A covariant fermionic path integral for scalar Langevin processes with multiplicative white noise

TL;DR

This work presents a covariant fermionic path-integral formulation for overdamped Langevin processes with multiplicative white noise, ensuring the generating functional remains form-invariant under non-linear coordinate changes through explicit transformations of auxiliary and Grassmann fields. By constructing a continuous-time framework with a delta-functional constraint and a Grassmann representation of the Jacobian, the authors derive a covariant action that encodes multiplicative noise and recover the Onsager-Machlup functional after integrating out auxiliary variables, matching results from higher-order discretization. The paper demonstrates two consistent routes to the same covariant Onsager-Machlup action, clarifying how quadratic terms emerge in a purely continuous-time approach and establishing measure invariance under $u=U(x)$ transformations. These results bridge discretization-based and functional approaches, enabling symmetry analyses and potential generalizations to general stochastic calculi ($\alpha$-family) and higher-dimensional settings, with implications for non-equilibrium field theories.

Abstract

We revisit the construction of the fermionic path-integral representation of overdamped scalar Langevin processes with multiplicative white noise, focusing on the covariance of the generating functional under non-linear changes of variables. We identify the transformations of the auxiliary (commuting and anticommuting) variables that ensure covariance under such transformations. The subtleties induced by the non-differentiable trajectories of the stochastic dynamics are encoded in the fermionic statistics. Upon integrating out the auxiliary variables, we derive the Onsager-Machlup formulation, which agrees with the one recently obtained using a higher-order discretization scheme. In contrast to the latter, the construction proposed here is formulated directly in continuous time.