Systems with outer constraints. Gupta-Bleuler electromagnetism as an algebraic field theory

Abstract

Since there are some important systems which have constraints not contained in their field algebras, we develop here in aC*-context the algebraic structures of these. The constraints are defined as a groupG acting as outer automorphisms on the field algebra ℱ, α:G ↦ Aut ℱ, α _G ⊄ Inn ℱ, and we find that the selection ofG-invariant states on ℱ is the same as the selection of states ω onM(G $M(G\mathop \times \limits_\alpha F)$ ℱ) by ω(U _g)=1∨g∈G, whereU _g ∈M (G $M(G\mathop \times \limits_\alpha F)$ ℱ)/ℱ are the canonical elements implementing α _g . These states are taken as the physical states, and this specifies the resulting algebraic structure of the physics inM(G $M(G\mathop \times \limits_\alpha F)$ ℱ), and in particular the maximal constraint free physical algebra ℛ. A nontriviality condition is given for ℛ to exist, and we extend the notion of a crossed product to deal with a situation whereG is not locally compact. This is necessary to deal with the field theoretical aspect of the constraints. Next theC*-algebra of the CCR is employed to define the abstract algebraic structure of Gupta-Bleuler electromagnetism in the present framework. The indefinite inner product representation structure is obtained, and this puts Gupta-Bleuler electromagnetism on a rigorous footing. Finally, as a bonus, we find that the algebraic structures just set up, provide a blueprint for constructive quadratic algebraic field theory.