Representation of Complex Probabilities

Let a ``complex probability’’ be a normalizable complex distribution $P(x)$ defined on ${\mathbb{R}}^D$. A real and positive probability distribution $p(z)$, defined on the complex plane ${\text{C}}^D$, is said to be a positive representation of $P(x)$ if $⟨Q(x){⟩}_P=⟨Q(z){⟩}_p$, where $Q(x)$ is any polynomial in ${\mathbb{R}}^D$ and $Q(z)$ its analytical extension on ${\text{C}}^D$. In this paper it is shown that every complex probability admits a real representation and a constructive method is given. Among other results, explicit positive representations, in any number of dimensions, are given for any complex distribution of the form Gaussian times polynomial, for any complex distributions with support at one point and for any periodic Gaussian times polynomial.