Diffusion of Brownian particles and Liouville field theory

TL;DR

This paper reviews a transformation simplifying exponential potentials and demonstrates how Liouville field theory can be mapped into a polynomial interaction field theory, illustrated with a random walk example.

Contribution

It introduces a transformation that simplifies non-polynomial exponential potentials and applies it to connect Liouville theory with polynomial field theories.

Findings

Liouville field theory can be mapped to polynomial interaction field theory.

The transformation simplifies analysis of exponential potentials.

Application demonstrated with a particle in delta function potentials.

Abstract

In this work we review a recently proposed transformation which is useful in order to simplify non-polynomial potentials given in the form of an exponential. As an application, it is shown that the Liouville field theory may be mapped into a field theory with a polynomial interaction between two scalar fields and a massive vector field. The used methodology is illustrated with the help of the simple case of a particle performing a random walk in a delta function potentials.