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Gauge Invariance and Symmetry Breaking by Topology and Energy Gap

Scuola Normale Superiore, 52126 Pisa, Italy
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Academic Editor: Palle E.T. Jorgensen
Mathematics 2015, 3(4), 984-1000; https://doi.org/10.3390/math3040984
Received: 16 July 2015 / Revised: 8 October 2015 / Accepted: 12 October 2015 / Published: 22 October 2015
(This article belongs to the Special Issue Mathematical physics)
For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra A, with a non-trivial center Z, describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations of A. In particular, this gives rise to a gauge symmetry described by the action of Z. A distinguished case is when the center of the observables arises from the fundamental group of the manifold of the positions of the quantum system. Symmetries that do not commute with the topological invariants represented by elements of Z are then spontaneously broken in each irreducible representation of the observable algebra, compatibly with an energy gap; such a breaking exhibits a mechanism radically different from Goldstone and Higgs mechanisms. This is clearly displayed by the quantum particle on a circle, the Bloch electron and the two body problem. View Full-Text
Keywords: Weyl-polymer quantization; symmetry breaking by topology; energy gap Weyl-polymer quantization; symmetry breaking by topology; energy gap
MDPI and ACS Style

Strocchi, F.; Heissenberg, C. Gauge Invariance and Symmetry Breaking by Topology and Energy Gap. Mathematics 2015, 3, 984-1000.

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