Algebraic quantization of systems with gauge degeneracy
type: Item
instance of: scholarly article
title: Algebraic quantization of systems with gauge degeneracy
zbMATH Open document ID: 0578.46064
DOI: 10.1007/BF01205789
author: H. Grundling
author: C. A. Hurst
published in: Communications in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: This paper is concerned with the algebraic structures which evolve from the imposition of constraints as occur in degenerate physical theories. Assuming the constraints to be specified as unitary elements \(U_ i\) in the field \(C^*\)-algebra \({\mathcal F}\), the physical states \({\mathfrak S}_ D\) are selected by the condition \(\omega (u_ i)=1\). Nontriviality conditions are given, and the largest \(C^*\)-algebra \({\mathcal D}\) annihilated by all \(\omega\in {\mathfrak S}_ D\) is obtained. The algebra of observables \({\mathcal O}\) is chosen as the multiplier algebra of \({\mathcal D}\) in \({\mathcal J}\), this choice being vindicated by showing that the traditional observables \(\{U_ i\}'\) are in \({\mathcal O}\). The nondegenerate physical algebra is defined as \({\mathcal R}:={\mathcal O}/{\mathcal D}\). The relation between the state spaces of the various algebras involved is explored, as well as the automorphisms of \({\mathcal J}\) compatible with the construction. The physically acceptable transformations \({\mathfrak P}\) consist of the automorphisms which conserve \({\mathcal D}\). If T:\({\mathfrak P}\to Aut {\mathcal R}\) is the canonical map, the gauge transformations is Ker T. Various automorphism groups derived from \(\{U_ i\}\) and \({\mathcal D}\) are shown to be in Ker T, and Ker T is also characterized in terms of \({\mathfrak S}_ D.\) The Dirac version of electromagnetism is developed as an example of a degenerate linear algebraic field theory to which the preceding structures are applicable. It turns out to be the prototype for any linear boson field theory with linear hermitian constraints. For such theories the structure simplifies in that \({\mathcal O}=\{U_ i\}'\), and that \(\{U_ i\}\) should commute.
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 81T08
zbMATH DE Number: 3925602
zbMATH Keywords: field \(C^*\)-algebra
zbMATH Keywords: algebra of observables
zbMATH Keywords: multiplier algebra
zbMATH Keywords: state spaces
zbMATH Keywords: gauge transformations
zbMATH Keywords: automorphism groups
zbMATH Keywords: Dirac version of electromagnetism
zbMATH Keywords: degenerate linear algebraic field theory
zbMATH Keywords: linear boson field theory with linear hermitian constraints
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3925602