Automorphisms of the canonical anticommutation relations and index theory |
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Authors: | AL Carey CA Hurst DM OBrien |
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Institution: | Department of Pure Mathematics, The University of Adelaide, Adelaide, South Australia 5001, Australia;Department of Mathematical Physics, The University of Adelaide, Adelaide, South Australia 5001, Australia |
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Abstract: | Let H be a complex Hilbert space, P+ an orthogonal projection on H, and P? the complementary projection. If is any symmetrically normed ideal in the ring of bounded operators on H, then we consider the group of unitary operators on H such that P+UP?and P?UP+ lie in . When is the Hilbert-Schmidt class, these unitaries define automorphisms of the C1-algebra of the canonical anticommutation relations over H which are implementable in the representation of determined by P?. We investigate the structure of the group , proving in particular that it has infinitely many connected components, k, labelled by the Fredholm index of P+UP+. The connected component of the identity, 0, is generated by unitaries of the form exp(iA), with A self-adjoint and P+AP? in . Finally we consider an application of these results to two dimensional field theory, showing in particular that the charge and chiral charge quantum numbers arise as the Fredholm indices of P±UP± for certain unitary U on L2(, 2) |
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