Positive representations of complex distributions on groups

TL;DR

This paper develops methods to construct positive representations of complex distributions on group manifolds, enabling Monte Carlo simulations for lattice gauge theories despite the sign problem.

Contribution

It introduces a framework for creating positive representations of complex weights on compact Lie groups, addressing the sign problem in lattice gauge theories.

Findings

Constructed positive representations for complex weights on Lie groups.

Analyzed the impact of representation localization on variance.

Provided techniques applicable to Abelian and non-Abelian groups.

Abstract

A normalizable complex distribution $P(x)$ on a manifold $\mathcal{M}$ can be regarded as a complex weight, thereby allowing to define expectation values of observables $A(x)$ defined on $\mathcal{M}$. Straightforward importance sampling, $x\sim P$, is not available for non positive $P$, leading to the well-known sign (or phase) problem. A positive representation $\rho(z)$ of $P(x)$ is any normalizable positive distribution on the complexified manifold $\mathcal{M}^c$, such that, $\langle A(x)\rangle_P = \langle A(z) \rangle_\rho$ for a dense set of observables, where $A(z)$ stands for the analytically continued function on $\mathcal{M}^c$. Such representations allow to carry out Monte Carlo calculations to obtain estimates of $\langle A(x) \rangle_P$, through the sampling $z \sim \rho$. In the present work we tackle the problem of constructing positive representations for complex weights defined on manifolds of compact Lie groups, both Abelian and non Abelian, as required in lattice gauge field theories. Since the variance of the estimates increase for broad representations, special attention is put on the question of localization of the support of the representations.