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Theoretical and Mathematical Physics

On the relation between quantum propagators and geometrical parallel transport

G. A. Vilkovyskii

Published Jul 1, 1973• 0 citations
Publisher
DOI
10.1007/bf01035622

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The problem of a geometric phase for an open quantum system is reinvestigated in a unifying approach. Two of the existing methods to define geometric phase, one by Uhlmann's approach and the other by a kinematic approach, which have been considered to be distinct, are shown to be related in this framework. The method is based upon purification of a density matrix by its uniform decomposition and a generalization of the parallel transport condition obtained from this decomposition. It is shown that the generalized parallel transport condition can be satisfied when Uhlmann's condition holds. However, it does not mean that all solutions of the generalized parallel transport condition are compatible with those of Uhlmann's. It is also shown how to recover the earlier known definitions of geometric phase as well as how to generalize them when degeneracy exists and varies in time.