Particle spin from representations of the diffeomorphism group
type: Item
instance of: scholarly article
title: Particle spin from representations of the diffeomorphism group
zbMATH Open document ID: 0536.58011
DOI: 10.1007/BF01210847
author: Gerald A. Goldin
author: David Sharp
published in: Communications in Mathematical Physics
publication date: 1983-01-01T00:00:00Z
review text: It is shown that one can obtain particle spin directly from irreducible representations of the diffeomorphism group, without the need for anticommuting field operators. The semidirect product \(S\wedge K\) of Schwartz's space of functions on \(\mathbb R^ 3\) with the group \(K\) of diffeomorphisms of \(\mathbb R^ 3\) provides a model for quantum field theory. The representations of \(\mathrm{SL}(3,\mathbb R)\) are used to induce representations of the diffeomorphism group \(K\). The local currents are interpreted in these representations; the local measurement operators are then constructed by taking appropriate limits in a representation of \(S\wedge K\). The Hilbert space decomposes into invariant subspaces of fixed spin under the local rigid rotations. The generalization of the results to the case of \(N\) particles is also given. Finally, the physical interpretation of the procedure is discussed. It is pointed out that the description of particles with spin in which the local \(\mathrm{SU}(2)\) current group is adjoined to the group \(S\wedge K\) can be obtained from a unified point of view. The unitarily inequivalent representations of the same group \(S\wedge K\) can describe many different physical systems and, in particular, particles with integers or half-integer spin. The universality of the given model is possible owing to the infinitely many degrees of freedom embodied in the group of diffeomorphisms.
Mathematics Subject Classification ID: 81R99
Mathematics Subject Classification ID: 81R05
Mathematics Subject Classification ID: 58D05
Mathematics Subject Classification ID: 22E65
zbMATH DE Number: 3852059
zbMATH Keywords: semidirect product, local current
zbMATH Keywords: rigid rotation
zbMATH Keywords: spin density
reviewed by: Gheorghe Zet
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01210847
OpenAlex ID: W2061848814
description: scientific article; zbMATH DE number 3852059