Quantum mechanics on manifolds and topological effects

TL;DR

This paper classifies topological effects in quantum mechanics on manifolds using invariance principles, linking observable algebras to the fundamental group, and showing the fundamental group as the sole source of topological effects without spin or external fields.

Contribution

It introduces a unique Lie-Rinehart C* algebra framework for quantum observables on manifolds, connecting representations to the fundamental group and topological effects.

Findings

Observable algebra is a Lie-Rinehart C* algebra.

Regular representations correspond to the fundamental group's unitary representations.

The fundamental group is the only source of topological effects without spin or external fields.

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.