Stochastic Mechanics as a Gauge Theory

TL;DR

This paper demonstrates that non-relativistic quantum mechanics can be modeled as a gauge-invariant classical stochastic process, revealing new insights into the gauge symmetry and the classical-quantum correspondence.

Contribution

It introduces a gauge-theoretic representation of quantum mechanics using classical diffusion with Z_4 symmetry and establishes the emergence of U(1) gauge symmetry in the continuum limit.

Findings

Quantum density matrices as averages over classical distributions

Quantum conditioning described via gauge-invariant variables

Quantum mechanics equivalent to gauge-invariant stochastic processes

Abstract

We show that non-relativistic Quantum Mechanics can be faithfully represented in terms of a classical diffusion process endowed with a gauge symmetry of group Z_4. The representation is based on a quantization condition for the realized action along paths. A lattice regularization is introduced to make rigorous sense of the construction and then removed. Quantum mechanics is recovered in the continuum limit and the full U(1) gauge group symmetry of electro-magnetism appears. Anti-particle representations emerge naturally, albeit the context is non-relativistic. Quantum density matrices are obtained by averaging classical probability distributions over phase-action variables. We find that quantum conditioning can be described in classical terms but not through the standard notion of sub sigma-algebras. Delicate restrictions arise by the constraint that we are only interested in the algebra of gauge invariant random variables. We conclude that Quantum Mechanics is equivalent to a theory of gauge invariant classical stochastic processes we call Stochastic Mechanics.