Algebraic quantization of systems with gauge degeneracy
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.
degenerate linear algebraic field theory
linear boson field theory with linear hermitian constraints
scientific article; zbMATH DE number 3925602
scientific article