# Positive Representations of a Class of Complex Measures

## TL;DR

This paper explores methods to construct positive probability representations of complex measures on products of U(1) groups, with applications to abelian lattice gauge theories, providing theoretical conditions and concrete realizations.

## Contribution

It identifies necessary and sufficient conditions for positive representations of complex measures and proposes several explicit realizations with practical examples.

## Key findings

- Derived conditions for positive representations of complex measures
- Proposed multiple concrete realizations of these representations
- Applied methods to examples in abelian lattice gauge theories

## Abstract

We study the problem of constructing positive representations of complex measures. In this paper we consider complex densities on a direct product of $U(1)$ groups and look for representations by probability distributions on the complexification of those groups. After identifying general necessary and sufficient conditions we propose several concrete realizations. Finally we study some of those realizations in examples representing problems in abelian lattice gauge theories.