Positive representations of complex distributions on 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.