Journal Article

Symmetries and Paraparticles as a Motivation for Structuralism

Adam Caulton and Jeremy Butterfield

in The British Journal for the Philosophy of Science

Published on behalf of British Society for the Philosophy of Science

Volume 63, issue 2, pages 233-285
Published in print June 2012 | ISSN: 0007-0882
Published online November 2011 | e-ISSN: 1464-3537 | DOI: http://dx.doi.org/10.1093/bjps/axr034
Symmetries and Paraparticles as a Motivation for Structuralism

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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.

1Introduction

2Structuralism Applied to General Relativity and Quantum Mechanics

  2.1Structuralism and individuality

  2.2The semantics of the structuralist

  2.3General relativity and Leibniz equivalence

3Anti-haecceitism and Quantum Mechanics

  3.1Pure and mixed states; permutations

  3.2Symmetrization and indistinguishability

  3.3Quantum statistics and the bad argument for anti-haecceitism

4The Generalized Hole Argument for Sets

  4.1Models and possible worlds

  4.2Permutations and permutes

  4.3State descriptions and structure descriptions

  4.4Theories; permutability, fixity, and general permutability

  4.5Stachel's argument for structuralism

5Fixed Theories and Metaphysical Under-determination

  5.1Pooley's objection; amending Stachel's premises

  5.2Is SP anti-haecceitistic?

6Superselection and Paraparticles

  6.1Quantum permutability

   6.1.1Symmetry types and symmetric operators

   6.1.2Statistics and symmetrization; superselection

   6.1.3The two arguments for quantum permutability

  6.2Quantum general permutability

  6.3Examples of quantum permutability

   6.3.1Two quantum coins, again

   6.3.2A generalized ray

Journal Article.  22215 words.  Illustrated.

Subjects: Philosophy of Science ; Science and Mathematics

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