The existence of positive representations for complex weights

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 Rd can be represented by real and positive weights p(z) on Cd, 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.