A Stochastic Origin of Spacetime Non-Commutativity

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 $\kappa$-Poincaré algebra. Our findings suggest that spacetime non-commutativity can be understood as a result of fundamental discreteness in temporal and/or spatial evolution.

23+10 pages; v2: extended discussion and references added

References in corpus (7)

• Anatomy of a deformed symmetry: field quantization on curved momentum space
• Geometric Event-Based Relativistic Quantum Mechanics
• Time Discretization From Noncommutativity
• Stochastic Mechanics and the Unification of Quantum Mechanics with Brownian Motion
• A group theoretic description of the $\kappa$-Poincaré Hopf algebra
• Relativistic Planck-scale polymer
• Quantum Mechanics from Stochastic Processes

Cited by in corpus (1)

• Striving to find a bridge between noncommutative and generalized entropy parameters in the harmonic oscillator dynamics