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Journal of Physics A Mathematical and General

The geometric phase on Kahler manifolds

A. Stanley

Published Dec 7, 1993• 1 citation
Publisher

Abstract

A new expression for the phase one-form is derived in terms of a derivative of the Kahler potential for the class of projective Hilbert spaces which are Kahler manifolds. The technique used is that of imaginary time-translation, previously introduced by the author.
DOI
10.1088/0305-4470/26/23/006

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In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.