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A Stochastic Origin of Spacetime Non-Commutativity

Michele Arzano, Folkert Kuipers

TL;DR

The paper addresses how spacetime non-commutativity can emerge from fundamental discreteness by recasting path integrals in a stochastic framework based on Wiener processes. It shows that continuous, nowhere-differentiable paths reproduce standard quantum commutators, while introducing discontinuous time paths yields κ-Minkowski-type spacetime with a Lie-algebra structure and deformed Leibniz rules tied to a κ-Poincaré Hopf algebra. The key results include a derivation of $[\hat{x}^0,\hat{x}^i] = i\,\hat{x}^i/\kappa$ from a discrete-time Poisson process and the demonstration that different left/right-continuous discretizations correspond to different coproduct bases, linking stochastic discreteness to deformed spacetime symmetries. The work suggests a broader framework in which spacetime discreteness, stochastic calculus, and non-commutative geometry inform quantum gravity phenomenology and offer new avenues to model spacetime at the Planck scale.

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 $κ$-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 non-commutativity can be understood as a result of fundamental discreteness in temporal and/or spatial evolution.

A Stochastic Origin of Spacetime Non-Commutativity

TL;DR

The paper addresses how spacetime non-commutativity can emerge from fundamental discreteness by recasting path integrals in a stochastic framework based on Wiener processes. It shows that continuous, nowhere-differentiable paths reproduce standard quantum commutators, while introducing discontinuous time paths yields κ-Minkowski-type spacetime with a Lie-algebra structure and deformed Leibniz rules tied to a κ-Poincaré Hopf algebra. The key results include a derivation of from a discrete-time Poisson process and the demonstration that different left/right-continuous discretizations correspond to different coproduct bases, linking stochastic discreteness to deformed spacetime symmetries. The work suggests a broader framework in which spacetime discreteness, stochastic calculus, and non-commutative geometry inform quantum gravity phenomenology and offer new avenues to model spacetime at the Planck scale.

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 -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 -Poincaré algebra. Our findings suggest that spacetime non-commutativity can be understood as a result of fundamental discreteness in temporal and/or spatial evolution.
Paper Structure (14 sections, 126 equations)

This paper contains 14 sections, 126 equations.

Table of Contents

  1. Introduction
  2. Non-commutative spacetime from fundamental discreteness: the path integral approach
  3. Operator ordering vs. time ordering
  4. Path integrals and Wiener processes
  5. Canonical commutation relation from non-differentiable paths
  6. Spacetime non-commutativity due to discontinuous paths
  7. Discontinuous paths and deformed Leibniz rule
  8. Differentiable Trajectories
  9. Continuous Trajectories
  10. Discontinuous trajectories
  11. Conclusions
  12. The Wiener process
  13. Velocity
  14. Quadratic Variation

Theorems & Definitions (2)

  • Definition
  • Definition