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Advances in Mathematical Physics
Volume 2015, Article ID 124393, 16 pages
http://dx.doi.org/10.1155/2015/124393
Research Article

The Global Symmetry Group of Quantum Spectral Beams and Geometric Phase Factors

1Department of Mathematics, University of Athens, Panepistimioupolis, 15784 Athens, Greece
2Parmenides Foundation, Kirchplatz 1, Pullach, 82049 Munich, Germany

Received 10 November 2014; Accepted 9 March 2015

Academic Editor: Soheil Salahshour

Copyright © 2015 Elias Zafiris. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose a cohomological modelling schema of quantum state spaces and their connectivity structures in relation to the formulation of global geometric phase phenomena. In the course of this schema, we introduce the notion of Hermitian differential line sheaves or unitary rays and classify their gauge equivalence classes in terms of a global differential invariant given by the de Rham cohomology class of the curvature. Furthermore, we formulate and interpret physically the curvature recognition integrality theorem for unitary rays. Using this recognition theorem, we define the notion of a quantum spectral beam and show that it has an affine space structure with structure group given by the characters of the fundamental group.