Beyond complex Langevin equations

TL;DR

This paper introduces a new method to represent complex weights with positive distributions, enabling potential direct Minkowski time path integrals, and demonstrates its application in quantum mechanics examples.

Contribution

It generalizes the complex Gaussian solution to arbitrary complex slopes and constructs explicit positive representations for complex path integrals.

Findings

Explicit positive representations for complex distributions are constructed.

The method is successfully applied to quantum mechanical examples including a magnetic field.

Continuum limits require specific tuning of couplings.

Abstract

A simple integral relation between a complex weight and the corresponding positive distribution is derived by introducing a second complex variable. Together with the positivity and normalizability conditions, this sum rule allows to construct explicitly equivalent pairs of distributions in simple cases. In particular the well known solution for a complex gaussian distribution is generalized to an arbitrary complex slope. This opens a possibility of positive representation of Feynman path integrals directly in the Minkowski time. Such construction is then explicitly carried through in the second part of this presentation. The continuum limit of the new representation exists only if some of the additional couplings tend to infinity and are tuned in a specific way. The approach is then successfully applied to three quantum mechanical examples including a particle in a constant magnetic field -- a simplest prototype of a Wilson line. Further generalizations are shortly discussed and an amusing interpretation of new variables is briefly mentioned.