and non-statistical interpretations of the uncertainty relations. M.Jammer, who clearly distinguished between these interpretations (I1 and I2 in his terminology), tried to explain the relation between them. He wrote (Jammer, 1974, pp. 80,81):

As intimated in our historical account, thought experiments, such as the Heisenberg gamma-ray microscope experiment, were to spurious to serve as rigorous arguments in support of I1. On the other hand, I2 being essentially a mathematical consequence of the basic formulae of quantum mechanics, could hardly be rejected without introducing major modifications into the whole theory. Does it therefore follow that I1 has no logical justification whatever?

In contrast to those who flatly reject I1 or claim that between I1 and I2 there is a logical gap which can never be bridged, we contend that I1 is a logical consequence of I2 if a certain measurement theoretical assumption A is accepted. According to this assumption every measurement involved in this context is a measurement of the first kind, that is every measurement is repeatable and, if immediately repeated, yields the same result as its predecessor.

It is interesting that Jammer’s explanation does not speak to Mandelstam’s particular reasoning (although it speaks to Kemble’s reasoning). Mandelstam also admits I2 as fundamental. He, however, does not attach importance to the results of single (individual) measurements in quantum mechanics and expresses a hostility to the problem of how to explain the fact that the second measurement on the same system, if it immediately followed the first measurement, yields the same result. According to Mandelstam, the repeatable measurement was rather an exemption. The theory of indirect measurement, the theory along which all his discussion develops, exposed the general case. The passage quoted below immediately follows the one cited in the previous section, which explains indirect measurement (Mandelstam, 1950a, p.370-371) : Let system 1 before the first measurement be represented by y 1(x). Then after the interaction [of system 1 and system 2] we have Y (x,y)= u1(x)j 1(y). Then for the second measurement we have the initial function u1(x), which is not generally speaking equal to y 1(x) . Therefore after the second measurement Y (x,y) will change, too.

In a word, one finds nowhere that a measuring device must be able to repeat a measurement yielding the same result. It is possible, but it occurs only in exceptional cases.

He refused to regard as postulates of the theory those statements (the "reduction of the wave packet" and the "projective postulate") which such measurements suggested.

Mandelstam supplements I2 with I1 to demonstrate the fundamental status of the uncertainty relations in quantum mechanics, that is, to demonstrate that quantum theory, in its contemporary form, is complete. Margenau, Kemble, and Nikol’skii share this position, but they do not go to demonstrations.

Mandelstam’s reasoning runs as follows (Mandelstam, 1950a, p.358, 395):

If it turned out that the uncertainty relations were presupposed by the mathematical technique, but measurement yielded precise values for x and p, the theory would be inconsistent. However, measurement disturbances come into play. We could precisely measure x and p if the measuring devices have not disturbed the wave function of a system. But this is not the case. The theory implies that these devices inevitably change the wave function. In this reasoning Mandelstam was apparently influenced by von Neumann, to whom Mandelstam referred at the beginning of his discussion of the EPR argument, and whose Mathematische Grundlagen he planned to present in the second half of his lecture cycle on quantum theory. Indeed, as is well-known among those who are interested in the philosophy of quantum mechanics, von Neumann starts his famous demonstration of the completeness of quantum mechanics by formulating a non-mathematical argument; he starts by distinguishing between two alternative interpretations of an ensemble (Neumann von, 1955, p.302): 1. The individual systems s1,...,sN of our ensemble can be in different states so that the ensemble [s1,...,sN] is defined by their relative frequencies. The fact that we do not obtain sharp values for the physical quantities in this case is caused by our lack of information: we do not know in which state we are measuring, and therefore we can not predict the result.

2. All individual systems s1,...,sN are in the same state, but the laws of nature are not causal.

J.von Neumann argued (and Mandelstam followed him) for the second interpretation by referring to the Copenhagen conception of the disturbance of the values of magnitudes in any measurement process. But he pointed to the inadequacy of this argumentation: "nevertheless, we could attempt to maintain the fiction that each dispersing ensemble can be divided into two (or more) parts, different from each other and from it, without a change of its elements" (Neumann von, 1955, pp. 304-305). To complete this argumentation von Neumann drew his celebrated demonstration that every pure ensemble is homogeneous, the demonstration which Mandelstam took for granted and illustrated by considering the EPR argument and the corresponding thought experiment. Actually, Mandelstam considered a development of the EPR thought experiment: he conducted the thought experiment with an ensemble of the couples of correlated particles (see below).

8. The reduction of the wave packet again.

As I said in sec. 5, Mandelstam did not attach much importance to the "reduction of the wave packet". As we have seen, his account of measurement did not require the concept. Heisenberg, Pauli, Dirac and von Neumann interpreted the transformation of formula 1 into formula 2 (section 6) as the reduction. For Mandelstam formula 2 was implied by the postulate of the measurement theory in quantum mechanics.

Moreover, as mentioned above, Mandelstam expressed a hostility to the problem of how to explain the fact that the second measurement on the same system, if it immediately followed the first measurement, yielded the same result, the problem which led the concept of the reduction.

Nevertheless Mandelstam kept the concept of the reduction as an alternative way to explain measurement in quantum mechanics. Like Kemble, Mandelstam regarded the reduction as a subensemble selection. However, his approach had specific. According to Mandelstam, the reduction consisted in selecting elements of the mixture and composing a new pure ensemble (the subensemble consisting of the systems on which the measurements had yielded the identical results). Mandelstam said (Mandelstam, 1950a, p.373):

After the measurement we have generally speaking a mixture (formula 1, section 6). However we can set off a subset (say, by fixing the definite µ=µ1), in which the wave function for system 1 exists. Mandelstam’s conception of the reduction is closer to that of d’Espagnat, who introduced the "reduction" as a special "quantum rule": "Let E0 be an ensemble of systems S. Immediately after simultaneous (ideal) measurements have been performed on a complete set of observables pertaining to the systems S, every subensemble E of E0 that is composed of systems S on which the measurements have produced identical sets of results can be described by a ket. This ket then necessarily is an eigenket common to the eigenvalues found as a result of the measurements" (D’Espagnat, 1976, p.19).

However, Mandelstam went further than d’Espagnat, who restricted himself to postmeasurement ensembles. In discussing the EPR argument Mandelstam kept the image of the ensemble in order to envisage the process of measuring. In Einstein’s style he claimed that the operation of measuring one of the conjugate observables narrows the ensemble under consideration. If we measure, say, momentum, we automatically narrow down the whole ensemble to a subensemble consisting of those systems which are not being subjected to the measurement of position. As A.Fine writes, if we measure A, we take a subensemble of systems that are not A-defective (Fine, p.52). As a result, Mandelstam proceeded from two processes of subensemble selection: 1. The choice of which kind of observable we measure (without added reference to the measurement’s outcome) already makes a reduction to a narrower ensemble. 2. If we specify a collection of identical outcomes of the measurement, say, a1, we produce the second subensemble selection: we narrow down our ensemble (which has become the mixed one) to a subensemble of the systems which have given the identical measured value a1. Thus we divide the mixed ensemble into the pure subensembles corresponding to the results of the measurement.

Mandelstam did not go further than the minimal ensemble interpretation allows. He did not specify the properties of the elements which are "A-defective" and which would give under the measurement the value a1. All the properties were distributed over the whole ensemble.

To conclude this section it is worth to noting that the postwar II discussions revealed the difficulties in the minimal ensemble approach to the reduction of the wave packet (Whitaker, pp. 286-89).

9. Einstein’s argument on action at a distance and the "statistical locality".

Let us turn to the thought experiment which was considered to be crucial when the problem of the reduction of the wave packet was taken under examination at that time. As was mentioned, in 1927 Einstein launched his criticism of the Copenhagen interpretation. His main objective was the concept of the reduction, which, according to him, led to the unacceptable notion of the instantaneous action-at-a distance. Einstein demonstrated a thought experiment which the Copenhagen authors dismissed as insufficient. Popper and Mandelstam discussed not this experiment but its modification proposed by Heisenberg. Here is a description of it (Heisenberg, 1930, p.39):

We imagine a photon which is represented by a wave packet built up out of Maxwell waves. By reflection at a semi-transparent mirror, it is possible to decompose it into two parts, a reflected and transmitted packets. There is then a definite probability of finding the photon either in one part or in the other part of the divided packet. After a sufficient time the two parts will be separated by any distance desired; now if an experiment yields the result that the photon is, say, in the reflected part of the packet, then the probability of finding the photon in the other part of the packet immediately becomes zero. The experiment at the position of the reflected packet thus exerts a kind of action (reduction of the wave packet) at the distant point occupied by the transmitted packet, and one sees that this action is propagated with velocity which greater than that of light. However it is also obviously that this kind of action can never be utilized for transmission of signals so that it is not in conflict with the postulates of the theory of relativity. In Popper’s opinion, his statistical interpretation makes this problem "perfectly clear, if not trivial" (Popper, 1959, p.235). He treats the reduction as a choice of a more narrow reference class, say, the reference class of those trials which indicate that a photon has passed through the mirror. It is quite clear that the probability of finding a photon passed through the mirror is becoming 1 in relation to this new reference class.

Popper gave an early analogue to what is later called the "statistical locality"(see, for example: (Redhead, 1987, p.113-116)). Popper wrote: "Saying of the logical consequences of this choice (or, perhaps, of logical consequences of this information) that they spread with super-lumenal velocity" is about as helpful as saying that twice two turns out with super-lumenal velocity into four" (Popper, 1959, p.236).

Mandelstam, who examined the problem of the reduction to defend quantum mechanics against Einstein’s 1927 attack, was not satisfied by Heisenberg’s account either. Mandelstam said: "The standard treatment of this issue strikes me as a kind of mysticism. Things are very simple really... There is no action here. After all is probability a thing which can be physically acted upon?" (Mandelstam, 1950a, p.371-372)

Mandelstam discussed this experiment within the framework of his theory of indirect measurement. To simplify the problem he considered the one dimension case and symbolized a system measured (system 1) by the coordinate x and a measuring system (system 2) by the coordinate y (Mandelstam, 1950a, p.372):

Before a measurement (with the aid of the system 2, symbolized by the coordinate y) we had Y 0(x,y) =y 0(x)j 0(y), where y 0(x) is the divided wave packet. After the measurement we obtain Y (x,y) = S ci ui (x)j i (y), that is, there is no y (x) and the wave packet does not exist. We can, however, select a subensemble, that is, select those cases where direct measurement on system 2 yields, say, µ =µ1. Within this subensemble the wave function exists, namely, u1(x). This separation of the wave function resulted from an additional classical measurement relating to y. This is exactly the reduction of the wave packet and the formation of the new wave packet. It is not surprising that this packet differs from the initial one. Here Mandelstam solved two problems: 1. He outlined the concept of the reduction as a selection of a subensemble of the systems for which the measurements yielded identical results (see above). 2. He demonstrated what was later called "statistical locality": after the "measurement" (the interaction of a measuring device and the wave packet) we have a superposition which does not represent the divided wave packet. He also pointed out that the "additional classical measurement" selecting those measurements that fixed a photon at the area of the reflected part would be effective in the transmitted part, too. However, this deals nothing with the divided wave packet. To make a proper selective measurement in the transmitted part we must be informed about the measurement in the reflected part, and the information can not be transferred with super luminal velocity. A simple example can illustrate the problem. When I lecture in Oxford, the audience there learn instantaneously that my room in London is empty, but to produce a physical change in London, for example to prevent students knocking on my door, the information that I have arrived in Oxford must be transmitted to London. This passage is borrowed not from Mandelstam’s lectures but from M.Redhead’s 1987 book (Redhead, 1987, p. 115). However if the words Oxford and London are replaced by the words Moscow and Leningrad it becomes identical with that in Mandelstam’s lectures.

The statistical locality, however, does not spread light on the behavior of the physical system represented by the divided wave packet. With this type of locality one cannot address the problem of what becomes of systems once the measurement is performed. Here one needs another type of locality, namely locality of the properties of physical systems themselves. Popper and Mandelstam attempted to answer this question in their discussion of the EPR argument.

10. The EPR argument.

In reacting to the EPR argument, Popper, Margenau, Nikol’skii, and Mandelstam seemed to have only one point in common: they did not subscribe to Bohr’s response culminating in the relational concept of the state. As for the EPR argument itself they reacted in very different ways.

Popper adopted the EPR argument, which was fully consistent with his anti-Copenhagen policy, and his PIV version of the statistical interpretation of quantum mechanics strengthened the EPR conclusion that quantum mechanics is incomplete.

As for Margenau, his version of the statistical interpretation seemed to force him to overlook the EPR argument. But Margenau commented on the Copenhagen response to this argument, which contained the idea of the reduction of the wave packet, which, in turn, implied nonlocality of the properties of the microsystems. According to Margenau, "no such event happens; it can only be shown… that there exists a peculiar correlation between the probabilities for the various results of measurements. It is this: when suitable measurements on the two particles are made, the probability for the occurrence of a pair of values is zero unless the pair is the one which corresponds to the two eigenfunctions of the reduced package" (Margenau, 1937, p.367-368). In a word, he claimed that the technique of quantum mechanics implied that the EPR pair of systems behaved itself, if the measurement is performed, as if it is represented by the reduced (factorized) wave function.

Kemble and Nikol’skii did not agree with the EPR argument, but they did not formulate their responses to the argument.

Mandelstam’s attitude to the EPR argument confirmed what was noticed above (section 2) about his interpretation of quantum mechanics: this interpretation was close to the Copenhagen one but not identical with it.

Mandelstam rejected the EPR argument (which is cited by him as Einstein’s argument). "Einstein", Mandelstam said, "poses the problem not very profoundly. Bohr objected to him at once and Einstein has recognized his mistake" (Mandelstam, 1950a, p.362).

However, although Mandelstam was following Bohr in his insistence upon the completeness of the quantum theory he adopted a slightly different mode of reasoning. If Bohr was concerned with the pair of correlated particles located very far from each other, Mandelstam considered the case of an ensemble of the pairs of correlated and distant from each other particles (or two correlated distant ensembles of particles). "The debate between N.Bohr and A..Einstein resulted from a crude mistake about the ensemble", Mandelstam told (Mandelstam, 1950a, p.333).

Both Bohr and Mandelstam rejected the EPR criteria of reality. However, Bohr insisted that the reality of the properties of system 1 is conditioned by the experiment performing on the distant system 2. He suggested violation of the locality principle which can be formulated as follows: "a previously undefined value for an observable cannot be defined by measurements performed 'at a distance'" (Redhead, 1987, p.77) In turn, Mandelstam emphasized that the measurement of the momentum and position was conducted by invoking different subensembles. If we were measuring the position of system 2, we automatically narrowed down the initial ensemble to the subensemble which was not subjected to the momentum measurement. At the same time we correspondingly narrowed down the ensemble of system 1. If we completed the measurement and took into account its specific results, we produce the second subensemble selection: we partitioned our narrowed ensembles of system 2 and correspondingly of system 1 in accord with the measured outcome. However, the partition of the ensemble on which the position measurement was conducted did not touch the ensemble on which momentum measurement was conducted. These were the different narrowed ensembles.

Mandelstam invoked the concept of nonlocality, which presupposed violation of the following locality condition: distant measurements cannot provide the selection of a subensemble.

Mandelstam wrote (Mandelstam, 1950a ,p.365) (he indicated the first systems by the position coordinates y and z, and the second system by the position coordinate x):

As long as I measure nothing I have got Y (x,z), where all z (positions) and all p (momenta) are represented. While performing measurements I can select a subensemble from the entire ensemble, a subensemble for which the measurement results on system 2 show a distinct value and system 1 is described by the wave function. For this subensemble, if the momentum has a distinct value, than the position is of all kinds, and vice versa. The thing is that by performing measurements for different magnitudes referring to the system 2, we select different subensembles... In each of the subensembles system 1 obeys the principle of uncertainty. But simultaneous precise values of position and momentum are quite admissible in the different subensembles. Mandelstam also emphasized his allegiance to von Neumann’s approach to the completeness of quantum mechanics. Having outlined the EPR argument he told (Mandelstam, 1950, p.370): Einstein apparently did not know that this question had been already considered. J.von Neumann examined whether it is possible to make the total description by the wave function so precise and supplement it by introducing "hidden variables" in order to suspend the uncertainty relations. It is probable that Mandelstam planned to provide more profound discussion of the EPR argument by using von Neumann’s idea of the completeness of quantum mechanics, that is, he was close to the idea of the statistical completeness of a theory, which (under standard conditions) implied its EPR completeness (see (Elby, Brown, Foster, 1993)). In any case, the above citation can be reread as an illustration of von Neuman’s prove of the completeness of quantum theory. In doing so, we replace the EPR argument by an argument for the thesis that "there exists a dispersion free ensemble". This argument attaches reality to a pure ensemble which existence can be predicted with certainty and without disturbing the systems. Mandelstam’s discussion has broken the argument. To reach a dispersion free ensemble we need to select an ensemble consisting of those systems yielding identical positions as the measured outcome and then from this ensemble we need to select a subensemble of those systems yielding identical mementums as the measured outcomes. This is impossible.

True, Mandelstam’s argumentation was a version of Bohr’s response to the EPR argument. However, this version was more removed from the associations with the classical "action-at-a distance" than the original Bohr's response.

11. Interim conclusion

So, it would be artificial to separate Nikol’skii’s and Mandelstam’s interpretations from the main stream of ideas in the history of the interpretation of quantum mechanics. Like Kemble, Mandelstam combined the Copenhagen approach with the statistical one. Like Popper, Margenau and Kemble, Mandelstam and Nikol’skii could not accept the "reduction of the wave packet" as an "acasual jump" resulted from an individual observation. Like Popper and Kemble, they regarded the reduction as a subensemble selection. Along Popper’s line, Mandelstam objected the violation of the statistical non-locality. Like Margenau and Kemble (and unlike Popper) Mandelstam and Nikol’skii rejected the EPR argument but could not accept N.Bohr’s response as it stood.

This similarity extends to their terminology. Margenau, Nikol’skii, and Mandelstam preferred to use the concepts of generalized position and of generalized momentum in quantum mechanics. Both Popper and Mandelstam were reluctant to accept Heisenberg’s reply to Einstein’s 1927 argument on action at a distance. As was mentioned, one could come to Popper’s characteristic phrase in Mandelstam’s Lectures. "After all is probability a thing which can be physically acted upon ?" Both Nikol’skii and Kemble used the term "Gibbsian ensembles" in their formulations of the statistical interpretation.

Let us look at the other side of a coin. Mandelstam did not referred to his predecessors. Lectures for students do not presuppose that the professor provide a complete bibliography. Apart from this, it is unlikely that Mandelstam was familiar with their writings. At least his papers and letters do not give a hint that he looked at these writings. As was mentioned Nikol’skii referred to Kemble’s 1937 book, but he did it in his 1940 book which summed up what he proposed in his 1936-1938 articles. In these articles he did not referred to Kemble. Popper, Margenau, and Kemble did not refer to each other either. There is no cross references in Mandelstam’s and Nikol’skii’s writings. True, from 1936 they worked at the same institution (the Physics Institution of the Academy of Sciences) and might communicate, However, their interpretations had been form before 1936. Nikol’skii published his first article in 1936, Mandelstam sketched his interpretation in his 1935 Lectures on optics. Moreover, Mandelstam and Nikol’skii used different terminology and different rhetoric. Unlike Nikol’skii, Mandelstam never used political or political-like arguments in his writings and lectures.

The statistical interpretations of quantum mechanics proposed by Popper, Margenau, Kemble, Nikol’skii, and Mandelstam belong to one stream of ideas stimulated by Einstein’s criticism of the Copenhagen interpretation. It is plausible that all five scientists arrived at their interpretations of quantum mechanics independently, and here history evokes five parallel lines of thought.

In the next section we shall extend the comparison of Popper’s, Maregenau’s, Kemble’s, Nikol’skii’s, and Mandelstam’s approaches to quantum mechanics to a comparison of their backgrounds.

11. The backgrounds.

1. Operationalism. Mandelstam and Kemble shared the operationalist philosophy of science. They emphasized that in quantum mechanics position and momentum must be subject to operational definitions which differ from the classical ones. Following Heisenberg’s 1927 article they stressed the operationalist meaning of the uncertainty relations. In fact, they justified their ensemble approach on an operationalist basis: since quantum mechanics was a statistical theory, it should refer to a set of elements that is an ensemble of repeatable experiments. Thus operationalism led them to combine the statistical (ensemble) and Copenhagen approaches in their interpretations of quantum mechanics.

Mandelstam’s operationalism has been outlined by the present author in (Pechenkin, 2000). This was a German kind of operationalism which differed from the "essentially American philosophy", as G.Holton put it, of the American physicist P.Bridgman, whose operationalism tended to be paradigmatic. Mandelstam’s operationalism was consistent with realism. Moreover, Mandelstam’s theory of indirect measurement led to the flexible version of operationalism. In his Lectures on Quantum Mechanics he formulated his notion of operations by including theoretical calculations as an essential part of the majority of operations and he admitted different operational definitions for the same concept. Mandelstam emphasised that if (at least) two operational definitions can be formulated for a physical concept, then one of them may be treated as an empirical sentence testable against observation and experiment, with the other regarded as a formal definition.

Kemble, who had been Bridgman’s graduate student, declared in the Preface to his book that "a feature of the present volume on the physical and philosophical side is its consistent emphasis on the operational point of view" (Kemble, 1937, p.vii). Unlike Mandelstam, he was influenced by the subjectivist tenor of Bridgman’s philosophy of science. Thus, by the end of his book he had concluded that the wave function is merely a subjective computational tool (Kemble, 1937, p.328) True, in spite of this conclusion, he did not discard his differentiation between the subjective and objective states.

Although Margenau was known as a critic of Bridgman’s operationalism, he included a good portion of this conception in his philosophy of science. True, Margenau consistently stressed that "operationalism... is not the whole criterion of scientific validity" (Margenau, 1978, p. 247). However, theoretical concepts ("constructs", as he called them) had to be subjected to the operational definitions. "In virtue of the epistemic (in particular, operational) definitions of our constructs we can measure them, and in virtue of the constitutive definitions we are able to reason about the constructs" (Margenau, 1935, p.170; 1978, p.63).

To some extent Mandelstam was close to Margenau in distinguishing between the operational and mathematical parts of a physical theory. They both justified the ensemble approach on operationalist grounds.

Popper, "a believer in PIV", was a strong antioperationalist. There was no operationalist tenor in Nikol’skii writings.

2. The frequency conception of probability. Popper and Margenau shared allegiance to the von Mises frequency (empirical) conception of probability, which they called the "objective probability" in contrast to classical Laplacian subjectivist probability. Their statistical interpretations of quantum mechanics proceeded from this conception of probability. Kemble, who proceeded from the Gibbsian "assemblages", confessed that the "concept of probability had no precise operational meaning"(Kemble, 1937, p.53). For him, probability was a theoretical concept. However, Kemble was close to the frequency interpretation, for he suggested that the concept of probability is meaningful when we have a "large number of experiments in which configuration and other properties of the system are observed". According to Kemble, probability was approximately defined as a "fraction of the whole numbers of cases" where the measurement results lay in the range under consideration.

Like Kemble, Nikol’skii proceeded from a theoretical concept of probability.

Mandelstam was not only a proponent of the frequency theory of probability, he was a life-long fried of the founder of this theory. Mandelstam met R.von Mises in Strasbourg and kept in touch even when they stranded on opposite sides of the border in later years (Pechenkin, 1999).

Mandelstam and his disciples enthusiastically greeted von Mises' book Wahrscheinlichkeit, Statistik, und Wahrheit (1928). Mandelstam contributed to the rapid publication of its Russian translation (Mises, 1930), and he and his disciples inspired its discussions of it at seminars and in journals in the USSR. In Mandelstam’s Lectures on Quantum Mechanics one can find a typical von Mises’ statement: "Wave mechanics is a statistical theory. However, to speak of statistics and probability is possible if there is a set of elements, to which the statistics refers." (Mandelstam, 1950, p.332). In essence, it is a quotation from Wahrscheinlichkeit, Statistik, und Wahrheit (Mises, 1930, p.15).

In the stile of their Western colleagues Mandelstam and his disciples highly appreciated von Mises’ elaboration on the objective conception of probability (in contrast to classical Laplacian subjectivist probability). B.M.Hessen, Mandelstam’s graduate student, tended to develop a "materialistic version" of von Mises’ conception. From the evidence of the Archive of Russian Academy of Sciences, in 1926-1928 Mandelstam advised B.M.Hessen's studies in the philosophy of probability and statistics at the Communist Academy. In fact, Mandelstam supported Hessen's planned trip to Germany to study probability under von Mises. However, Mandelstam himself did not resort to a materialist rhetoric.

3.The emphasis on statistical (collective) measurement. The orthodox treatment of the reduction of the wave packet (section 6) is conditioned by the believe that a single act of observation is of cognitive importance, that "a single picture can serve as evidence for a new entity or effect" (Galison, 1997, p.20). Heisenberg, Pauli, and Dirac illustrated their approach to the reduction of the wave packet by conducting thought experiments in the style of the "image tradition," which was at its early stage at that time. Their experiments (the diffraction of electrons, ?-microscope) produced images of microparticles and microprocesses restricted to wave-particle dualism or complementarity.

To this day the "image tradition" remains influential in explanations of the reduction of the wave packet. Braginskii and Khalili illustrate the reduction by referring to Pavel Cherenkov’s discovery: "he detected individual photons with his eyes" (Braginsky, Khalili, 1992, p.390).

Popper, Margenau, Kemble, Nikol’skii, and Mandelstam expressed a hostility to "images" and "pictures". Arguing against the orthodox concept of the reduction of the wave packet, they gave preference to the statistical (collective) measurement and tended to neglect the individual (single) one.

According to Popper, the objectivity of scientific statements lies in the fact that they can be intersubjectively tested. This means that they can be tested against "reproducible effects" which are "events" rather than "occurrences". The term "event" denotes what may be typical or universal about an occurrence.

The statistical interpretation of quantum mechanics is objective, as far as it allows us to regard this theory as intersubjectively testable.

Although Margenau’s philosophy of science drastically differred from that of Popper, he also emphasize uniformity of experience. In agreement with Popper, Margenau stated that a single measurement could not give objective information about the quantum state. Together with other proponents of the statistical interpretation, he insisted that only statistical (ensemble) treatment of measurement could provide an objective basis for quantum mechanics. The train of thought was as follows: Each single experiment disturbs a microsystem. The Copenhagen authors are correct in emphasizing this peculiarity of measurement in the realm of microphysics. However, the statistical description of measurement "permitted us to forget about these difficulties completely by incorporating a suitable compensation mechanism in its fundamental structure" (Margenau, 1937, p.364).

Kemble differed from Margenau by postulating that two exact successive individual measurements must give the same result. However, he also wrote that from the point of view of empirical verification of quantum mechanics, "statistical experiments are of primary concern" (Kemble, 1937, p.319). One finds Margenau’s style of explanation, that an ensemble of measurements guarantees an objective description in quantum mechanics (Kemble, 1937, p.53).

Mandelstam’s preference for collective measurement was connected with his specific approach to physics. This approach can be called "oscillatory". He clearly expressed this approach in his final 1944 Lectures on Some Issues in the Theory of Oscillations, a cycle of lectures which was left unfinished. "It is clear", he said, "that periodic, repeatable processes are easier to study than those which occur only once. In general, I think that phenomena that are non-repeatable in principle, that occur only once, cannot be a subject of study. Meteorology is in a bad state now because its phenomena contain no periodicity" (Mandelstam, 1950, pp. 437-438).

This approach is implicitly present in his earlier lectures and articles. Mandelstam repeatedly spoke and wrote about the "oscillatory mutual assistance" between different fields of physics. Mandelstam was inclined to admit that the theory of oscillations provided a universal language of physics, or, as he said, an "international language" of physics which also used the "national languages" (he meant languages of disciplines) of acoustics, optics, atomic physics, etc. He argued for this by emphasizing that the theory of oscillations was being constituted by studying regular, repeatable phenomena which provided the grounds for any scientific description.

To explain his argument, Mandelstam cited the English philosopher A.N.Whitehead, who wrote that the rise of theoretical physics was attained by applying the concept of periodicity to different issues (incidentally, Whitehead was the only philosopher who Mandelstam referred to in his published writings). Mandelstam said that he did not want to go as far as Whitehead did and thought that physics was concerned not only with periodical but with repeatable phenomena in general. But namely repeatable phenomena are under study in the theory of oscillations.

Apart from the "oscillatory approach", Mandelstam came out with the "statistical approach" to physics. For him, every physical theory was statistical. The reason was the same: every theory must be confirmed by regular, repeatable phenomena (Mandelstam, 1950, p.355).

Nikol’skii regarded as a weak point of quantum mechanics that this theory could not provide the description of an individual process. (see sec. 3). However, he wrote on individual processes in a vague and philosophical stile which conflicts with his elaborate treatment of quantum mechanics as a theory of ensembles. Moreover, as we have seen, Nikol’skii objected against Heisenberg’s thought experiments with an individual electron. He claimed that such experiments were based on "an erroneous transfer of the characteristics of an ensemble to the individual process".

In the stile of Margenau and Kemble Nikol’skii took ensembles as a vehicle of scientific objectivity. Let us recall section 3. According to Nikol’skii, a microscopic system is included into the macroscopic body which behavior is classical. However, the "determination of a quantum part by the classical mechanism" is generally speaking uncertain. "However it is possible to avoid this uncertainty resulting from the use of the classical means in quantum realm. To do this the problem must be set in a statistical way. A statistical treatment does not imply an elimination of uncertainty. But it is a method to describe quantum processes as the objective reality in spite of the uncertainty" (Nikol’skii, 1937, p.554).

In his 1940 book Nikol’skii directly referred to Kemble’s objective states. By formulating quantum mechanics as a theory of an individual atomic system, we inevitably come to the conclusion that the "wave function is a notebook of an observer". The quantum ensembles allow us to restore objectivity (Nikol’skii, 1940, p. 150).

By reviewing Kemble’s, Nikol’skii, and Mandelstam’s excursion to experimental physics and Kemble’s and Mandelstam’s experimental writings it is not difficult to comprehend that they belong to the classical tradition of macroscopic experimentation.

So, Popper, Margenau, Kemble, Nikol’skii, and Mandelstam had much in common in their approach to measurement and experiment. All of them were affected by (in P.Galison’s terms) the culture of macroscopic experimentation which emphasized repeatable (collective) measurement (Galison, 1987). In his 1997 book Galison sees the development of this culture in the "logic tradition" which is competitive to the "image tradition". This is a tradition to "sacrifice the details of the one for the stability of many" (Galison, 1997, p.20).

4. Struggle for objectivity. The preceding subsection shows that struggle for the ensemble interpretation of a quantum state have been strongly connected with the struggle for objectivity. Here I shall only describe the different contexts of the struggle. Before War II Popper was not an articulated realist. His attack on the Copenhagen interpretation was basically an attack against the psychologism (and hence against subjectivism) in science. Margenau was very far from scientific realism. However, by attacking the Copenhagen approach he also struggled against psychologism.

As physicists Kemble, Nikol’skii, and Mandelstam respected scientific objectivity as it was. As a matter of fact they did not need any additional motivation for their struggle for objectivity. Nevertheless their philosophical positions were different. Kemble, the former P.Bridgman’s student, sometimes develops his interpretation in a subjectivist tenor. Nikolsk’ii was an alleged materialist who used materialistic rhetoric. Mandelstam, who was educated in Germany and communicated with von Mises, sympathized with positivism. Nevertheless, as was noted, he developed his version of operationalism consistent with realism. Moreover, sometimes he allowed himself to speculate about deterministic metaphysics. We mentioned that Mandelstam joined von Neumann's discussion of the completeness of quantum theory. This does not mean that Mandelstam shared von Neumann's philosophy of causality. He tended to avoid the indeterminism which von Neumann proclaimed. Thus von Neumann wrote that "es gibt gegenwartig keinen Anlass und keine Entschuldigung dafur, von der Kausalität in der Natur zu reden" (von Neumann, 1932, s.167). Mandelstam said however the following:

They say that von Neumann demonstrated that the construction of the theory on the base of determinism is impossible. I think that such phrases say next to nothing (Mandelstam, 1950a, p.403)

If they sometimes say that von Neumann demonstrated that the causal theory of the atom phenomena is impossible then this is not the case (Mandelstam, 1950a, p.414).

So, in struggling for objectivity the five scientists expressed themselves in different ways and had different motivations.


In the previous section we learned something about the backgrounds of the statistical interpretations proposed by five people in the thirties. Kemble, Mandelstam, and partially Margenau were influenced by operationalism. Popper, Margenau, and Mandelstam shared von Mises’ frequency concept of probability. All five made emphasis on statistical (collective) measurement. All five struggled for objectivity.

Nevertheless there was no isomorphism between the "backgrounds" and the interpretative ideas. Both Kemble and Mandelstam combined the Copenhagen and statistical attitudes in their interpretations of quantum theory. They, however, disagreed on the concept of probability. Kemble’s book sometimes shows subjectivism which was hostile to Mandelstam. Kemble and Nikol’skii arrived at very similar conclusions on the nature of probability and of quantum mechanics. However, Nikol’skii showed hard materialism from which Kemble was very far. Both Mandelstam and Popper were inclined to scientific realism. Mandelstam, however, would not agree with Popper’s PIV interpretation. Margenau and Popper agreed on the von Mises’ conception of probability, but they disagreed on the interpretation of uncertainty relations, on the conception of the reduction of the wave packet and on the EPR argument.

Therefore, the "backgrounds" only partially explain why a transnational scientific community which put forward the statistical interpretation arose in the 1930s. Metaphysical explanations -- like Hegel’s "spirit is its own community" (Hegel, p.778) -- are unlikely helpful. We must take as a historical fact that in the thirties the similar ideas were pushed in the different countries and on the different grounds. Scientists, some of whom did not even know about the existence of each other, proposed ideas which supplemented and clarified each other. These ideas had the future (see a discussion of the postwar II debates in (Home&Whitaker, 1992)).

Let me make two comments more. With respect to the problems of quantum physics the ‘movement" for the statistical interpretation of quantum mechanics was a periphery. The physicists, who much contributed to quantum theory, proceeded from the Copenhagen interpretation. The Soviet physicists Tamm and Landau provided a good example.

From a physical point of view Mandelstam’s interpretation is most interesting. This interpretation is connected with his theory of indirect measurement and it should be regarded as a base of his article with Tamm on the energy-time uncertainty relation.

Appendix. Konstantin Vjacheslavivich Nikol’skii was born in 1905. He graduated from the North-Caucasus State University in Nal’chik 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 under Fock his Doctor Science Dissertation at the Institute of mathematics and mechanics of the Leningrad University. The extended text of this dissertation was published as a book (Nikol’skii, 1934). Since 1936 he had worked at the Lebedev Physics Institute of the Academy of Sciences. Judging by the Archives of this Institute and S.M.Rytov’s comments (interviewed by the present author in 1992) his work on the foundations of quantum mechanics was supported by Director Academician S.I.Vavilov. He had been suffering from a mental disease resulting in his retirement in 1946. In 1978 he was still alive: he asked the administration of the Physics Institution to give him a notification that the institute was his employer.

Nikol’skii was the first physicist who proclaimed the statistical interpretation of quantum mechanics in the USSR. He was also the first who used philosophical and political rhetoric in the debates on the interpretation of quantum. Having been criticized by V.A.Fock (1937) Nikol’skii claimed that N.Bohr's conception of quantum mechanics was incompatible with "progressive physics" and represented idealism, namely Machism in physics (Nikol’skii, 1937, p.555). K.V.Nikol’skii called Tamm, Fock, Landau, and M.Bronstein the "Soviet Branch of the Copenhagen School" (Nikoskii, 1937, p.555). He wrote that the "Soviet Branch of the Copenhagen School" came to positivism (Nikol’skii, 1938, p.147). One can meet such a claim in the Western literature (for example, in Popper’s Logik der Forschung). However, in the USSR such a claim tended to be a kind of political accusation.

In spite of his hard attack on the Copenhagen faction Nikol’skii had not been supported by a group which opposed to the fundamentals of quantum mechanics. This was a professor of the Physics Faculty of the Moscow University A.K.Timerizev, the outstanding specialist in electric engineering V.F.Mitkevich, and some other physicists distant from the modern trends in theoretical physics (see in Delakorov, 1982, p.318).

Although Leonid Isaakovich Mandelstam (1879-1944) was not a great one for quantum mechanics, he was an outstanding physicist contributing to radio-physics, the theory of oscillations and optics.

Mandelstam did much to ensure that the Soviet scientific community would adopt the non-classic theories. As Tamm, who regarded Mandelstam as his mentor, wrote: "Leonid Isaakovich was a pioneer in the theory of oscillations and in several areas of optics but in the area of quantum mechanics his primary goal was to provide maximal consistency in the fundamental concepts and principles of the theory"(Tamm, 1991, p.273).

Judging by his letters and his closest friends' recollections, Mandelstam considered it one of his important tasks to elaborate his views concerning the foundations of quantum mechanics. In 1928 he wrote an article "On the theory of the Schrödinger equation" with his graduate student M. A. Leontovich in which they discussed the behavior of a particle in the presence of a potential barrier (in subsequent terminology) and a potential well. In Mandelstam's biography his closest friends and collaborators wrote that after the publication of this article, Mandelstam "did not write any new articles on quantum mechanics for a considerable period of time although he thought very much about it and touched upon it in his lectures and seminars" (Mandelstam, 1948, p.52). Mandelstam struggled for clear understanding of the foundations of quantum mechanics in that period. This resulted in his 1939 Lectures and his article with I.E.Tamm on the energy-time uncertainty relation.

Although Mandelstam never maintained any official position he was very authoritative and influential as a leader of the powerful scientific community in Soviet physics. His attitude toward the foundations of science extended over a broad group of scholars. To this day some physicists of the older generation refer to Mandelstam's Lectures as an authoritative explanation of the Bohr-Einstein controversy. A.D.Sakharov, for example, wrote in his Recollections that, as a result of his first visit to his teacher I.E.Tamm, he received two books and a manuscript for "essential reading". These books were W.Pauli’s The Theory of Relativity and Quantum Mechanics and the manuscript was L..I.Mandelstam’s Lectures on Quantum Mechanics. A.D.Sakharov believed that these Lectures "had been magnificently and profoundly written" (Sakharov, 1996, vol.1, p.106).


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