On Complex Langevin Dynamics and the Evaluation of Observables

TL;DR

This paper reviews complex Langevin dynamics for complex-valued actions, confirming its limitations in certain parameter regimes and revealing subtle issues in the relationship between the original measure and the complexified probability distribution.

Contribution

It provides a detailed analysis of complex Langevin dynamics applied to a U(1) one link model, highlighting its parameter-dependent accuracy and uncovering nuanced effects in observable estimation.

Findings

Complex Langevin works only within specific parameter ranges.

Moments of stochastic variables are misestimated, yet some derived observables are accurate.

The relationship between the original measure and the complexified distribution remains subtle.

Abstract

In stochastic quantisation, quantum mechanical expectation values are computed as averages over the time history of a stochastic process described by a Langevin equation. Complex stochastic quantisation, though theoretically not rigorously established, extends this idea to cases where the action is complex-valued by complexifying the basic degrees of freedom, all observables and allowing the stochastic process to probe the complexified configuration space. We review the method for a previously studied one-dimensional toy model, the U(1) one link model. We confirm that complex Langevin dynamics only works for a certain range of parameters, misestimating observables otherwise. A curious effect is observed where all moments of the basic stochastic variable are misestimated, although these misestimated moments may be used to construct, by a Taylor series, other observables that are reproduced correctly. This suggests a subtle but not completely resolved relationship between the original complex integration measure and the higher-dimensional probability distribution in the complexified configuration space, generated by the complex Langevin process.