Abstract

The purpose of this article is to give a general overview of permutations in physics, particularly the symmetry of theories under permutations. Particular attention is paid to classical mechanics, classical statistical mechanics and quantum mechanics. There are two recurring themes: (i) the metaphysical dispute between haecceitism and anti-haecceitism, and the extent to which this dispute may be settled empirically; and relatedly, (ii) the way in which elementary systems are individuated in a theory's formalism, either primitively or in terms of the properties and relations those systems are represented as bearing. Section 1 introduces permutations and provides a brief outline of the symmetric and braid groups. Section 2 discusses permutations in the general setting provided by model theory, in particular providing some definitions and elementary results regarding the permutability and indiscernibility of objects. Section 3 lays some philosophical groundwork for later sections, in particular articulated the distinction between haecceitism and anti-haecceitism and the distinction between transcendental and qualitative individuation. Section 4 addresses classical mechanics and introduces the procedure of quotienting, under which permutable states are identified. Section 5 addresses classical statistical mechanics, and outlines a number of equivalent ways to implement permutation invariance. I also briefly outline how particles may be qualitatively individuated in this framework. Section 6 addresses quantum mechanics. This contains an outline of: the representation theory of the symmetric groups; the topological approach to quantum statistics, in which the braid groups become relevant; and a brief proposal for qualitatively individuating quantum particles, and its implications for entanglement. Section 7 concludes with a discussion of equilibrium ensembles in the classical and quantum theories under permutation invariance. A (much) shorter version of this paper was published as a chapter in E. Knox & A. Wilson (eds), the Routledge Companion to Philosophy of Physics (Routledge, 2021), pp. 578-594.