Generally covariant quantum mechanics on 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., C∞(Q)⊗Mn(C) which leads to non-Abelian Yang-Mills theories, and of noncommutative tori Tθd. Moreover, we examine models over the Moyal-deformed plane Rθ2. Assuming the conservation of electrical charges, we show that in this case the canonical uncertainty relation [xk,ẋl]=igkl with metric gkl is only consistent if gkl is constant.