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 _g , 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 ℜ_C for finite open space-time regionsC are assumed, with the usual transformation properties of$\mathfrak{G}$ underU(a, Λ). The collection of allX∈ℜ_C 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 _g — 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.