Symmetries and Paraparticles as a Motivation for Structuralism

Abstract

This paper 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 symmetry, in which states (vectors, rays or density operators) are not fixed by all permutations (called 'paraparticle states'). Thus Stachel's analogy is revived.

Entities Without Intrinsic Physical Identity

“…This last point shows in particular that primitive contextual diversity does not imply haecceitistic differences (it is meaningless to consider contextual objects independently of the actual qualitative relations they enter into, independently of the actual qualitative structures they are part of), whereas haecceitistic diversity does (Ladyman 2007), at least within the general understanding (but see below). Indeed primitive contextual diversity is more naturally associated with the denial of such haecceitistic differences, which is in line with fundamental physical features such as general permutation invariance in quantum theory and general relativity (see the introduction and in particular the recent discussion in Caulton and Butterfield 2012b).…”

The role of symmetry in the interpretation of physical theories

“…3). 12 An argument to this effect in the case where AΦ ∼ = Sn, the group of permutations on n objects, is given in Caulton & Butterfield (2012).…”

“…Besides, it seems to me that there are both analytic symmetries that are not gauge and gauge symmetries that are not analytic. Permutations in many-particle quantum mechanics are surely an example of the former (see Caulton & Butterfield 2012); the revaluation of a currency in a foreign-exchange market may be an (admittedly recherché!) example of the latter.…”

The Conventionality of Parastatistics

“…In recent debates regarding structuralism and haecceities, Caulton and Butterfield [2012] employ paraparticles to find a way around Oliver Pooley's criticism of Stachel [2002]. Stachel has argued that the general relativistic hole argument possesses an analogue in quantum theory where permutation invariance plays the role of diffeomorphism invariance.…”

“…(ii) Structuralism and Haecceities -Paraparticles have also featured in philosophical arguments concerning identical particles and structural realism, most recently by Caulton and Butterfield [2012]. They claim that the possibility (in principle) of paraparticles undermines the "quantum hole argument" of Stachel [2002], and supports anti-haecceitism about fundamental particles.…”