THE EARLY HISTORY OF QUANTUM STATISTICS
Determining the origins of a revolution is always a tricky business. Most commentators take Planck's famous paper of 1900 1 as marking the birth of quantum theory, although Dorling, in his presentation of the history of quantum mechanics, 2 suggested that Planck failed to understand what he had initiated and Kuhn 3 subsequently argued that the revolution actually began with Einstein's explanation of the photoelectric effect, since he took the new 'quanta' as more than mere calculational devices. In either case, Schrödinger's claim above still stands, as both Planck's and Einstein's work were intimately bound up with the application of statistics.
Like Boltzmann, Planck had grappled with the physical basis of the Second Law of Thermodynamics, but rejected Boltzmann's statistical mechanics in favour of an approach based on the foundations of electromagnetism. 4 Although this turned out to be less than fruitful in the thermodynamical context, it placed him in a excellent position from which to tackle a critical problem in late-nineteenth-century physics. This was the problem of 'black-body radiation': a 'black body' is a perfect (and hence idealized) absorber and emitter and the radiation emitted depends, not on the nature of the body, but only on its temperature. Consideration of a black body, whether theoretical or experimental (in the form of close physical approximations to this idealization), then allowed physicists to study the nature of electromagnetic radiation and in particular, the distribution of the energy over the various frequencies of the radiation. Unfortunately, however, the observed distribution did not mesh with the theoretical expressions obtained from classical electromagnetic theory. Through a process of what might broadly be described as 'phenomenological construction', Planck obtained a radiation law which was empirically adequate. 5 In order to ground this law in a micro-physical description, in an act
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of desperation, he turned to Boltzmann's Combinatorial Approach, applied now to the oscillators of the black body. 6 However, the details of the statistics used by Planck turned out to be radically and fundamentally different from Boltzmann's.
The crucial step in Planck's derivation of his law came from thinking about how a given total energy could be distributed over N linear oscillators, all vibrating with frequency ν. He divided this energy into a finite number of energy elements, P, which were then distributed over the oscillators. The number of ways of doing this, that is, the number of 'complexions', is 454h76e given by 7
It is not immediately clear how Planck arrived at this expression although Rosenfeld has suggested that he discovered it by working backwards from his radiation law. 8
Planck himself considered the derivation of his law to be based on only two theorems: the well-established relation between the energy density of the radiation and the average energy of the oscillators; and Boltzmann's relationship between the entropy of a system and the logarithm of the total number of possible complexions (as presented in Chapter 2). Planck split the latter into two parts:
(1) The entropy of the system in a given state is proportional to the logarithm of the probability of that state and (2) The probability of any state is proportional to the number of corresponding complexions, or, in other words, any definite complexion is equally as probable as any other complexion. 9
The first part simply expresses Boltzmann's crucial insight that, as Planck subsequently put it in his Nobel Prize acceptance speech, 'entropy is a measure for physical probability'. The second was regarded by Planck as the core of his whole theory and the proof of this part he took to rest ultimately on empirical grounds. He considered it to be a more elaborate and detailed expression of
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the hypothesis that the energy of the radiation is randomly distributed over the various partial vibrations of the oscillators.
Thus the statistical component of Planck's work followed
Boltzmann's Combinatorial Approach in broad outline but differed from it in
significant respects. First of all, whereas Boltzmann let the size of his
energy elements decrease to zero, as we have seen, Planck did not, but left
them as finite and related to the frequency via the famous expression 
= hν.
10 Secondly, and more importantly from our point of view,
Planck's combinatorial expression above is very different from Boltzmann's.
Planck considered, at least implicitly, only the number of energy quanta
assigned to each oscillator and not which quanta were possessed by which
oscillators. Thus the numerator of his expression gives simply the total number
of ways of arranging the quanta and oscillators, regarded as N + P
− 1 distinct elements (we shall discuss the significance of the −1
when we come to consider Ehrenfest's analysis of this result).
The division by P! implies that permutations of the quanta are not regarded as giving rise to new, countably different arrangements. As we have already indicated, it was that which led to the conclusion that the quanta should be regarded as not only indistinguishable but also unlabelled and devoid of individuality. 11 Thus whereas Boltzmann considered the distribution of indistinguishable but individual atoms over energy states or cells in phase space, Planck was effectively concerned with the distribution of indistinguishable but non-individual quanta over oscillators. 12 The non-classical nature of Planck's expression and the questions surrounding its interpretation became a focus of early criticism, particularly from Ehrenfest, as we shall see, who undertook to expose the conceptual foundations of the new quantum theory.
Finally, we recall that Planck's second sub-theorem above, regarding the probability of a state, is simply a restatement of the crucial assumption that equal a priori weights should be assigned to the various complexions. It was again Ehrenfest who subjected this assumption to critical examination and who first discussed the way in which the weighting assignments of classical statistical mechanics had to be revised in the light of the new physics.
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Ehrenfest's first thoughts on this subject were set down in his 1905 paper in which he expressed his puzzlement over the exact nature of the relationship between Planck's work and Boltzmann's Combinatorial Approach. 13 In particular, he noted that Planck's choice of those states to be given equal a priori probabilities was different from Boltzmann's. This difference arises from the different choice of events to be counted: whereas Boltzmann attached equal a priori probabilities to each region of phase space, Planck assigned them to each energy distribution. 14 Ehrenfest was well aware that a definite hypothesis concerning such assignments was crucial to the Combinatorial Approach and six years later he presented a generalized weight function suitable for the new statistics. 15
1905 also saw the next major advance in the history of quantum physics, with the publication of Einstein's famous paper in which he proposed the hypothesis of independent energy quanta in radiation. 16 Interestingly, the central argument of this paper is based, not on Planck's law, but on the classical expression for the distribution of radiation energy (Wien's law, to be specific) which is strictly empirically inadequate. Nevertheless, Einstein was able to obtain the correct results because he treated his quanta as Maxwell-Boltzmann particles instead of-what came to be known as-bosons. 17 It is precisely in the limit of the classical expression that the latter behave like the former. It turns out that Einstein's result can also be derived using the correct distribution law-namely Planck's-and the correct quantum statistics-namely, Bose-Einstein, as one would nowadays expect. 18 In other words, if the quanta are treated as statistically independent, an empirically inadequate expression is obtained. The fact that Planck's law is actually the correct one therefore suggested that the light quanta were not statistically independent. This point, and the related question of the nature of the interdependence between quanta, was subsequently the subject of much discussion by Ehrenfest and others, as we shall see.
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By the following year Einstein had realized that Planck's theory pre-supposed the existence of light quanta 19 and in 1907 he noted that if this theory were correct, then energy values other than those given by hν are not accessible to the oscillators. 20 His analysis of energy fluctuations in black body radiation in 1909 produced an expression for the mean square energy fluctuation which was the sum of two terms. 21 The classical wave theory of light would give the second term, associated with wave phenomena, only, and the first, the particle term, corresponded to " . independently moving pointlike quanta with energy hν" 22
This is the first intimation of 'wave-particle duality' but it is interesting that Einstein still regarded the quanta as statistically independent here. Their non-classical interdependence could then be explained by referring to their wave-like aspect. Regarded as crests in a system of waves, quanta would become properties of the whole system and the probability of a quantum existing in a particular state would then be dependent on the other quanta in the system. As we shall see, Einstein subsequently justified the counting procedure that has come to be associated with his name and that of Bose, by referring to de Broglie's suggestion that particles are analogous to crests in a system of waves; a suggestion that was subsequently taken up by Schrödinger and elaborated into his so-called 'wave mechanics'.
The second point to note concerning Einstein's statement
above is that he referred to the light quanta as point-like. It is clear that
by 1909 he was thinking of quanta as particles. 23 This view
was reinforced by his work on spontaneous and induced radiative transitions in
the course of which he derived the result that a light quantum carries momentum
24
Thus there emerged the concept of a photon as a particle carrying both
energy and momentum.
Planck went on to present a rearranged version of his 1900 derivationand introduced an alternative approach involving a phase-space description of the equiprobable regions accessible to an oscillator. These regions were elliptical rings of area h on the energy hypersurface, within which lay the oscillators, and consideration of the distribution of the oscillators over these areas then allowed him to compute directly the number of complexions corresponding to a given state. This formed the basis for a quantized statistical treatment of the distribution of entities-particles or resonators-over certain regions of phase space, which was much more akin to the classical combinatorial procedure. Quantal results were ensured, however, by the fixed size given to these regions as determined by h. 26
This method was subsequently generalized and a natural unit for phase extension of h for each degree of freedom established. This meant that each cell in phase space could be given the volume h, thus giving an absolute value for the entropy. 27 These results implied that the classical assignment of equal a priori weights to equal volumes of phase space had to be abandoned and only certain regions of this space given non-zero weight. Thus for Planck's oscillators these regions were those particular ellipses of constant energy whose enclosed areas were integral multiples of h. As we have said, it was Ehrenfest who was virtually alone in recognizing that the basis of Boltzmann's proof of the Second Law had been lost in principle as soon as physics had followed Planck in abandoning the classical weighting assignment 28 and he went on to present a suitably generalized weight function for quantum statistics which did not rely on the notion of an oscillator. 29 This was part of a move to a more general conception of quantum statistics which was applicable to both light quanta and material particles.
Returning to Planck's distribution law, Ehrenfest showed that it could be derived without recourse to oscillators as well and also that the entropy function that was obtained was common to Boltzmann and Planck. 30 He thus concluded that the source of the difference between classical and quantum statistical mechanics must lie, not in the entropy equations themselves but in the imposed constraints, that is, in the quantization requirement. Subsequently, he examined the difference between Planck's quantized oscillators and Einstein's light quanta. The prevailing opinion at the time was that the main difference lay in the separate existence of quanta in empty space. However, Ehrenfest pointed out that the statistical independence of quanta was another source of contention, with Einstein affirming and Planck denying it. As he noted, Planck had not assumed this in his derivation and it led, in fact, to Wien's law, which Einstein had used. This problem of the interdependent statistical behaviour of Planck's quanta clearly troubled Ehrenfest and not only was it
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an issue to which he himself would return, as we shall see, but it also lay at the centre of an early debate over the individuality of quanta.
Like Ehrenfest, Natanson was also interested in the differences between Planck's and Einstein's conceptions. In particular he examined the assumptions underlying Planck's consideration of the distribution of energy elements over 'receptacles', and identified three different 'modes of distribution', depending on whether the quanta or receptacles or both were regarded as individuals. 31 Thus he used the term 'mode of distribution' for the correlation of energy elements with receptacles in which the distribution is characterized only by the number of receptacles containing a given number of quanta. In this case no account is taken of the possible individuality (or 'identifiability') of receptacles or energy elements. However, if the former are regarded as individuals then every 'mode of distribution' branches out into a number of 'modes of collocation', which specify the number of energy elements in each individual receptacle. This, he argued, corresponded to Planck's microstate. If energy elements are also regarded as individuals then each 'mode of collocation' splits up into a number of 'modes of association', which associate individual energy elements with individual receptacles. This corresponds to Boltzmann's notion of a complexion. 32
As Natanson noted, the thermodynamic probability of a given mode of distribution depends on whether all modes of collocation or all modes of association are regarded as equally probable. Whereas Einstein took the latter view, Planck took the former and thus, according to Natanson, implicitly regarded his receptacles, the oscillators, as individuals but not his energy elements, the quanta. The odd terminology is an attempt to distinguish the meaning of the term 'complexion' as used in quantum and classical statistics. In the former, a complexion specifies only how many elements are in each receptacle and this is called a mode of collocation. A permutation of two elements between receptacles does not lead to a new mode of collocation. In the latter, a complexion specifies which individual element is in which individual receptacle, and this is termed a mode of association. A permutation of elements between receptacles does lead to a different mode of association.
Joffe also attempted to modify Einstein's idea of quanta so as to reconcile it with Planck's law, 33 and a controversy subsequently arose between Krutkow
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and Wolfke over just this question. Krutkow, following Ehrenfest, demonstrated that if classical statistics are used, then the assumption of independent quanta leads to Wien's law. Planck's law can only be obtained, he argued, if this assumption is abandoned. Wolfke, however, argued that one must distinguish between two meanings of the word 'independent'. If the quanta exist independently then, he claimed, Planck's law is obtained, but if, in addition, the quanta are regarded as spatially independent then Wien's law will result. 34 Thus he concluded that Einstein had implicitly assumed the quanta to both exist independently and to be spatially independent whereas Planck has assumed only the former, suggesting that some sort of spatial correlation had to exist between the quanta. It is in these debates and the work of Ehrenfest that we see the first explorations of the metaphysical implications of quantum statistics. As we shall see, these concerns regarding individuality, re-identifiability and independence were to arise again and again in the development of the field.
Further light was shed on these questions by an important paper of Ehrenfest and Kammerlingh Onnes, 35 which gave an intuitive way of understanding Planck's combinatorial formula. Ehrenfest had not been satisfied with the standard derivations given in the textbooks as they appealed to proof by induction and offered no insight into the peculiar structure of the result. Thus he set out to provide his own.
The problem he and Kammerlingh Onnes considered was to find the number of ways in which P objects could be placed in N containers where only the number of objects in each container is of importance. Two distributions were called identical when corresponding containers, that is, oscillators, in each distribution possessed the same number of objects, that is, energy elements. A distribution was represented symbolically thus
where the Π's are the 'fixed boundaries', represents
the energy units, and the ο's separate these units in successive
oscillators, numbered from left to right. 36 With general
values of N and P the symbol will contain P
times and ο N−1 times, and they asked how many different
symbols for the distribution could
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be formed from the given number of and
ο. The answer was
This is derived as follows. There are N + P
− 1 quantities and
ο which, if regarded as distinguishable, can be arranged in (N + P
− 1)! ways between the ends Π. However, the s
and οs can be permuted among themselves P! and (N −
1)! times respectively and two distributions differing only by such
permutations are identical. Therefore as there are P!(N −
1)! symbols corresponding to these identical distributions one must divide (N
+ P − 1)! by P!(N − 1)! to obtain the number
of distinct distributions. The result is Planck's combinatorial formula above.
This derivation allowed some important conclusions to be
drawn regarding the status of Planck's quanta and their relationship with
Einstein's. It confirmed Ehrenfest in his old opinion that the energy elements ,
and hence also Planck's quanta, were merely a formal device, imbued with no
more physical significance than the divider elements. 37 The
failure to realize this, he believed, had led to the mistaken interpretation
that these quanta were mutually independent and identical to Einstein's. The
difference between the two was demonstrated by comparing their different statistical
behaviour. It was noted that the number of ways of distributing a number P
of Einstein's light quanta, regarded as independent, over N 1
and then over N 2 cells in space stand to each other in the
ratio
If Planck's quanta were also regarded as mutually independent, then in passing from N 1 to N 2 oscillators, the number of possible distributions would increase in the same ratio. This is clearly not true, as Planck's formula gives
According to Ehrenfest and Kammerlingh Onnes, the explanation was simple: Einstein's quanta could be regarded as existing independently of each other, whereas Planck's could not 38 and were no more than a formal device:
The real object which is counted remains the number of all
the different distributions of N resonators over the energy grades 0, , 2,.
with a given total P. 39
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Finally, they gave an example of the quantal reduction in the number of possible arrangements which illustrated the classical nature of Einstein's quanta and the non-classical behaviour of Planck's.
The claim, then, was that if Planck's quanta were to be considered as statistically independent then the traditional classical counting procedure should be applied rather than Planck's own. 40 Since the number of distributions obtained by Planck differed from that obtained by treating the quanta as independent the entropy change in any appropriately specified process would also have to be different. It is ironic that although the above derivation is now almost universally given as an intuitive underpinning of Planck's procedure, its original purpose was to argue that this procedure cannot be correct if the quanta are to be independent. This was also the substance of Ehrenfest's criticism of Einstein's 1924 paper proposing Bose-Einstein statistics, as we shall see.
It follows, then, in a reciprocal manner, that if Planck's procedure is correct then the quanta cannot be regarded as statistically independent and must exhibit some sort of non-classical correlation. Furthermore, the fact that a permutation of Einstein's quanta gives a new distribution whereas a permutation of Planck's does not implies that the former can be regarded as individuals whereas the latter cannot. Klein has remarked that the blurring of the concept of particle when it comes to light quanta was already implicit in Einstein's results for energy and momentum fluctuations in black body radiation. 41 These results furthered the identification between Einstein's and Planck's quanta because they effectively allowed Einstein to explain quantum phenomena in terms of the wave-like aspect associated with his, individual, quanta, whereas Planck effectively had to appeal to the non-classical correlations between his, non-individual, quanta.
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