Stochastic origin of spacetime noncommutativity

Abstract

We propose a stochastic interpretation of spacetime noncommutativity starting from the path integral formulation of quantum mechanical commutation relations. We discuss how the (non)commutativity of spacetime is inherently related to the continuity or discontinuity of paths in the path integral formulation. Utilizing Wiener processes, we demonstrate that continuous paths lead to commutative spacetime, whereas discontinuous paths correspond to noncommutative spacetime structures. As an example we introduce discontinuous paths from which the κ-Minkowski spacetime commutators can be obtained. Moreover we focus on modifications of the Leibniz rule for differentials acting on discontinuous trajectories. We show how these can be related to the deformed action of translation generators focusing, as a working example, on the κ-Poincaré algebra. Our findings suggest that spacetime noncommutativity can be understood as a result of fundamental discreteness in temporal and/or spatial evolution. Published by the American Physical Society 2025