Algebras of observables with continuous representations of symmetry groups

Abstract

Concrete C*-algebras, interpreted physically as algebras of observables, are defined for quantum mechanics and local quantum field theory. Aquantum mechanical system is characterized formally by a continuous unitary representation up to a factorU g of a symmetry group $$\mathfrak{G}$$ in Hilbert space ℌ and a von Neumann algebra ℜ on ℌ invariant with respect toU g . The set $$\mathfrak{A}$$ of all operatorsX∈ℜ such thatU g X U −1 , as a function ofg∈ $$\mathfrak{G}$$ , is continuous with respect to the uniform operator topology, is aC*-algebra called thealgebra of observables. The algebra ℜ is shown to be the weak (or strong) closure of $$\mathfrak{A}$$ . Infield theory, a unitary representation up to a factorU(a, Λ) of the proper inhomogeneous Lorentz group $$\mathfrak{G}$$ and local von Neumann algebras ℜC for finite open space-time regionsC are assumed, with the usual transformation properties of $$\mathfrak{G}$$ underU(a, Λ). The collection of allX∈ℜC giving uniformly continuous functionsU (a, Λ)X U −1 (a, Λ) on $$\mathfrak{G}$$ is then a localC*-algebra $$\mathfrak{A}_C $$ , called thealgebra of local observables. The algebra $$\mathfrak{A}_C $$ is again weakly (or strongly) dense in ℜ c . The norm-closed union $$\mathfrak{A}$$ of the $$\mathfrak{A}_C $$ for allC is calledalgebra of quasilocal observables (or quasilocal algebra). In either case, the group $$\mathfrak{G}$$ is represented by automorphisms V g resp. V(a, Λ) — with V g X=U g X U −1 — of theC*-algebra $$\mathfrak{A}$$ , and this is astrongly continuous representation of $$\mathfrak{G}$$ on the Banach space $$\mathfrak{A}$$ . Conditions for V (a, Λ) can then be formulated which correspond to the usualspectrum condition forU (a, Λ) in field theory.