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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The density manifold and configuration space quantization


Author: John D. Lafferty
Journal: Trans. Amer. Math. Soc. 305 (1988), 699-741
MSC: Primary 58F06; Secondary 58B25, 60H07, 81C20, 81C25
DOI: https://doi.org/10.1090/S0002-9947-1988-0924776-9
MathSciNet review: 924776
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Abstract: The differential geometric structure of a Fréchet manifold of densities is developed, providing a geometrical framework for quantization related to Nelson's stochastic mechanics. The Riemannian and symplectic structures of the density manifold are studied, and the Schrödinger equation is derived from a variational principle. By a theorem of Moser, the density manifold is an infinite dimensional homogeneous space, being the quotient of the group of diffeomorphisms of the underlying base manifold modulo the group of diffeomorphisms which preserve the Riemannian volume. From this structure and symplectic reduction, the quantization procedure is equivalent to Lie-Poisson equations on the dual of a semidirect product Lie algebra. A Poisson map is obtained between the dual of this Lie algebra and the underlying projective Hilbert space.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0924776-9
Article copyright: © Copyright 1988 American Mathematical Society