Satisfying positivity requirement in the Beyond Complex Langevin approach

Adam Wyrzykowski,Błażej Ruba Ruba

doi:10.1051/epjconf/201817511026

Adam Wyrzykowski, Błażej Ruba Ruba

Open Access

https://doi.org/10.1051/epjconf/201817511026

Copy DOI

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.

Highlights

Our goal is to represent integrals of complex densities over a real measure by integrals of real, positive probability distributions over a complex measure
The aim of this paper is analysis of a one-dimensional problem [15,16,17,18] to find a positive and normalizable probability distribution P(x, y) such that: f (x)ρ(x) dx = f (x + iy)P(x, y) dxdy for a given complex function ρ(x) and every f (x). This approach to the sign problem, related to the Complex Langevin method [1, 2], does not require introduction of the corresponding stochastic process. This method focuses only on satisfying the so-called matching conditions which follow from eq (1)
The two methods proposed in the previous section were used to find a positive probability distribution

Summary

Introduction

Our goal is to represent integrals of complex densities over a real measure by integrals of real, positive probability distributions over a complex measure. Complex Langevin method [1, 2] has become a popular, and in many cases successful [3,4,5,6], approach to perform this task. For a given complex function ρ(x) and every f (x) This approach to the sign problem, related to the Complex Langevin method [1, 2], does not require introduction of the corresponding stochastic process. Instead, this method focuses only on satisfying the so-called matching conditions which follow from eq (1).

Objectives

Results

Conclusion