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Superselection theory for subsystems. (English) Zbl 1049.81044

The authors examine the notion of subsystem in algebraic quantum field theory [R. Haag, Local quantum physics. Fields, particles, algebras. 2nd., ed., Texts and Monographs in Physics. Berlin: Springer (1996; Zbl 0857.46057)]. Starting with inclusions of observable nets \(\mathfrak{A}_1\subset\mathfrak{A}_2\) and corresponding field nets \(\mathfrak{F}_1\subset\mathfrak{F}_2\), they show using net cohomology that under usual assumptions this inclusion induces an inclusion functor of the associated tensor categories of transportable morphisms. They proceed to consider inclusions of field nets and give conditions for the existence of associated conditional expectations from the larger onto the smaller net. These are important in the proof that an inclusion of observable algebras gives rise to an inclusion of the corresponding canonical field nets. The final and central result of the paper is the proof that every intermediate net, i.e. a local net between an observable net and its canonical field net, arises as the fixed point net for the action of a closed subgroup of the global gauge group. Also, the superselection sectors of the intermediate net are shown to correspond to equivalence classes of irreducible representations of that subgroup. For this result, infinite statistics for the observable net has to be excluded. Related work: D. R. Davidson, Classification of subsystems of local algebras. Ph.D. Thesis, University of California at Berkeley, 1993) and the first two authors, Commun. Math. Phys. 217, No.1, 89–106 (2001; Zbl 0986.81067).

MSC:

81T05 Axiomatic quantum field theory; operator algebras
46L60 Applications of selfadjoint operator algebras to physics
81T13 Yang-Mills and other gauge theories in quantum field theory
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