A Stochastic Origin of Spacetime Non-Commutativity

TL;DR

This paper offers a stochastic interpretation of spacetime non-commutativity, linking it to the continuity of paths in quantum path integrals and demonstrating how discontinuous paths relate to non-commutative geometries like $2$-Minkowski.

Contribution

It introduces a novel stochastic framework connecting path discontinuities with spacetime non-commutativity and explores implications for deformed translation symmetries.

Findings

Discontinuous paths lead to non-commutative spacetime structures.

Continuous paths correspond to commutative spacetime.

Modified Leibniz rule relates to deformed translation generators.

Abstract

We propose a stochastic interpretation of spacetime non-commutativity 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 non-commutative spacetime structures. As an example we introduce discontinuous paths from which the $\kappa$-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 $\kappa$-Poincar\'e algebra. Our findings suggest that spacetime non-commutativity can be understood as a result of fundamental discreteness in temporal and/or spatial evolution.