Generally covariant quantum mechanics on noncommutative configuration spaces

TL;DR

This paper extends algebraic formulations of Feynman's proof to noncommutative spaces, providing an axiomatic approach to quantum mechanics without relying on specific coordinates, and explores examples like noncommutative tori and Moyal planes.

Contribution

It introduces a coordinate-independent axiomatic framework for nonrelativistic quantum mechanics on noncommutative spaces, generalizing previous algebraic approaches and analyzing specific models.

Findings

Canonical uncertainty relation requires constant metric g_{kl} on Moyal plane

Provides detailed examples including nonabelian Yang-Mills theories and noncommutative tori

Extends Feynman's proof to noncommutative configuration spaces

Abstract

We generalize the previously given algebraic version of "Feynman's proof of Maxwell's equations" to noncommutative configuration spaces. By doing so, we also obtain an axiomatic formulation of nonrelativistic quantum mechanics over such spaces, which, in contrast to most examples discussed in the literature, does not rely on a distinguished set of coordinates. We give a detailed account of several examples, e.g., of nonabelian Yang-Mills theories, and of noncommutative tori. Moreover we, examine models over the Moyal-deformed plane. Assuming the conservation of electrical charges, we show that in this case the canonical uncertainty relation [x_k, \dot{x}_l] = ig_{kl} with metric g_{kl} is only consistent if g_{kl} is constant.