Satisfying positivity requirement in the Beyond Complex Langevin approach

TL;DR

This paper investigates methods to find positive distributions matching complex densities by solving quadratic moment-matching conditions, using Groebner basis techniques, with a focus on Gaussian cases.

Contribution

It introduces a novel approach to satisfy positivity in the Beyond Complex Langevin method by solving quadratic moment-matching conditions with Groebner basis methods.

Findings

Approximate solutions closely match the exact Gaussian solution.

Groebner basis effectively solves quadratic moment conditions.

Method provides a pathway to positive distributions in complex density problems.

Abstract

The problem of finding a positive distribution, which corresponds to a given complex density, is studied. By the requirement that the moments of the positive distribution and of the complex density are equal, one can reduce the problem to solving the matching conditions. These conditions are a set of quadratic equations, thus Groebner basis method was used to find its solutions when it is restricted to a few lowest-order moments. For a Gaussian complex density, these approximate solutions are compared with the exact solution, that is known in this special case.