Comment on Geometric Interpretation of Ito Calculus on the Lattice

TL;DR

This paper clarifies the covariant geometric interpretation of Ito calculus in the context of lattice gauge theories, providing a coordinate covariant formulation of Langevin and Fokker-Planck equations on group manifolds.

Contribution

It introduces a geometric, covariant formulation of Langevin dynamics in lattice gauge theories, enhancing understanding of stochastic quantization on group manifolds.

Findings

Covariant form of Langevin equation for gauge theories

Geometric interpretation of Ito calculus on group manifolds

Manifestly covariant collective field theory formulation

Abstract

A covariant nature of the Langevin equation in Ito calculus is clarified in applying stochastic quantization method to U(N) and SU(N) lattice gauge theories. The stochastic process is expressed in a manifestly general coordinate covariant form as a collective field theory on the group manifold. A geometric interpretation is given for the Langevin equation and the corresponding Fokker-Planck equation in the sense of Ito.