Density manifold and configuration space quantization
This thesis develops the differential geometric structure of a Frechet manifold of densities, 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 Schroedinger 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 volume preserving diffeomorphisms. Using this structure, it is shown how a momentum map for the action of an isotropy subgroup on the cotangent bundle of the diffeomorphism group leads to Lie Poisson equations on the dual of a semidirect product Lie Algebra. A Poisson map is then obtained between the dual of this Lie algebra and the symplectic structure.
- Research Organization:
- Princeton Univ., NJ (USA)
- OSTI ID:
- 6680104
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICAL MANIFOLDS
MATHEMATICAL SPACE
PARTIAL DIFFERENTIAL EQUATIONS
QUANTIZATION
RIEMANN SPACE
SCHROEDINGER EQUATION
SPACE
STOCHASTIC PROCESSES
VARIATIONAL METHODS
WAVE EQUATIONS