Representation of Complex Probabilities

Abstract

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 $\C^D$, 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 $\C^D$. 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.

REVTeX, 15 pages, no figures, uuencoded

References in corpus (1)

• On the Segregation Phenomenon in Complex Langevin Simulation

Cited by in corpus (22)

• Complex coacervation: A field theoretic simulation study of polyelectrolyte complexation
• Complex Langevin and other approaches to the sign problem in quantum many-body physics
• Complex Langevin Equations and Schwinger-Dyson Equations
• Status of Complex Langevin
• Does the complex Langevin method give unbiased results?
• Complex Probabilities on R^N as Real Probabilities on C^N and an Application to Path Integrals
• Combining the complex Langevin method and the generalized Lefschetz-thimble method
• Existence of positive representations for complex weights
• Gibbs sampling of complex valued distributions
• Beyond complex Langevin equations II: a positive representation of Feynman path integrals directly in the Minkowski time
• Beyond Complex Langevin Equations: positive representation of a class of complex measures
• Positive Representations of a Class of Complex Measures
• Reconciling fractional entropy and black hole entropy compositions
• Beyond complex Langevin equations I: two simple examples
• Representation of complex probabilities and complex Gibbs sampling
• Positive representations of complex distributions on groups
• Explicit positive representation for weights on $R^d$
• Necessary and sufficient conditions for correctness of complex Langevin
• The Role of Integration Cycles in Complex Langevin Simulations
• Satisfying positivity requirement in the Beyond Complex Langevin approach
• Beyond Complex Langevin Equations: a Progress Report
• Correctness criteria for complex Langevin