Quantum mechanics on manifolds and topological effects

Abstract

A unique classification of the topological effects associated to quantum mechanics on manifolds is obtained on the basis of the invariance under diffeomorphisms and the realization of the Lie-Rinehart relations between the generators of the diffeomorphism group and the algebra of infinitely differentiable functions on the manifold. This leads to a unique ("Lie-Rinehart") C* algebra as observable algebra; its regular representations are shown to be locally Schroedinger and in one to one correspondence with the unitary representations of the fundamental group of the manifold. Therefore, in the absence of spin degrees of freedom and external fields, the first homotopy group of the manifold appears as the only source of topological effects.

A few comments have been added to the Introduction, together with related references; a few words have been changed in the Abstract and a Note added to the Title

References in corpus (5)

• Lie-Rinehart algebras, descent, and quantization
• Lie Groupoids and Lie algebroids in physics and noncommutative geometry
• Topological Factors Derived From Bohmian Mechanics
• Quantization of kinematics on configuration manifolds
• Connes-Lott model building on the two-sphere

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