On the explanation for quantum statistics

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Abstract

The concept of classical indistinguishability is analysed and defended against a number of well-known criticisms, with particular attention to the Gibbs’ paradox. Granted that it is as much at home in classical as in quantum statistical mechanics, the question arises as to why indistinguishability, in quantum mechanics but not in classical mechanics, forces a change in statistics. The answer, illustrated with simple examples, is that the equilibrium measure on classical phase space is continuous, whilst on Hilbert space it is discrete. The relevance of names, or equivalently, properties stable in time that can be used as names, is also discussed.

Introduction

Einstein's contributions to quantum theory, early and late, turned on investigations in statistics—most famously with his introduction of the light quantum, now in its second century, but equally with his penultimate contribution to the new mechanics (on Bose–Einstein statistics) in 1924. Shortly after, his statistics was incorporated into the new (matrix and wave) mechanics. But there remained puzzles, even setting to one side the question of his last contribution (on the completeness of quantum mechanics). A number of these centre on the concept of particle indistinguishability, which will occupy us greatly in what follows.

To keep the discussion within reasonable bounds, and for the sake of historical transparency, I shall use only the simplest examples, and the elementary combinatoric arguments widely used at the time. For similar reasons, I shall largely focus on Bose–Einstein statistics (and I shall neglect parastatistics entirely). Hence, whilst not a study of the history of quantum statistics, I shall be speaking to Einstein's time.

But if not anachronistic, my way of putting things is certainly idiosyncractic, and calls for some stage-setting.

Section snippets

The puzzle

These are the puzzling features to be explained: distinguishable particles, classically, obey Maxwell–Boltzmann statistics, but so do indistinguishable (permutable) particles. In quantum mechanics, distinguishable particles also obey Maxwell–Boltzmann statistics; but not so indistinguishable ones. There is evidently something about the combination of permutation symmetry and quantum mechanics that leads to a difference in statistics. What, precisely?

Were the concept of indistinguishability

Against classical indistinguishability

The clearest argument in the literature against classical indistinguishability is that the principle is not needed (what I shall call the ‘dispensability argument’); a thesis due to Ehrenfest and Trkal (1920), and subsequently defended by van Kampen (1984). The objection that classical indistinguishability is incoherent is more murky, and has rarely been defended explicitly; for that I consider only van Kampen (1984) and Bach (1997).

Ehrenfest and Trkal considered the equilibrium condition for

Demystifying classical indistinguishability

The answer, presumably, is that we surely can single out classical particles uniquely, by reference to their trajectories. But there is a key objection to this line of thinking: so can quantum particles, at least in certain circumstances, be distinguished by their states. No matter whether the state is localized or not, the ‘up’ state of spin, for example, is distinguished from the ‘down’, and may well be distinguished in this way over time. In such cases, particle properties can be used as

The explanation for quantum statistics

To begin at the beginning:

The distribution of energy over each type of resonator must now be considered, first, the distribution of the energy E over the N resonators with frequency υ. If E is regarded as infinitely divisible, an infinite number of different distributions is possible. We, however, consider—and this is the essential point—E to be composed of a determinate number of equal finite parts and employ in their determination the natural constant h=6.55×10-27ergs. This constant,

Addenda

What explains the difference between classical and quantum statistics? The structure of their state spaces: in the quantum case the measure is discrete, the sum over states, but in the classical case it is continuous.20 This makes a difference when one passes to the quotient space

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