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 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 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.
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Work supported in part by the Deutsche Forschungsgemeinschaft.
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Kraus, K. Algebras of observables with continuous representations of symmetry groups. Commun.Math. Phys. 7, 99–111 (1968). https://doi.org/10.1007/BF01648329
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DOI: https://doi.org/10.1007/BF01648329