Quantum Mechanics on Manifolds

TL;DR

This paper proposes a framework for defining quantum mechanics on manifolds, demonstrating its application to homogeneous spaces and exploring the existence of multiple inequivalent realizations.

Contribution

It introduces a method to realize quantum mechanics on manifolds as unitary representations, applicable to homogeneous spaces, and analyzes the multiplicity of inequivalent realizations.

Findings

Quantum mechanics can be formulated on manifolds using unitary representations.

Multiple inequivalent realizations exist for quantum mechanics on certain manifolds.

Examples include quantum mechanics on spheres, tori, and projective spaces.

Abstract

A definition of quantum mechanics on a manifold $ M $ is proposed and a method to realize the definition is presented. This scheme is applicable to a homogeneous space $ M = G / H $. The realization is a unitary representation of the transformation group $ G $ on the space of vector bundle-valued functions. When $ H \ne \{ e \} $, there exist a number of inequivalent realizations. As examples, quantum mechanics on a sphere $ S^n $, a torus $ T^n $ and a projective space $ \RP $ are studied. In any case, it is shown that there are an infinite number of inequivalent realizations.