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Routledge eBooks

Permutations

Adam Caulton

Published Aug 11, 2021• 3 citations
Publisher

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. A recurring theme is the metaphysical dispute between haecceitism and anti-haecceitism, and the extent to which this dispute may be settled empirically.

DOI
10.4324/9781315623818-54

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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. A recurring theme is the metaphysical dispute between haecceitism and anti-haecceitism, and the extent to which this dispute may be settled empirically.
The permutation symmetry of quantum mechanics is widely thought to imply a sort of metaphysical underdetermination about the identity of particles. Despite claims to the contrary, this implication does not hold in the more fundamental quantum field theory, where an ontology of particles is not generally available. Although permutations are often defined as acting on particles, a more general account of permutation symmetry can be formulated using superselection theory. As a result, permutation symmetry applies even in field theories with no particle interpretation. The quantum mechanical account of permutations acting on particles is recovered as a special case.
The permutation symmetry of Newtonian stationary-state functions – such as the total momentum and the Hamiltonian – is crucial for correct state counting in classical statistical mechanics, but plays virtually no role for classical many-body systems. In contrast, it yields novel effects for quantum many-body systems of equal/identical particles. In order to avoid the misunderstandings Jaynes (1992) has rightly stressed, basic notions, such as equal and identical particles, (in)distinguishability and exchangeability (the really relevant notion) are discussed first within classical mechanics. Then, they are applied to the weight (limiting) functions. The permutation symmetry of the wave functions follow largely from that of the weight functions. The different effects for bosons and fermions are mentioned and illustrated. For the generalization towards anyons and the fractional quantum Hall effect, some ideas for further exploration are proposed. Entanglement is treated as a consequence of the conservation laws, in particular, for the angular momentum.
This article develops an analogy proposed by Stachel between general relativity (GR) and quantum mechanics (QM) as regards permutation invariance. Our main idea is to overcome Pooley's criticism of the analogy by appeal to paraparticles. In GR, the equations are (the solution space is) invariant under diffeomorphisms permuting spacetime points. Similarly, in QM the equations are invariant under particle permutations. Stachel argued that this feature—a theory's ‘not caring which point, or particle, is which’—supported a structuralist ontology. Pooley criticizes this analogy: in QM the (anti-)symmetrization of fermions and bosons implies that each individual state (solution) is fixed by each permutation, while in GR a diffeomorphism yields in general a distinct, albeit isomorphic, solution. We define various versions of structuralism, and go on to formulate Stachel's and Pooley's positions, admittedly in our own terms. We then reply to Pooley. Though he is right about fermions and bosons, QM equally allows more general types of particle symmetry, in which states (vectors, rays, or density operators) are not fixed by all permutations (called ‘paraparticle states’). Thus Stachel's analogy is revived. 1 Introduction2 Structuralism Applied to General Relativity and Quantum Mechanics 2.1 Structuralism and individuality 2.2 The semantics of the structuralist 2.3 General relativity and Leibniz equivalence3 Anti-haecceitism and Quantum Mechanics 3.1 Pure and mixed states; permutations 3.2 Symmetrization and indistinguishability 3.3 Quantum statistics and the bad argument for anti-haecceitism4 The Generalized Hole Argument for Sets 4.1 Models and possible worlds 4.2 Permutations and permutes 4.3 State descriptions and structure descriptions 4.4 Theories; permutability, fixity, and general permutability 4.5 Stachel's argument for structuralism5 Fixed Theories and Metaphysical Under-determination 5.1 Pooley's objection; amending Stachel's premises 5.2 Is SP anti-haecceitistic?6 Superselection and Paraparticles 6.1 Quantum permutability 6.1.1 Symmetry types and symmetric operators 6.1.2 Statistics and symmetrization; superselection 6.1.3 The two arguments for quantum permutability 6.2 Quantum general permutability 6.3 Examples of quantum permutability 6.3.1 Two quantum coins, again 6.3.2 A generalized ray
This study examines the epistemological implications of competinginterpretations of quantum states, focusing on the distinction between epistemicand ontological accounts of the wave function (Harrigan & Spekkens, 2010).Rather than treating interpretative disputes as purely metaphysical, this studyanalyses the extent to which quantum theory constrains, enables or underminesclaims to knowledge about physical reality (van Fraassen, 2008). By situatingthe epistemic–ontological debate within broader discussions of realism,objectivity and scientific explanation, this study argues that quantum theorynecessitates a reconceptualisation of epistemic access to reality (O’Neill,1997). Particular attention is given to the roles of uncertainty, contextuality andobserver-dependence in shaping the epistemic limits of quantum physics (Bohr,1935). The article concludes by proposing a moderate epistemic realism thatpreserves empirical objectivity while acknowledging principled limits onontological transparency (Healey, 2017).