Localization via Automorphisms of the CARs: Local Gauge Invariance |
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Authors: | Hendrik Grundling and Karl-Hermann Neeb |
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Institution: | 1.Department of Mathematics,University of New South Wales,Sydney,Australia;2.Mathematisches Institut,Friedrich-Alexander-Universit?t Erlangen-Nürnberg,Erlangen,Germany |
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Abstract: | The classical matter fields are sections of a vector bundle E with base manifold M, and the space L
2(E) of square integrable matter fields w.r.t. a locally Lebesgue measure on M, has an important module action of Cb¥(M){C_b^\infty(M)} on it. This module action defines restriction maps and encodes the local structure of the classical fields. For the quantum
context, we show that this module action defines an automorphism group on the algebra of the canonical anticommutation relations,
CAR(L
2(E)), with which we can perform the analogous localization. That is, the net structure of the CAR(L
2(E)) w.r.t. appropriate subsets of M can be obtained simply from the invariance algebras of appropriate subgroups. We also identify the quantum analogues of restriction
maps, and as a corollary, we prove a well–known “folk theorem,” that the CAR(L
2(E)) contains only trivial gauge invariant observables w.r.t. a local gauge group acting on E. |
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Keywords: | |
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