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The density manifold and configuration space quantization. (English) Zbl 0642.58009

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.

MSC:

58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
60H07 Stochastic calculus of variations and the Malliavin calculus
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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