Effective action for stochastic partial differential equations

TL;DR

This paper develops a functional integral formalism for stochastic partial differential equations (SPDEs), enabling the calculation of one-loop effective actions and potentials, and highlights the analogy between noise amplitude and Planck's constant.

Contribution

It introduces a minimalist approach to derive the one-loop effective action for arbitrary SPDEs with Gaussian noise, simplifying previous formalisms and establishing a noise-amplitude-based loop expansion.

Findings

Derived a general one-loop effective potential for SPDEs with translation-invariant Gaussian noise.

Showed the noise amplitude acts as a loop-counting parameter analogous to Planck's constant.

Compared the SPDE effective potential with quantum field theory results.

Abstract

Stochastic partial differential equations (SPDEs) are the basic tool for modeling systems where noise is important. In this paper we set up a functional integral formalism and demonstrate how to extract all the one-loop physics for an arbitrary SPDE subject to arbitrary Gaussian noise. It is extremely important to realize that Gaussian noise does not imply that the field variables undergo Gaussian fluctuations, and that these non-quantum field theories are fully interacting. Experience with quantum field theories (QFTs) has taught us that one-loop physics is often quite adequate to give a good description of the salient issues, and furthermore offers marked technical advantages: We can sidestep the complications inherent in the Martin-Siggia-Rose formalism (the SPDE analog of the BRST formalism used in QFT) and instead focus attention on a minimalist approach that uses only the physical fields (this ``direct approach'' is the SPDE analog of canonical quantization using physical fields.) We show how to define the effective action to all loops, and then focus on the one-loop effective action, and its specialization to constant fields: the effective potential. An important result is that the amplitude of the two-point function governing the noise acts as the loop-counting parameter and is the analog of Planck's constant hbar in this SPDE context. We derive a general expression for the one-loop effective potential of an arbitrary SPDE subject to translation-invariant Gaussian noise, and compare this with the one-loop potential for QFT.