Existence of positive representations for complex weights

TL;DR

This paper demonstrates that complex weights in high-dimensional integrals can be represented by positive real weights on complexified spaces, enabling Monte Carlo methods for complex-weighted integrals.

Contribution

It introduces a general method to represent complex weights as positive real weights on complex manifolds, extending to compact Lie groups.

Findings

Complex weights can be represented by positive weights on C^d.

The method applies to high-dimensional integrals and Lie groups.

Enables Monte Carlo techniques for complex-weighted integrals.

Abstract

The necessity of computing integrals with complex weights over manifolds with a large number of dimensions, e.g., in some field theoretical settings, poses a problem for the use of Monte Carlo techniques. Here it is shown that very general complex weight functions P(x) on R^d can be represented by real and positive weights p(z) on C^d, in the sense that for any observable f, <f(x)>_P = <f(z)>_p, f(z) being the analytical extension of f(x). The construction is extended to arbitrary compact Lie groups.