On the Correct Convergence of Complex Langevin Simulations for Polynomial Actions

TL;DR

This paper investigates the conditions necessary for complex Langevin simulations to accurately compute integrals with complex polynomial actions, ensuring proper convergence in field theory applications.

Contribution

It defines the conditions for correct convergence of complex Langevin methods applied to polynomial actions in field theory.

Findings

Identifies key conditions for convergence in complex Langevin simulations

Clarifies the stationary limit criteria for proper convergence

Provides theoretical insights into complex Langevin process behavior

Abstract

There are problems in physics and particularly in field theory which are defined by complex valued weight functions $e^{-S}$ where $S$ is a polynomial action $S: R^n \rightarrow C $. The conditions under which a convergent complex Langevin calculation correctly simulates such integrals are discussed. All conditions on the process which are used to prove proper convergence are defined in the stationary limit.