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Studies in History and Philosophy of Modern Physics 40 (2009) 383-394 " Studies in History and Philosophy of Modern Physics A note on the prehistory of indistinguishable particles Daniela Monaldi Max Planck Institute for the History of Science, Berlin, Gennany ARTICLE INFO Article history: Received 21 April 2009 Received in revised form 5 September 2009 Keywords: Indistinguishability Base-Einstein statistics Light quanta Ideal gas Einstein Planck ABSTRACT In modern terms, quantum statistics differs from classical statistics for the indistinguishability of its elementary entities. An historical investigation of the emergence of Bose-Einstein statistics, however, shows that quantum statistics was initially interpreted as a statistics of non-independence, for it extended to gas particles the statistical correlation that was a long-recognized characteristic of light quanta. At the same time, the development of a quantum-statistical theory of the ideal gas was riddled with the question of the statistical significance of the exchange symmetry of a system of equal particles. Indistinguishability combines exchange symmetry and statistical correlation, and relates them to the loss of identity of particles in quantum mechanics. It is instructive, however, not to conflate these properties when analysing the historical emergence of quantum statistics. The statistical correlation of light quanta and the exchange symmetry of gas molecules remained two separate problems even though quantum gas theory and Bose-Einstein statistics were born from gas-radiation analogies in statistical theory. © 2009 Elsevier Ltd. All rights reserved. When citing this paper, please use the full journal title Studies in History and Philosophy of Modem Physics 1. Introduction The indistinguishability of particles is the modern interpreta- tion of the difference between quantum statistics and classical statistical mechanics. A large body of historical scholarship has dealt with the emergence of the notion of indistinguishable particles in the first decades of the twentieth century (Bergia, 1987; Darrigol, 1991 ; Delbruck, 1980; Kastler, 1983 ; Pesic, 1991 ). The resulting picture is that, although the idea became estab- lished with the formulation of the quantum mechanics of multi- particle systems, its roots were already present in certain earlier applications of statistical mechanics. One root can be traced to Max Planck's law of thermal radiation, which, when interpreted in terms of light quanta, revealed the existence of statistical correlations between these entities. A second root reaches back to a modification of the classical calculation of the entropy of the monatomic ideal gas, namely, the subtraction of a term depending on the number of possible permutations of equal particles, which is still today frequently justified as a correction required by the symmetry of multi-particle systems under the exchange of equal particles. The aim of this paper is to suggest that, if in hindsight we can identify these two issues as two roots of indistinguish- ability, for the purpose of reconstructing the history of quantum statistics it is worth noticing that the roots are not the plant. E-maii address:[email protected] In the modern interpretation of quantum statistics, exchange symmetry and statistical correlation are two facets of a single property, the indistinguishability of particles. Historically, how- ever, they were two separate problems, differently formulated and belonging in different theoretical areas. The problem of statistical correlations concerned the hypothesis of light quanta in radiation theory, while the problem of exchange symmetry arose in the quantum statistical theory of the ideal gas. The first instance of quantum statistics, the Base-Einstein statistics, originated from Albert Einstein's application to the ideal gas of a procedure for the statistical calculation of entropy that Satyendra Nath Base had invented for radiation. This paper revisits the two roots of indistinguishability up to the birth of Base-Einstein statistics. I will examine first the issue of statistical correlation in radiation theory and then the issue of exchange symmetry in gas theory, paying attention to the ways in which they related to one another in the context of evolving gas-radiation analogies. My goal is to lay the basis for an investigation of the historical process that transformed older concepts into the notion of indistinguishable particles. 2. Boltzmann's statistics and quantum statistics It may be useful to recall that the difference between quantum statistics and classical statistical mechanics lies in the rule for counting the configurations of a system (microstates, in the

A Note on the Prehistory of Indistinguishable Particles

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Page 1: A Note on the Prehistory of Indistinguishable Particles

Studies in History and Philosophy of Modern Physics 40 (2009) 383-394

"Studies in History and Philosophy

of Modern Physics

A note on the prehistory of indistinguishable particles

Daniela MonaldiMax PlanckInstitute for the History of Science, Berlin,Gennany

ARTICLE INFO

Article history:Received 21 April 2009Received in revised form5 September 2009

Keywords:IndistinguishabilityBase-Einstein statisticsLight quantaIdeal gasEinsteinPlanck

ABSTRACT

In modern terms, quantum statistics differs from classical statistics for the indistinguishability of itselementary entities. An historical investigation of the emergence of Bose-Einstein statistics, however,shows that quantum statistics was initially interpreted as a statistics of non-independence, for itextended to gas particles the statistical correlation that was a long-recognized characteristic of lightquanta. At the same time, the development of a quantum-statistical theory of the ideal gas was riddledwith the question of the statistical significance of the exchange symmetry of a system of equal particles.Indistinguishability combines exchange symmetry and statistical correlation, and relates them to theloss of identity of particles in quantum mechanics. It is instructive, however, not to conflate theseproperties when analysing the historical emergence of quantum statistics. The statistical correlation oflight quanta and the exchange symmetry of gas molecules remained two separate problems eventhough quantum gas theory and Bose-Einstein statistics were born from gas-radiation analogies instatistical theory.

© 2009 Elsevier Ltd. All rights reserved.

When citing this paper, please use the full journal title Studies in History and Philosophy of Modem Physics

1. Introduction

The indistinguishability of particles is the modern interpreta­tion of the difference between quantum statistics and classicalstatistical mechanics. A large body of historical scholarship hasdealt with the emergence of the notion of indistinguishableparticles in the first decades of the twentieth century (Bergia,1987; Darrigol, 1991 ; Delbruck, 1980; Kastler, 1983 ; Pesic, 1991 ).The resulting picture is that, although the idea became estab­lished with the formulation of the quantum mechanics of multi­particle systems, its roots were already present in certain earlierapplications of statistical mechanics. One root can be traced toMax Planck's law of thermal radiation, which, when interpreted interms of light quanta, revealed the existence of statisticalcorrelations between these entities. A second root reaches backto a modification of the classical calculation of the entropy of themonatomic ideal gas, namely, the subtraction of a term dependingon the number of possible permutations of equal particles, whichis still today frequently justified as a correction required by thesymmetry of multi-particle systems under the exchange of equalparticles. The aim of this paper is to suggest that, if in hindsightwe can identify these two issues as two roots of indistinguish­ability, for the purpose of reconstructing the history of quantumstatistics it is worth noticing that the roots are not the plant.

E-maiiaddress:[email protected]

In the modern interpretation of quantum statistics, exchangesymmetry and statistical correlation are two facets of a singleproperty, the indistinguishability of particles. Historically, how­ever, they were two separate problems, differently formulated andbelonging in different theoretical areas. The problem of statisticalcorrelations concerned the hypothesis of light quanta in radiationtheory, while the problem of exchange symmetry arose in thequantum statistical theory of the ideal gas. The first instance ofquantum statistics, the Base-Einstein statistics, originated fromAlbert Einstein's application to the ideal gas of a procedure for thestatistical calculation of entropy that Satyendra Nath Base hadinvented for radiation. This paper revisits the two roots ofindistinguishability up to the birth of Base-Einstein statistics. Iwill examine first the issue of statistical correlation in radiationtheory and then the issue of exchange symmetry in gas theory,paying attention to the ways in which they related to one anotherin the context of evolving gas-radiation analogies. My goal is to laythe basis for an investigation of the historical process thattransformed older concepts into the notion of indistinguishableparticles.

2. Boltzmann's statistics and quantum statistics

It may be useful to recall that the difference between quantumstatistics and classical statistical mechanics lies in the rule forcounting the configurations of a system (microstates, in the

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modern terminology) in order to calculate the thermodynamicprobability (W) of an energy distribution (describing the macro-state of which the microstates are possible realizations), so thatthe entropy (S) can be evaluated through the Boltzmann principle,S=k log W. The method of calculating probability by countingconfigurations originates from the combinatorial argument thatLudwig Boltzmann formulated in 1877 to demonstrate that thestate of equilibrium of an ideal gas, characterized by the Maxwell–Boltzmann distribution, was the most probable state. Boltzmannused the artifice of approximating the energy distribution of a gasof N molecules by a set of integers, w0, w1, w2,y, wp (occupationnumbers), representing the numbers of molecules with energies0, e, 2e,y, Pe, under the fictitious assumption that the moleculescould only assume discrete energy values. He named ‘‘complex-ion’’ a configuration in which each molecule was assigned anenergy value, and surmised that the probability of an energydistribution was the number of complexions that realized it, B,divided by the total number of possible complexions, J. Thedetermination of probability was in this way reduced to a problemof combinatorial calculus. B corresponded to the number ofpermutations of N ‘‘elements’’ of which w0, w1, w2,y, wp were‘‘equal to one another’’, B=N!/(w0!w1!w2!y, wp!), a quantitythat Boltzmann called the ‘‘permutability’’ of the distribution(Boltzmann, 1877, p. 169, 170 and 176). This determination clearlyrests on the assumption of equal probability for the complexions.It can be justified as the joint probability of N equal andstatistically independent molecules: each molecule has the sameprobability of falling into an energy region of size e, regardless ofhow many other particles are in the same region. Boltzmann’scombinatorial method became the vehicle for the application ofstatistical mechanics to quantum systems, for which the discreteenergy states and the occupation numbers were no longerapproximations but literal descriptions. Although the expression‘‘quantum statistics’’ was initially used to denote these applica-tions, quantum statistics in the modern sense began when, inaddition to the quantization of energies, Boltzmann’s count ofcomplexions was replaced by a new rule that assigned the sameprobability to all the possible energy distributions, regardless oftheir permutability. It is this new counting rule that defines thestatistics of indistinguishable particles in contrast with Boltz-mann’s statistics. It leads to the Bose–Einstein distribution in thecase in which there is no restriction on the number of particlesthat can occupy the same state, and to the Fermi–Diracdistribution in the case of particles obeying the exclusionprinciple.

From a combinatorial point of view—that is, for what concernsthe counting of possible configurations of a set of elements intosubsets—the new rule dictates that all the configurationsobtained from one another by exchange of equal particles are tobe counted as one. Its physical interpretation is that two suchconfigurations do not just realize the same macrostate, as they didin the Maxwell–Boltzmann case, but also represent the samemicrostate. Configurations that assign individual particles todefinite single-particle states, though still theoretically definable,become physically meaningless. This rule is considered appro-priate to entities governed by quantum mechanics, to which nopermanent individual labels can be assigned. In contrast,Boltzmann’s rule is seen as the statistical application of whatBoltzmann, in another context, called the ‘‘first fundamentalassumption’’ of classical mechanics, namely, that material pointscan be distinguished from one another by their continuoustrajectories (Bergia, 1987, p. 226; Boltzmann, 1897, p. 230; Pais,1979, p. 893). From a statistical or, which is the same in physics,probabilistic point of view—that is, for what concerns theassignation of probabilities to configurations—the rule assignsequal probability to all the energy distributions. If translated into

single-particle probabilities, it implies that the particles are notstatistically independent: the probability for one of them tooccupy a single-particle energy state depends on the number ofother particle in the same state. Exchange symmetry andstatistical correlation merge into the notion of indistinguishableparticles that working physicists today embrace to explainquantum statistics and its startling consequences (Delbruck,1980; Ketterle, 2007).

The early history of quantum theory was in large measure ahistory of cross-fertilization between radiation theory and gastheory. It was driven by the diffusion of the statistical method,which in turn was enabled by the construction of analogiesbetween the paradigmatic model of the molecular-kinetic theoryof the ideal gas and corresponding models of other physicalsystems. The kernel of statistics was to describe a complex systemnot by tracing the individual histories of its components but bydetermining the numerical distributions of the components intheir dynamical variables, as for example Maxwell’s distributionof molecular velocities, in order to find observable quantities asaverages. The kinds of objects that corresponded through theseanalogies to the gas molecules represented, at least for thepurpose of statistical analysis, the elementary components of thesystem under consideration; for brevity, I shall refer to them as‘‘statistical entities’’. The emergence of the notion of indistin-guishable particles required as a precondition the invention of aclass of theoretical objects that performed the same role as gasmolecules in statistical analysis but differed from traditionalmolecules in that they were described by a different rule for thedetermination of probabilities, and hence did not follow theMaxwell–Boltzmann statistics. In other words, it required a newkind of statistical entities.

3. Planck’s law and the statistics of radiation

3.1. Planck’s gas-radiation analogy

The first root of indistinguishability is found in the statisticalproperties of Planck’s law of thermal radiation, un ¼ ð8pn2=c3Þ

ðhn=ðehn=kT � 1ÞÞ (where uv is the energy density of heat radiationin the frequency interval v,v+dv at equilibrium at temperature T).This law can be regarded as composed of two parts, which werederived by Planck at different times and in different ways(Darrigol, 1988, 1992b; Kuhn, 1978). Planck first calculated therelation of proportionality between the energy density of theradiation and the time average of the energy (Uv) of an idealresonator of the same frequency immersed in the radiation field,uv=(8pn2/c3)Uv. Having then found the empirically correctexpression for Uv, Uv=hv/(ehv/kT

�1), he finally succeeded inderiving it ‘‘in a deductive way’’ by adopting Boltzmann’s relationbetween entropy and probability and transforming it into thegeneral equation known as the Boltzmann principle, S=k ln W

(Boltzmann, 1877; Planck, 1900, P. 238).Planck assumed that Uv was equal to the instantaneous average

of the energy over a system of N independent copies of theresonator in equilibrium with the same radiation field. He thenproceeded to calculate it from the entropy of such a system,following closely Boltzmann’s statistical calculation of the entropyof an ideal gas of N molecules. He equated the ‘‘probability’’ of astate of energy E of his system of N resonators to the number ofcomplexions that realized the state. In his case, this number wasthe number of ways in which the resonators could be distributedover the energies 0, e, 2e,y, Pe, the total energy being E=Pe; it wastherefore the sum of the permutabilities of all the energydistributions of total energy E. To arrive at it ‘‘in a faster andeasier way’’, Planck took a shortcut offered by combinatorial

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calculus. Considering E as composed of a number P of ‘‘energyelements’’ e, one could reckon the number of complexions as the‘‘number of ways in which P energy elements can be distributedover N resonators’’ if—as Planck specified in the first edition of hisbook The Theory of Heat Radiation—‘‘no regard is given to whichenergy elements, but only to how many energy elements areassigned to a given resonator’’. This formal expedient turned thecomputation into a standard combinatorial question, to whichthe answer was ‘‘the number of possible ‘combinations withrepetition of N elements of class P’: W=(N+P�1)!/((N�1)!P!)’’(Planck, 1900, p. 240, 1906, p. 152).

At this point, Planck had to part ways with Boltzmann. ForBoltzmann, the quantity e had been only a mathematicalimplement that could be eliminated from the final result torecover the continuous energy distribution of the gas molecules.Planck found, instead, that his entropy formula met the experi-mental data if the energy elements had a definite value fixed bythe relation e=hv, where h was a new ‘‘constant of nature’’, or‘‘universal constant’’. Planck named this quantity ‘‘elementaryquantum of action, or element of action’’, and sought a physicalmeaning for it in his idea of elementary disorder (Planck, 1900,p. 239, 1906, p. 154). His entire derivation, he explained, rested onthe Boltzmann principle and on the assumption that the complex-ions were cases of equal probability. Unlike the case of the idealgas, the theory of radiation afforded no means to defineprobability other than the Boltzmann principle. The elementaryquantum determined an energy interval inside which all thepossible different states of a resonator functioned as if they werethe same state as far as the observable quantities were concerned.Since for Planck the values of the observable quantities at a giventime determined univocally their future course, the effect of theelementary quantum was to make the exact energy value ofthe resonators at once unknowable and irrelevant to the timeevolution of the system. Planck did not suppose that all thedifferent possible states inside an interval of size e were reducedto one; he did not regard the elementary quantum as restrictingthe possible values of resonator’s energy (Kuhn, 1978). On thecontrary, he saw it as sharpening his hypothesis of ‘‘naturalradiation’’, which he had introduced as the electromagneticmanifestation of the ‘‘elementary disorder’’ described by Boltz-mann’s hypothesis of molecular chaos in mechanics (Needell,1988; Planck, 1900, p. 238 and 242). The quantum of actionconstituted the measure of elementary disorder in the sense thatit enabled univocal observable phenomena to arise from theunderlying unobservable multiplicity.

Planck’s probability, W=(N+P�1)!/((N�1)!P!), is sometimessingled out as the prime example of indistinguishability, for it canbe characterized as ‘‘the standard expression for the number ofways in which P indistinguishable elements can be distributedover N distinguishable boxes’’ (Kuhn, 1978, p. 101). Conversely,Boltzmann’s permutability, B=N!/(w0!w1!w2!,y, wp!), is taken tobe the standard expression of distinguishability because, in thesame combinatorial language, it is said to represent the number ofways in which N distinguishable elements can be distributed overP+1 distinguishable boxes, w0 elements being in the first box, w1

in the second, and so on. This terminology, however, does notpoint to where Planck’s definition of entropy diverged fromBoltzmann’s, and is consequently misleading to understand theemergence of the notion of indistinguishable particles. It onlyrefers to the difference between two levels of description of theways in which a set of generic elements can be arranged intosubsets in abstract combinatorial calculus. In his application ofcombinatorial calculus, Planck employed the same level ofdescription employed by Boltzmann: molecules and resonatorswere the combinatorial elements to be grouped in the two cases,and in both cases they were treated as ‘‘distinguishable’’. Only

when he resorted to the expedient of describing the complexionsusing the energy elements as combinatorial elements did Planckswitch to the level at which the combinatorial elements weretreated as ‘‘indistinguishable’’. Although he exploited the inter-changeability of the two descriptions in the combinatorial part ofthe calculation, he was clear that resonators and energy elementsdid not play interchangeable roles in his statistical problem,which required the complexions to be cases of equal probability.

Planck developed his theory from an analogy between idealgas and thermal radiation that hinged on his idea of elementarydisorder (Darrigol, 1992b). As Planck explained at length in The

Theory of Heat Radiation, the source of disorder, and hence thereason for introducing probability in physical theory, was thecircumstance that the exact states of complex physical systemswere underdetermined by their observable properties. Forsystems consisting of numerous components, such as a gasmodelled as an assembly of molecules or a radiation fielddecomposed into its partial vibrations, a state defined by a setof values of the observable variables was completely determinedin its time evolution, yet it could be realized by many differentexact states of its unobservable components. The probability of agiven state of the system was for Planck the number of exactstates of the components that realized it. Thanks to this generaldefinition of probability, he could extend the Boltzmann principlealong the track of a specific correspondence: the coordinates andmomenta that defined the exact state of the gas moleculescorresponded to the amplitudes and phases that defined the exactstate of the partial vibrations.

In Planck’s gas-radiation analogy the role of statistical entitieswas played by the partial vibrations of the field and, hence, by theresonators that represented them in the auxiliary theoreticalmodel used to work out the mean energy. The energy elementswere only mathematical entities that served, initially, a support-ing function in the count of complexions as they did in the case ofBoltzmann’s gas, and then acquired a physical role in the dis-analogy that resulted. It was not, however, the role of statisticalentities and, in Planck’s conception, it could not have been. Itwould not have been possible to picture them as objectsdistributed over resonators in a plurality of exact states analogousto the states of molecules distributed over energy regions.

3.2. Einstein’s critique of complexions and the quantum theory of

radiation

Planck’s derivation of the radiation law came under thescrutiny of the then young Einstein, who was developing hisown generalization of Boltzmann’s statistical theory. In 1905,Einstein set forth his ‘‘heuristic point of view’’ on the structure ofradiation, now famous as the hypothesis of light quanta, accordingto which the energy of radiation consisted of localized andindivisible ‘‘energy quanta’’ moving independently through space,entities analogous, in a specific sense, to the particles of an idealgas (Einstein, 1905). Starting discretely on that occasion andcontinuing more openly in 1906 and 1909, he also analyzedPlanck’s gas-radiation analogy from the viewpoint of his statisticalstudies, and delivered a criticism of Planck’s determination ofprobability that was the beginning of quantum theory. Theanalysis rested on a general formulation of the Boltzmannprinciple for systems of ‘‘molecular entities’’ described by a setof ‘‘state variables’’, p1,y, pn, which represented the general-ization of Boltzmann’s gas molecules. Einstein identified themolecular entities of radiation with Planck’s resonators, which heinterpreted as ‘‘bound electrons’’ or ‘‘ions’’, capable of emittingand absorbing radiation but otherwise analogous to ‘‘point masses(atoms)’’ oscillating harmonically around their equilibrium

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positions. He used the position and velocities of the resonators asstate variables, and found that the entropy calculated in this wayled to a spectral distribution of energy, today known as theRayleigh–Jeans law, which was empirically inadequate andtheoretically untenable. How could, then, Planck’s way ofcomputing the probability by the number of complexions producePlanck’s law instead? The answer was Planck’s count did notinclude all the possible configurations of state variable represent-ing equiprobable ways of distributing energy over the resonators.It included ‘‘only a vanishing small portion’’ of them, namely,those in which the resonator energies had values equal to integralmultiples of the energy quantum, e=hv. (Einstein used a notationdifferent from Planck’s, and wrote e=(R/N)bv, with R/N=k andb=h/k.) Planck’s law, therefore, rested on the hidden assumptionthat the energy of a resonator was discontinuous, and that it couldchange through absorption and emission only by whole numbersof energy quanta (Einstein, 1905, p. 134, 1906, p. 199–203, 1909a,p. 187 and 1909b, p. 822).

Having arrived at an understanding of the energy elementsdifferent from Planck’s, Einstein concluded that Planck’s theorywas not in conflict with his ‘‘hypothesis of light quanta’’ butimplicitly made use of it (Einstein, 1906, p. 199). This does notmean, however, that he identified Planck’s energy elements withhis light quanta. On the contrary, he offered a re-formulation ofthe hypothesis of light quanta that concerned only the emissionand absorption of energy in discrete units. He evidently knew thathis original hypothesis that radiation energy propagated throughspace in independent quanta led necessarily to Wien’s law, whichwas only the high-frequency limit of Planck’s law (Renn, 1993;Stachel, 2000). This may explain why he omitted the remark thatPlanck’s complexions, in addition to being only a selected few, didnot correspond to cases of equal probability for independentquanta distributed over resonators.

The view that Planck’s law demanded the energy of resonatorsto be restricted to whole numbers of energy quanta becamemainstream in 1908–1911. Processing the break with classicalphysics demanded by the quantum discontinuity, physiciststurned to the question of whether the discontinuity arose fromthe features of matter–radiation interactions or belonged to thenature of radiation itself. The resonators were a theoreticalconstruct waiting be replaced by a consistent picture of theinteraction between matter and radiation. Meanwhile, thequantum discontinuity gave rise to a quantum theory of radiation,known as the Ehrenfest–Debye scheme, that eliminated the needto take the resonators into account. In this scheme, Planck’s lawwas derived from a model of quantized waves, that is, Maxwell’swaves that could only assume values of energy equal to wholenumbers of quanta. The coefficient of Planck’s equation, 8pv2/c3,could be calculated as the number of standing waves, or normalmodes of vibrations, in a given volume, and the average energy ofa resonator corresponded to the average energy of a mode ofvibrations. The modes of vibration replaced Planck’s resonators asstatistical entities, and the statistical part of Planck’s derivationcould be simply imported into the new scheme (Darrigol, 1986;Debye, 1910). In this scheme, the energy elements, also calledenergy quanta, were units of radiant energy, objects of a differentclass from atoms and molecules. Like Planck’s resonator model,the quantized wave model needed no consideration of thestatistical distributions of energy quanta, and the tacit taxonomyon which it rested suggested none.

In 1916, Einstein found a new derivation of Planck’s law byintegrating Niels Bohr’s hypothesis of state transitions in astatistical theory of quantized emission and absorption ofradiation by gas molecules (Einstein, 1916a). This theory was abreakthrough in the study of matter–radiation interactions, but itdid not change the statistical status of the energy quanta because

it was based on an explicit analogy between the quantized gasmolecules and Planck’s resonators. Nonetheless, it advanceddecisively the possibility to treat the energy quanta as objectsanalogous to particles because it showed that to each quantumthere was associated a momentum equal to hv/c in the direction ofpropagation of the radiation.

3.3. Einstein’s gas-radiation analogy and the statistics of

light quanta

Einstein and a few other physicists explored also the possibilityto interpret the quantization of radiation in terms of light quanta.The light quanta were imagined to transport energy through spacein a finite number of localized, indivisible amounts, in the sameway as ‘‘atoms and electrons’’ did (Einstein, 1905, p. 132). For thisreason, the light quantum interpretation was called the atomisticor corpuscular view of the structure of radiation. There was also amore specific reason for classifying light quanta as hypotheticalparticles. The core part of Einstein’s argument was a statisticalanalogy between the ideal gas and heat radiation of sufficientlyhigh frequency, that is, radiation for which the Wien law wasvalid. Einstein first argued on thermodynamic ground that whenmonochromatic Wien radiation fluctuated from a volume V0 to avolume V, its entropy changed according to the relationS�S0=k(E/hv)lg(V/V0). He then noted that this relation expressedthe same dependence of entropy on volume as that of an ideal gas,and offered an interpretation of this formal analogy in terms ofthe Boltzmann principle. The difference in entropy between twostates of the same system was related to the relative probability ofthe two states, W, according to the equation S�S0=k lg W. Therelative probability could be evaluated in a simple way for aspecial case, the case of a system of N material points that movedindependently from one another and fluctuated from a volume V0

to a volume V. A system of this kind could be an ideal gas, a dilutesolution, or any system composed of particles, as long as theinteractions among the particles were negligible. In this case, therelative probability to find in V any particle that was initially in V0

was given by the ratio of the two volumes, and the relativeprobability for all the particles, which Einstein called a ‘‘statisticalprobability’’, was W=(V/V0)N. Accordingly, the entropy formula forthe Wien radiation could be interpreted through the Boltzmannprinciple as demonstrating that the relative probability of tworadiation states of different volumes was W=(V/V0)N, if N=E/hv.Hence, monochromatic radiation in the region of validity of theWien law behaved thermodynamically as if it was composed of‘‘mutually independent’’ light quanta of energy hv (Einstein, 1905,pp. 139–143). In short, the theory of light quanta began with ananalogy that assigned them the role of statistical entities. Thestructure of the analogy dictated which features of the gasmolecules were to be transferred to the new hypothetical entities.The statistical entities of Wien radiation did not have to beconserved in number as molecules were, but they had to bemutually independent (Norton, 2006).

Planck’s law, however, entailed a more complex dependence ofentropy on volume than that of independent light quanta. In 1909,Einstein calculated the average fluctuation of radiation energy andmomentum from Planck’s law, and found that the resultingformulae were composed of two terms. The first term alone wouldbe obtained in the limit of high frequency, and it corresponded tothe fluctuation of the number of independent light quanta in agiven volume. Only the second term would be obtained in thelimit of low frequency, and it corresponded to the fluctuation dueto interference in a system of waves. Interference, the wavephenomenon that defeated the corpuscular theory of light,appeared in this way associated to the loss of independence of

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the light quanta outside the region of validity of the Wien law.Therefore, Einstein set forth the argument that Planck’s lawrevealed both a corpuscular and an undulatory structure ofradiation. He hoped to find a suitable field-theoretical frameworkfor this dualistic view. If Einstein modelled his light quanta uponmaterial corpuscles, he was already looking at the possibilityof modifying the mechanical model of corpuscles by means ofconcepts derived from field theory (Einstein, 1909a, 1909b;Howard, 1990; Klein, 1964; Kojevnikov, 2002; Pais, 1979; Stachel,1986, 2000).

While Einstein pursued his vision of the wave-particle dualityof light, other physicists endeavoured to reconcile the light quantawith Planck’s law within a more conventional view of thecorpuscular model. Paul Ehrenfest, a disciple of Boltzmann whobecame Einstein’s closest interlocutor on statistical matters,analyzed Einstein’s hypothesis into three parts. First, a resonatorcould only assume discrete energy values. Second, the energyvalues were made up by the aggregation of ‘‘mutually indepen-dent’’ quanta. Third, the quanta were not only a property of theinteractions with matter but existed also in free space. Ehrenfestclarified what he meant by ‘‘mutually independent’’ in thefollowing discussion, in which he stressed the need to distinguishthe condition of mutual independence from the other parts of thehypothesis in order to correct two related misapprehensions. Onthe one hand, it was a widespread opinion that only the existenceof light quanta in free space overstepped Planck’s theory, but infact already their mutual independence did so. On the other hand,the condition of mutual independence was often overlookedbecause of a statistical mistake, namely, that the distribution ofresonators over energy regions was equivalent to the distributionof individual energy quanta over resonators with equal probabil-ities. The distribution of individual quanta over resonators withequal probabilities—in other terms, the distribution of statisti-cally independent quanta—would not lead to Planck’s law(Ehrenfest, 1911, pp. 110–113; Klein, 1959; Navarro & Perez,2004). These observations might have been directed to Einstein,who understood the non-independence of light quanta demandedby Planck’s law mainly as correlation in the spatial distribution.Einstein’s view left open the possibility that the light quanta werephysical entities having a spatial behaviour different from that ofmechanical particles. In contrast, Ehrenfest was concerned withthe lack of statistical independence even in the distribution ‘‘overresonators’’, which impaired the possibility of treating the lightquanta as statistical entities even before worrying about aphysical interpretation.

Following Ehrenfest’s analysis, other physicists arrived at theconclusion that the assumption of statistically independentquanta, to which they referred as the hypothesis of ‘‘light atoms’’,led necessarily to Wien’s law. They also suggested that one couldrecover Planck’s law by admitting that the light atoms aggregatedinto groups, sometimes called ‘‘light molecules’’, composed of one,two, three, etc. quanta. This idea found heuristic support in thepower expansion of Planck’s formula:

rn ¼8pn2

c3hn e�

hnkTþe�

2hnkT þe�

3hnkT þ . . .

� �

the terms of which could be read as partial radiations havingspectral distributions similar to Wien’s law. Even though theformal correspondence was not exact, the expansion suggestedthat heat radiation could be viewed as gas mixture of differentmolecular species, each analogous to Boltzmann’s gas (Bergia,Ferrario, & Monzoni, 1985; Darrigol, 1991; Howard, 1990;Kojevnikov, 2002; Krutkow, 1914; Wolfke, 1914). The idea of lightmolecules was revived in 1922–1923, when the hypothesis of acorpuscular structure of radiation began receiving experimentalsupport (de Broglie, 1922, 1923; Wolfke, 1921). In 1923, the Berlin

physicist Walther Bothe re-interpreted the light moleculesthrough Einstein’s 1916 quantum theory of radiation. He arguedthat the probability of stimulated emission explained theformation of multiple quanta without the intervention of forces,so that the dissociation work of the multiple quanta would be zero(Bothe, 1923).

3.4. Indistinguishable energy elements

To what extent did the early statistical analyses of Planck’s lawpoint to the existence of a new kind of statistical entities? Twoepisodes are illuminating in this regard. In 1911, LadislasNatanson, a theoretician at the University of Cracow, analyzedwhat he took to be Planck’s problem, ‘‘in how many ways n unitsof energy can be contained in N receptacles,’’ where by‘‘receptacles’’ he meant material particles capable of emittingand absorbing radiation energy (Natanson, 1911, p. 135). Althoughhe added the statistical clause, ‘‘assuming that the chance of anyone unit getting into any given receptacle is exactly the same’’(which was not part of Planck’s problem) and was aware that the‘‘ways’’ to be counted had to be cases of equal probability, hediscussed the question from a purely combinatorial point of view.Combinatorial calculus admitted several answers, correspondingto different levels of description for receptacles and energy units,which Natanson classified according to whether one presumed tobe able or unable to distinguish individual receptacles and energyunits. The answer chosen by Planck corresponded to the level inwhich the receptacles were ‘‘distinguishable’’ and the energy unitswere ‘‘all regarded as being undistinguishably alike [sic].’’ Butsince it was possible to imagine also the energy units as‘‘separately sensible to us’’, Natanson concluded that Planck’schoice could not be accepted unless the receptacles could ‘‘betreated as distinguishable’’ and the energy units, ‘‘being in allrespects precisely alike’’, could not be so treated. He added that,‘‘by a distinct appeal to experience we find a posteriori that theproposition is true’’ (Natanson, 1911, p. 136 and 139).

More than one modern commentator has read in these wordsan early statement of indistinguishability, and has wondered whyother physicists did not follow Natanson’s insight (Bergia, 1987;Kastler, 1983). Yet, Natanson did not present the case as a choicebetween two different kinds of statistical entities for the simplereason that he did not regard the energy units as statisticalentities. He did not even realize that the choice of configurationsin which energy units were treated as indistinguishable wasincompatible with his statistical clause. His gas-radiation analogywas identical to Planck’s and, correspondingly, he considered thestatistics of radiation as no different from Boltzmann’s statistics.This was explicit in the second part of his paper, where hecompared Boltzmann’s gas and Planck’s radiation as ‘‘two extremecases of particular importance’’ of the same kind of systems. Inboth cases, units of energy were distributed over receptacles,which were gas molecules in one case, radiators in the other. Theonly difference was that the gas was ‘‘a system upon which unitsof energy have been abundantly bestowed’’, while the radiatorswere a system ‘‘poorly endowed with energy’’ (Natanson, 1911,p. 142).

In reply, Planck objected that the answer to his question wasunivocal, since he had formulated the question in terms of energyelements only as a means to find the number of equally probabledistributions of resonators over energy (Planck, 1911, p. 277). Thealternative raised by Natanson was an artefact of havingabstracted the question from its statistical context. This exchangeshows that questions about the statistics of energy elements couldonly arise in contexts where the energy elements were regardedas statistical entities. But even from the perspective of thecorpuscular interpretation Natanson’s analysis did not bring

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new insights. It was in fact quoted as revealing the same pointalready uncovered by Ehrenfest: Planck’s law diverged fromEinstein’s light quantum hypothesis because it did not positindependent energy elements (Krutkow, 1914, p. 134).

Another instance frequently quoted as an early statement ofindistinguishability is a paper published in 1914 by Ehrenfest andHeike Kamerlingh Onnes. In it, the authors show that Planck’scombinatorial expression, W=(N+P�1)!/((N�1)!P!) is easilyobtained if the energy distribution of the energy E=Pe over N

resonators is graphically represented by a string of symbols of thefollowing kind:

II e e e e 0 e e 0 0 e II

where the first group of e symbols represents the energy elementsallocated on the first resonator, the second group those allocatedon the second resonator, and so on, and the symbols 0 separateone resonator from the next (in the example shown, P=7 andN=4). Planck’s W is the number of different strings that can beobtained by permuting the symbols. Strange as it may seem,Ehrenfest and Kamerlingh Onnes did not use their permutation-based demonstration to show that configurations differing only bypermutation of energy elements represented the same complex-ion. That point, already made clear by Planck, was evidentlyirrelevant for them. They used their demonstration to cautiononce again against the identification of Planck’s energy elementswith Einstein’s light quanta. Planck’s distribution of energyelements was a formal artifice just as their string of symbolswas, even though, they lamented, repeatedly a physical inter-pretation had been ‘‘mistakenly foisted upon it’’. The realdifference between Planck’s energy elements and Einstein’s lightquanta was not that the former existed only inside matter and thelatter also in empty space, but that Einstein dealt with ‘‘equalquanta independent of one another’’ (Ehrenfest & KamerlinghOnnes, 1915, p. 1023. Emphasis in the original). For Ehrenfest andKamerlingh Onnes, Planck’s W did not signal the appearance of anew kind of statistical entities, differing from Boltzmann’smolecules in that their exchanges did not produce new complex-ions. It signalled that the category of statistical entities could bestretched to include Planck’s energy elements only at the price oflosing the essential condition of statistical independence. In theirview, this simply excluded that the energy elements could beanything more than formal devices.

Despite superficial similarities with the modern interpretation,the early studies of Planck’s law did not point to the existence ofan alternative kind of statistical entities. Furthermore, no onedrew any connection between the non-independence of lightquanta and their exchange symmetry. This is noteworthy, sinceexchange symmetry was at that time becoming an issue in thetheory of the ideal gas, an issue that is considered in hindsight thesecond root of indistinguishability.

4. The quantum theory of the ideal gas and the exchangesymmetry of equal molecules

4.1. Gibbs’s generic phases

Many authors consider J. Willard Gibbs the father of the idea ofindistinguishable particles because he was the first scientist togive consideration to the significance of the symmetry of a systemunder the exchange of equal particles (Kastler, 1983; Pesic, 1991).Despite his declared avoidance of hypotheses about the constitu-tion of matter and his choice of an axiomatic definition ofensembles as the foundation of his statistical mechanics, Gibbsrecognized the physical importance of ‘‘systems composed of agreat number of entirely similar particles’’ and devoted the last

chapter of his book to them (Gibbs, 1902, pp. 187–207). He openedthe chapter explaining that he had to be very specific about whathe meant by phase. The reason was that he defined statisticalequilibrium as the stability of the number of systems of anensemble that fell ‘‘within any given limit with respect to phases’’,and for systems of equal particles the phases could be defined intwo ways. One could regard as distinct phases that differed onlyby the exchange of equal particles, or one could regard them asidentical. Gibbs named ‘‘specific phases’’ the former kind and‘‘generic phases’’ the latter. If N was the number of equal particles,each generic phase encompassed N! specific phases that wereobtained from one another by permutation of particles. Both kindsof phases were needed because analytical calculations weresimpler in terms of specific phases, but generic phases alloweda generalization of the condition of statistical equilibrium toensembles of systems with variable numbers of particles (Gibbs,1902, p. 197). Gibbs defined the statistical analogue of the entropyas the logarithm of the density in phase of an ensemble inequilibrium, and this could be defined either with respect tospecific phases or with respect to generic phases. He then showedthat the entropy analogue defined with respect to generic phasesdiffered from the corresponding quantity defined with respect tospecific phases only by a term that, in our notation, correspondedto �k lg N!. This difference was insignificant for systems withconstant N, while for systems with variable N it meant that theappropriate entropy analogue was to be defined with respect togeneric phases. Gibbs illustrated this point with the example ofthe mixing of two fluid masses. Since he considered it self-evidentthat the diffusion of two fluids into one another should producean increase in entropy if the fluids were different but no increase ifthey were equal, in his view the example plainly showed that ‘‘it isequilibrium with respect to generic phases, and not with respectto specific, with which we have to do in the evaluation of entropy’’(Gibbs, 1902, p. 207).

After the establishment of quantum mechanics, Gibbs’s method ofstatistical ensembles was found to be suitable to quantum statisticaladaptations. Gibbs’s prescription to evaluate entropy with respect togeneric phases, in particular, is commonly regarded as the precursorto quantum statistical rule of counting as one all the configurationsdiffering by particle exchange. The physicist Paul S. Epstein, in his1936 commentary to Gibbs’s works, maintained that the definition ofgeneric phases led necessarily to quantum statistics (Epstein, 1936a, p.502, 1936b, p. 521). Nevertheless, Gibbs’s entropy analogue, definedon a continuous phase space, corresponds to the entropy of aquantum system of indistinguishable particles only in the high-temperature limit. The reason is not just that in this limit thequantization of the phase space can be neglected, but also that only inthis case the �k lg N! term approximates the effects of the quantumstatistical rule. Gibbs’s prescription to use generic phases for thedefinition of entropy (which he considered insignificant if the numberof particles did not change) translates into the quantum statistical ruleonly if it is construed as the thesis that all the specific phases obtainedby particle permutations represent the same microstate or, in otherwords, that only generic phases are physically meaningful. At no pointdid Gibbs state that the reason to define the analogue of entropy withrespect to generic phases was that specific phases are physicallymeaningless. He did take care to consider this thesis, but only to rejectit. He laid out the proposition that, if the particles are ‘‘regarded asindistinguishable’’, then all the phases differing by particle exchangemust be regarded ‘‘as identical’’, but only to reject it as a matter ofprinciple. The proposition would hold in principle from the strictstandpoint of ensemble theory, in which to regard the particles ‘‘asindistinguishable’’ means to have no way to identify a particle of onesystem with a particle of another system. But Gibbs deliberatelyrelaxed this stand in his study of systems composed of equal particles.He pointed out that ensembles are imaginary constructs whose

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purpose is to represent real systems in their different states of motion,and in a real system the perfect equality of the particles does ‘‘not inthe least interfere’’ with their identification in different states ofmotion (Gibbs, 1902, pp. 187–188). For this reason, his argument forthe use of generic phases cannot be read as the thesis that specificphases are physically meaningless without the intervention of furtherconsiderations that are specific to the context of quantum theory.

4.2. Absolute entropy and exchange symmetry

The symmetry of a system under exchange of equal particles cameto the attention of other theorists when the quantum hypothesis wasextended from the theory of radiation to the theory of the ideal gas.The early forms of quantum gas theory originated as attempts togeneralize to all matter the theory of quantized molecular motion thatEinstein first proposed for solid bodies in 1907, on the model ofPlanck’s radiation theory. As this theory successfully predicted thelow-temperature deviation of the specific heat of solids from classicalkinetic theory, Walther Nernst embraced it as the theoreticalfoundation to his heat theorem, to which he was determined to giveuniversal validity. He intended to use the quantum hypothesis toexplain also the effects of degeneracy in gases, that is, deviations fromthe classical gas law that were predicted by his heat theorem anddetected by experiments at low temperatures. Planck, Nernst, andother Berlin physicists came to share the conviction that quantumtheory could be the foundation of a novel theory of gases in the sameway as it was the foundation of the new theory of radiation. A newparallel between radiation and matter took shape. As Planck put it inthe second edition of his book on heat radiation, the extension of thequantum hypothesis to material systems allowed a general definitionof absolute entropy by means of the Boltzmann principle that led, ‘‘asregards radiant heat, to a definite law of distribution of black bodyradiation, and, as regards heat energies of bodies, to Nernst’s heattheorem’’ (Planck, 1913, p. 134). This generalization of absoluteentropy rested on the analogy between molecules and resonators;therefore, it did not suggest any change in Boltzmann’s statisticsbeside the quantum hypothesis itself.

Defining entropy as absolute, however, made it necessary tofind a physical meaning for a term, �k ln N!, that had the purposeof making the entropy of an ideal gas if N equal molecules anextensive quantity (Darrigol, 1991; Desalvo, 1992). To solve thisproblem, in 1912 the young Dutch theoretician Hugo Tetrodetransposed Gibbs’s argument for defining entropy with respect togeneric phases into a quantum version of Boltzmann’s count ofcomplexions. He interpreted a complexion as a possible combina-tion of values of the phase variables, but with the specificationthat ‘‘two complexions should not be regarded as different’’ iftheir values of the phase variables were within an ‘‘elementaryregion’’ of fixed size s. In this way, the phase space available to thesystem was subdivided into ‘‘cells’’ of size s, the number of whichdetermined the probability. For a gas of N equal molecules, it wass=(zh)N, where z was a integer and h was the elementaryquantum of action. The number of complexions so defined,denoted by Wspec, was still not the correct thermodynamicprobability. In order to produce an additive entropy, it had to bereplaced by Wgen, which Tetrode defined as ‘‘the number of allpossible complexions, in which, however, two complexions aretreated as identical and therefore counted as one if they differ onlyby the exchange of equal molecules.’’ According to Tetrode, fromthe definition of Wgen and Wspec there followed the relation,Wgen=Wspec/N! (Tetrode, 1912, pp. 435–436). Even though Tetrodedid not quote Gibbs directly at this juncture, his suffixes clearlyreferred to Gibbs’s specific and generic phases, and his relationbetween Wgen and Wspec presumably derived from Gibbs’sobservation that each generic phase corresponded to N! specific

phases. Tetrode did not notice that in his case the relation was notcorrect in general, for configurations in which the variables of twomolecules were within an elementary domain were alreadycounted as identical in the calculation of Wspec. For the relationto be exact, Wspec would have to be divided by the permutability,N!/(w0!w1!w2!,y, wp!), where the numbers w0, w1, w2,y, wp

represented the occupation numbers of the phase space cells. Thiswould change the relative probabilities of the different energydistributions, leading to an equilibrium distribution differentfrom the Maxwell–Boltzmann distribution. The division by N!,instead, left relative probabilities and equilibrium distributionunchanged.

The entropy formula obtained by Tetrode proved to beempirically correct for high temperatures. It was also found,independently and with a different argument for the 1/N! factor,by Otto Sackur, and became famous as the Tetrode-Sackurformula. Starting with the second edition of the Theory of Heat

Radiation, Planck became the major advocate of a fundamentalconnection between a universal definition of absolute entropy anda universal quantum hypothesis, understood as the hypothesis ofa physical structure of the phase space determined by theelementary region of probability (Planck, 1913, 1916). He sooncame to adopt not only the 1/N! factor in the thermodynamicprobability of the ideal gas but also Tetrode’s rationalization of iton the basis of exchange symmetry. He proposed several versionsof this argument over the years. In the most extensive formula-tion, presented in the final chapter of the fourth edition of The

Theory of Heat Radiation, Planck argued that exchange symmetrywas best implemented by adopting Gibbs’s method of thecanonical ensemble. He introduced a simplifying alternative tothe calculation of entropy, which consisted of deriving all thethermodynamic functions from the ‘‘sum over states’’,

Pe�ðen=kTÞ,

where the summation ran over the elementary regions of thesingle molecule, and en was the energy inside the nth elementaryregion. For the monatomic ideal gas, the calculation could becarried out by approximating the sum by an integral over thephase space of the molecule, with the element of phase spacedivided by the extension of the elementary region, h3. In the limitin which the elementary region was infinitely small compared tothe total phase space of the single molecule, this calculationproduced the classical gas law. When the size of the elementaryregion was not negligible, there occurred instead the degeneracyexpected by Nernst (Planck, 1921, pp. 133–136). Nonetheless, evenin the classical limit this calculation did not produce empiricallycorrect results. The reason, according to Planck, was that the sumover states did not describe a system of N equal atoms enclosed ina volume V. It described either N equal atoms each moving in avolume V of its own, or N atoms in a common volume V, providedthat the atoms could be individually differentiated despite havingequal masses. For Planck, this meant that the molecules were‘‘taken to be more independent from one another than theyactually are in a gas, in which they exchange places through theirmotions and can replace one another so completely that theycannot afterwards be individually recognized’’. In such circum-stances of ‘‘complete exchangeability’’, the only strategy possiblewas that of ensemble theory, in which the whole gas was regardedas a ‘‘single entity’’ (Planck, 1921, p. 205). The sum over states ofindividual molecules had to be replaced by the sum over states ofthe whole gas,

Pe�ðEn=kTÞ, where En was the total energy of the

gas and the summation run over the elementary regions of the6N-dimensional phase space of the whole gas. A few pages later,Planck returned on the difference between a gas and othermolecular systems.

One notices the fundamental difference from the previously

considered case of oscillators or rotators. There, the molecules

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were entirely separate from one another, each having its definite

permanent place, and when two molecules exchanged their

energies, this produced a new complexion. Here, in contrast, we

do not have a system of mutually separated molecules, but a single

entity endowed with symmetries, and these symmetries consist of

the fact that there is no feature that allows one to recognize a

given atom if one considers the gas first in one state and then in

another state. Therefore, two states of the gas that differ only by

the exchange of two atoms do not produce a new complexion

(Planck, 1921, p. 209).

This argument resembles modern formulations of indistin-guishability because it relates what Planck calls ‘‘exchangeability’’with the loss of something he calls ‘‘independence’’. Neithernotion, however, was well defined. On the one hand, thepresumed impossibility to recognize particles when they ex-changed places ‘‘through their motion’’ conflicted with theclassical idea of motion. On the other hand, by ‘‘independence’’Planck evidently did not mean statistical independence. This wasmade clear in his meticulous illustration of the transition from theold method (sum over states of individual molecules) to the new(sum over states of the whole system), where he stressed preciselythe precondition of statistical independence. In the case of N equaland separate molecules, statistical independence permitted toreduce the sum over states of the system to the Nth power of the

sum over states of a single molecule, ðP

e�ðen=kTÞ�N

. The case of the

gas differed only for the exchangeability of the molecules; in thiscase, all the terms obtained form one another by a permutation ofmolecules had to enter the summation only once. To this end,according to Planck, all one had to do was to divide the sum overstates of the previous case by N!. Therefore, the new method ledto the same result as the inclusion of the 1/N! factor in thethermodynamic probability. Since this overall division left thestatistical distribution unchanged, the only difference betweenthe classical and the quantum gas was a low-temperaturedegeneracy caused by the quantization of the phase space.

Planck oddly propagated Tetrode’s oversight. He did point outthat the overall N! division was mathematically inexact, for onlythe terms in which all the molecules had different energies wouldoccur exactly N! times in the sum over states of the gas. In general,a term would describe w0, w1,y, wp particles with energies e0,e1,y, ep, and hence would recur N!/(w0!w1!,y, wp!) times. Inorder for the terms differing only by particle exchange to occuronce, each term would have to be divided by its permutability.Unlike the overall division by N!, this operation would make itimpossible to reduce the sum over states of the gas to the productof the sum over states of the single molecules. Had Planckperformed the exact calculation, he would have been confrontedwith a conflict between his demand of exchange symmetry andthe condition of statistical independence. But he stuck to theoverall division by N! as a generally valid approximation, holdingthat the cases in which two or more molecules were in the samesingle-molecule state could always be neglected due to theirinfrequency (Planck, 1921, p. 210). The approximation automati-cally preserved statistical independence, and Planck saw noreason to put it in question.

The exchange-symmetry argument, in its various forms, metthe skepticism of Hendrik A. Lorentz, Otto Stern, Ehrenfest, andEinstein, among others. In an unpublished manuscript presum-ably prepared for a lecture to the Deutsche PhysikalischeGesellschaft at the beginning of 1916, Einstein presented hisown understanding of the quantization of the phase space,stripping the ‘‘theory of Sackur and Tetrode for the entropyconstant’’ of what he regarded as ‘‘all the distracting accessories’’.He treated the 1/N! factor as an ad hoc implement, ‘‘meaningless

in itself’’, that only served the habit of expecting entropy to beextensive. Given that only entropy differences were thermodyna-mically defined and that no reversible process could augment thequantity of matter in a system, this habit was not ‘‘free ofarbitrariness’’ (Einstein, 1916b, p. 250 and 255). Ehrenfest, in apaper in which he objected to extensive entropy in terms similarto Einstein’s, remarked that he found Planck’s exchange-symme-try argument incomprehensible despite his best efforts (Ehrenfest& Trkal, 1921). Planck replied after evidently having re-read the1911 essay on the foundations of statistical mechanics in whichEhrenfest and his wife, Tatiana Afanassjewa had analyzedBoltzmann’s statistical model and had laid out the preconditionsof Boltzmann’s definition of complexions (Ehrenfest & Ehrenfest-Afanassjewa, 1912). Yet, Planck did not clarify why a particleexchange should not produce a new complexion but justrephrased his previous argument, this time dropping even theineffective warning about the approximate validity of the 1/N!factor (Planck, 1922).

5. A new statistics of non-independence

5.1. The birth of quantum statistics

The quantum theory of radiation and the quantum theory ofthe ideal gas developed along parallel tracks, propelled by partialmutual analogies, until June 1924, when Einstein received amanuscript from a young Indian physicist, Satyendra Nath Bose,containing a new derivation of Planck’s radiation law based on thehypothesis of light quanta. Einstein translated Bose’s article intoGerman, had it promptly published in Zeitschrift fur Physik, andimmediately authored the first of a series of three papers in whichhe applied Bose’s method to a new calculation of the entropy ofthe ideal gas (Bose, 1924; Einstein, 1924c; Pais, 1979; Stachel,1994). He thus turned Bose’s method into a new form of statisticalmechanics applicable to light quanta and also, potentially, tomaterial particles. We now know it as Bose–Einstein statistics, thebranch of quantum statistics that describes the behaviour ofindistinguishable particles not subjected to the exclusion princi-ple. Notably, however, neither Bose nor Einstein showed anyawareness that they were inaugurating the statistics of indis-tinguishable particles.

The result that Bose announced was that he had been able toderive the coefficient of Planck’s relation between field energy andresonator energy without recourse to classical electrodynamics.Einstein, who had always seen the classical origin of that relationas a difficulty for the light quantum hypothesis, praised the sameachievement in his reply. Recent events had stirred up renewedinterest in light quanta in the physics community. On the onehand, Arthur H. Compton’s experiment had been received bymany as the demise of the wave theory of light. On the other hand,Niels Bohr and his assistant, Hendrik Kramers, using an idea by ayoung American theorist, John Slater, had just published a paperin which they attempted to preserve the wave theory byproposing the non-conservation of energy as an alternative tothe existence of light quanta. Einstein reacted to the Bohr–Kramers–Slater theory with a list of objections. He had recentlyreported to a newspaper that Compton’s experiment proved ‘‘thatradiation behaves as if it consisted of discrete energy projectiles’’,and that therefore there existed now ‘‘two theories of light, bothindispensable and—as must be admitted despite twenty yearsof strenuous effort by theoretical physicists—without logicalconnection’’. He had also recently discussed at the Berlin Academythe statistical properties of radiation in connection with Bothe’stheory of multiple quanta and his own field-theoretical studies(Einstein, 1924a, 1924b; Klein, 1970; Hendry, 1981; Howard, 1990).

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Thus immersed in the problem of the duality of light, hewelcomed Bose’s derivation as a success for light quanta, butconfessed to Ehrenfest that its essence remained ‘‘obscure’’(Einstein to Ehrenfest, Einstein Archives, 24 July 1924).

At first sight, Bose succeeded by treating the light quanta asstatistical entities according to Einstein’s gas-radiation analogy,without any complication about duality or multiple quanta.He only took for a model a quantum gas instead of the classicalgas; in his words, all he did was to apply ‘‘statistical mechanics (inthe form adapted by Planck to the needs of quantum theory)’’(Bose, 1924, p. 179). Using the definition of the momentum of alight quantum given by Einstein in 1916, p=hn/c, he defined for thefist time the phase space of light quanta, then quantized it intocells of size h3, as Planck did for the gas molecules. He was thusable to show—with only the further adjustment of a factor twothat in the published translation of his paper is attributed to thetwo directions of polarization of light—that Planck’s coefficientwas equal to the number of phase-space cells in the unitaryvolume.

Bose completed the derivation by way of what he thought wasthe same analogy. Apparently unaware that the application ofBoltzmann’s method to light quanta had been repeatedly provento produce Wien’s law, he stumbled into a procedure thatcircumvented the obstacle (Delbruck, 1980; Bergia, 1987). Heused Boltzmann’s formula for the probability of an energydistribution of the ideal gas, but replaced the numbers ofmolecules having energies 0, e, 2e,y, with the numbers ofphase-space cells containing 0, 1, 2,y light quanta. By doing so,he inadvertently gave the formal role of statistical entities to the

Fig. 1. Einstein’s comparison of the ‘‘Bose statistics’’ with the statistics of ‘‘independe

together relatively more often than under the hypothesis of statistical independence’’. (A

the Hebrew University of Jerusalem.)

phase-space cells and automatically treated the light quanta asPlanck did the energy elements. As he admitted in a laterinterview, he did not realize that he was doing anything differentfrom what Boltzmann would have done (Pais, 1979, p. 893).

Without pausing to question Bose’s operation, Einstein quicklyexploited it to reverse the gas-radiation analogy: he followedBose’s recipe and treated the gas molecules as Bose did the lightquanta. In the first of his quantum gas papers, Einsteinemphasized the suitability of Bose’s method to generic ‘‘elemen-tary entities’’, which could be either gas molecules or light quanta.He then showed that, if applied to gas molecules, the methodreproduced the classical energy distribution and the Sackur–Tetrode formula in the limit of high temperature and lowfrequency, while also using a definition of absolute entropy thatsatisfied Nernst’s theorem. He finally showed how the new theorydiverged from the classical theory outside the classical limit,without addressing the question of what aspects of thenew method were responsible for the degeneracy (Einstein,1924c, p. 261, 265 and 266). At this stage, Einstein rehearsedpragmatically the basics of the ‘‘quantum statistics’’ that Bose hadlearned from Planck. He presented it as an algorithm thatconsisted of calculating the ‘‘‘probability’ of a macroscopicallydefined state (in Planck’s sense)’’ as ‘‘the number of differentmicroscopic states’’ by which the macroscopic state could ‘‘bethought to be realized’’. Perhaps, he presumed that this prob-ability ‘‘in Planck’s sense’’ (not his sense, evidently) accounted forBose’s anomalous use of Boltzmann’s permutabiliy. It appears thathe had not grasped yet the shift implicit in that use, for he wrotethat a microscopic state was ‘‘characterized by the way in which

nt molecules’’. Einstein explained that ‘‘According to Bose, the molecules crowd

. Einstein to E. Schrodinger, Berlin, 28 Feb. 1925. Courtesy of the Einstein Archives at

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the elementary entities are distributed’’ over the phase-spacecells—a description reminiscent of Boltzmann’s complexions butinapt for the ‘‘microscopic states’’ actually counted by Bose(Einstein, 1924c, p. 261).

Bose’s operation evidently provoked the protests of those who,like Ehrenfest, had been wrestling for over a decade with thestatistics of light quanta. Einstein addressed their criticism in thesecond paper, which he wrote 6 months later while in Leiden withEhrenfest. He recognized that Bose’s procedure was unorthodox,but not in the terms that we would expect. He wrote (Einstein,1925, p. 5)

Ehrenfest and other colleagues have objected, regarding Bose’s

theory of radiation and my analogous theory of the ideal gas,

that in these theories the quanta or molecules are not treated as

statistically independent entities, without this circumstance being

specifically pointed out in our articles. This is entirely correct.

Shortly thereafter, in giving a masterfully simple explanation ofBose’s method to Erwin Schrodinger, he would elevate it to a newstatistics (Fig. 1):

In Bose’s statistics, which I have used, the quanta or molecules are

not treated as independent from one another. [y] According to

this procedure, the molecules do not seem to be localized

independently from each other, but they have a preference to be

in the same cell with other molecules. [y] According to Bose, the

molecules crowd together relatively more often than under the

hypothesis of statistical independence.’’ (Einstein to Schrodinger,

Berlin, 28 Feb. 1925.)

Einstein made the argument that Bose’s method, with itsviolation of the hitherto basic assumption of statistical mechanics,was not an aberration but constituted an alternative kind ofstatistics. It deserved consideration because it was based on a‘‘far-reaching formal similarity’’ between gas and radiation: thedegenerate gas to which it led was related to the classical gas inthe same way as Planck’s radiation was related Wien’s radiation.Classical gas law and Wien’s radiation law were recovered if theelementary entities were treated as statistically independent,even if one proceeded in any other respect according to Bose’smethod. Aiming now at maximum clarity, Einstein carried out acomparison between the new statistics and the prior quantumstatistics of independent molecules. In both cases, a macroscopicstate was defined by the numbers of molecules in the elementaryregions of energy, and the ‘‘Planck probability’’ was ‘‘the numberof possibilities of realization’’ of the state. What differentiated thetwo statistics was the definition of the possibilities of realization,which coincided with the microscopic states and represented ageneralization of Boltzmann’s complexions. In the case of Bose,they were defined by stating ‘‘how many molecules are in eachcell’’; in the case of independent molecules, they were defined bystating ‘‘in which cell each molecule sits’’. The different definitionsled to different entropy expressions and to different equilibriumdistributions. In the case of independent molecules, the entropycontained a term depending only on the total number ofmolecules, N, which had to be dropped if one wanted to have anextensive entropy. Einstein recalled that this term was customa-rily eliminated by dividing the probability by N!, and that suchoperation was justified by the argument that two complexionsdiffering only by exchange of equal molecules should be treated asidentical (Einstein, 1925, p. 3, 5 and 6). But such operation,Einstein now pointed out, resulted in a negative entropy atabsolute zero. Bose’s case, in contrast, automatically verifiedNernst’s theorem. In sum, Einstein spelled out the quantumstatistical rule that defines the statistics of indistinguishableparticles, yet he did not interpret it as expressing indistinguish-

ability. He interpreted it as expressing non-independence, andexplicitly contrasted it to the exchange symmetry of equalparticles. His failure to see in Bose’s microscopic state a realizationof exchange-symmetry, as we do today, would be inexplicable ifwe did not recognize that for him the exchange-symmetryrequirement was nothing more than a specious excuse for themisguided N! division.

Einstein went on to suggest that the lack of independence wasthe symptom of a ‘‘mutual influence’’ of the elementary entitiesthat was ‘‘of an entirely mysterious kind for the time being’’.Calculating the energy fluctuations, Einstein found a formulaanalogous to the one that he had found for Planck’s radiation, andinterpreted it in a corresponding manner: the formula revealed adual structure, corpuscular and undulatory, for the gas as it did forradiation. This kinship in duality, Einstein believed, was more thana ‘‘mere analogy.’’ Having just read Louis de Broglie’s thesis, hesuggested that the undulatory term of the fluctuation formulamight be explained by de Broglie’s theory of matter waves(Einstein, 1925b, p. 3 and 6).

5.2. First reactions to the new statistics

Einstein’s new statistics directed also Planck to the issue ofstatistical independence. Planck compared his quantum gastheory to Einstein’s and concluded that the two theories, althoughsomewhat similar, were fundamentally different. Einstein hadtransposed the energy distribution of resonators onto the phase-space cells, leaving the molecules to be ‘‘distributed’’ as energyelements. Planck commented

This peculiar result can only be understood if one assumes that the

individual molecules are not statistically independent from one

another, in such a way that if one distributes the molecules one

after the other over the cells, the probability that a molecule falls

in a given cell depends on the number of molecules that the cell

has already received (Planck, 1925, p. 56).

By identifying the distinctive feature of Einstein’s theory,Planck indirectly declared statistical independence a fundamentalcondition of his theory, and confirmed that he did not perceiveany contradiction between this condition and the exchangesymmetry that he propounded. Einstein presumed molecularcorrelations in a system that otherwise conformed the ideal gasmodel. If these correlations were experimentally confirmed, inPlanck’s view they would demand ‘‘a fundamental modification ofthe ordinary conception of the nature and mode of interaction ofthe molecules’’ (Planck, 1925, p. 57).

Planck’s remark obviously referred to the theoretical traditionaccording to which material particles could affect one anotheronly by means of mechanical interactions. For any physicisteducated in classical mechanics, an understanding of the newstatistics would have to start from a modification of that view. Forexample, in a presentation to the Kapitza Club in Cambridge in1925, Paul Dirac explained that the fundamental assumption ofBose–Einstein statistics was that all the values for the number ofgas molecules (or light quanta) in a phase-space cell, orequivalently, all the values for the number of molecules associatedwith a de Broglie wave, had the same a priori probability. Thisrequired that the molecules were ‘‘not distributed independentlyfrom one another, so that there must be ‘‘some kind of interactionbetween them’’ (Dirac, 1925, p. 7).

The first to associate the Bose–Einstein statistics to exchangesymmetry was Erwin Schrodinger. After some reflection on theexplanation that he received from Einstein (Fig. 1), he delivered athorough criticism of Planck’s argument in a paper presented tothe Berlin Academy in July 1925 (Darrigol, 1991, 1992a; Hanle,

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1977; Schrodinger, 1925). He first compared the various statisticaldefinitions of entropy found in the literature, then examined the1/N! factor that, he explained, Planck wanted to ground on astatistical argument. Schrodinger found that accepting Planck’sargument and following it through would lead to the ‘‘at first sightmost peculiar kind of statistics’’ recently proposed by Einstein. Atnot too low temperatures, the entropies of Einstein and Planckwere approximately equal because the number of molecules inany energy level became either zero or one in most cases. But ifPlanck’s argument were valid, it would have to apply also attemperatures so low that the multiple occupations of quantumlevels was no longer negligible, and it would result in Einstein’sdefinition of entropy. Nevertheless, Schrodinger was neithersatisfied with Planck’s argument, nor did he welcome Einstein’sinnovation. Bose’s procedure represented ‘‘a radical departurefrom the Boltzmann–Gibbs kind of statistics’’. Turning now toEinstein’s explanation of it, Schrodinger objected that he did notsee ‘‘for the time being any possibility of understanding theremarkable kind of interaction among the molecules’’ by which itcould be justified (Schrodinger, 1925, pp. 437 and 440–441).

As is well known, Schrodinger ended up espousing de Broglie’stheory and turning it into wave mechanics. This does not meanthat he changed his mind about the Bose–Einstein statistics. Hewrote a new paper to argue that the ‘‘natural sense’’ was reluctantto accept the new statistics as something primary, which couldnot be further explained. In order to restore Boltzmann’s statistics,which he called ‘‘natural’’, Schrodinger added one more frame tothe dual gas-radiation analogy set up by Einstein. Bose’s statisticsyielded Planck’s law for light quanta and Einstein’s gas theory formaterial particles. But also natural statistics, if applied to thenormal modes of vibration of the radiation according to theEhrenfest–Debye scheme, produced Planck’s law. Hence, naturalstatistics must yield the new gas theory if applied to the gas,provided that the gas was modelled as a system of waves inanalogy with the normal modes of radiation, which would thenplay the role of the statistical entities of the gas. Schrodingermapped for the first time the formal relation between exchangesymmetry, statistical correlation, and the new quantum statisticalrule, yet he rejected the new rule as the foundation of analternative statistics. He chose to preserve exclusively the oldmodel of statistical entities, even at the price of drasticallyreplacing the physical objects that instantiated them (Dieks, 1990;Schrodinger, 1926).

6. Conclusion

The two issues that we now see as the two roots of indistin-guishability belonged in separate spheres before the advent ofquantum mechanics. In the case of radiation, the statisticalcorrelation demanded by Planck’s law was not associated to theobvious exchangeability of light quanta. In the debates over theentropy of the ideal gas, exchange symmetry was never regardedas implying the loss of statistical independence of the molecules.Although the two problems sprung from the common ground ofthe generalized Boltzmann principle and the generalized quantumhypothesis, neither of them was seen as illuminating the other,and no common solution was sought. The evolution of theoreticalmodels through analogies proceeded from a unique template ofstatistical entities, the molecules of Boltzmann’s ideal gas.Considerations of exchange symmetry for equal molecules didnot touch the assumption of statistical independence, whichwas a basic and unquestioned trait of the template. Correspond-ingly, the lack of statistical independence constituted a severehindrance to the admission of light quanta into the class ofstatistical entities.

Einstein’s second paper on the quantum gas was a turningpoint because it marked the invention of a new kind of statisticalentities, differing from the traditional kind in that it followedwhat Einstein called ‘‘a new statistics’’. Bose’s deceivingly simpleprocedure convinced Einstein that the light quanta did not have tobe modelled upon particles. The particles could, at least in theory,be modelled upon light quanta. Still, the birth of quantumstatistics did not coincide with the emergence of the theoreticalcategory of indistinguishable particles. In the immediate after-math of Bose’s and Einstein’s papers, the new statistics was notinterpreted as a statistics of indistinguishability. It was eitherviewed as a dispensable second-order description of a system ofwaves that could be treated as traditional statistical entities, or itwas interpreted as a theory of non-independent statisticalentities. A study of the history of indistinguishable particle willhave to attend to the shaping of a new interpretation.

Acknowledgements

The research for this paper was supported by a postdoctoralfellowship of the Max Planck Institute for the History of Science inBerlin. I am very grateful to the History and Foundations ofQuantum Physics group at the Max Planck Institute for providingan excellent research environment. A special thank to AriannaBorrelli and Massimilano Badino for enlightening conversationsand untiring advice. I also wish to express my appreciation to ananonymous referee for insightful and constructive criticism.

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