Relating Field Theories via Stochastic Quantization

TL;DR

The paper demonstrates that stochastic quantization can relate a broad class of models by mapping a $d$-dimensional classical theory to a $d+1$-dimensional quantum system whose ground-state wavefunction squared reproduces the classical partition function: $ig angle ext{ground}| ext{ground} angle = Z_{ ext{cl}}$. It systematically reviews Langevin, Fokker–Planck, and supersymmetric formulations, and introduces a discrete analog that underpins the quantum dimer model and quantum crystal melting. Through concrete examples—from zero-dimensional SUSY QM to bosonic fields and gauged WZW/topologically massive gauge theory, and extending to the quantum crystal/melting problems—the authors connect these theories within a common stochastic-quantization framework and explore holographic/string-theory interpretations, including Wheeler–DeWitt perspectives and links to the topological A-model and Kähler gravity. The work argues for a practical regularization lens and proposes that stochastic quantization could illuminate a path toward a topological M-theory-like description of crystal-based and geometric quantum systems.

Abstract

This note aims to subsume several apparently unrelated models under a common framework. Several examples of well-known quantum field theories are listed which are connected via stochastic quantization. We highlight the fact that the quantization method used to obtain the quantum crystal is a discrete analog of stochastic quantization. This model is of interest for string theory, since the (classical) melting crystal corner is related to the topological A-model. We outline several ideas for interpreting the quantum crystal on the string theory side of the correspondence, exploring interpretations in the Wheeler-De Witt framework and in terms of a non-Lorentz invariant limit of topological M-theory.