Representation of Complex Probabilities

Let a ``complex probability'' be a normalizable complex distribution $P(x)$ defined on $\R^D$. 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 $\langle Q(x)\rangle_P = \langle Q(z)\rangle_p$, where $Q(x)$ is any polynomial in $\R^D$ 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.Comment: REVTeX, 15 pages, no figures, uuencode

Representation of complex probabilities and complex Gibbs sampling

Complex weights appear in Physics which are beyond a straightforward importance sampling treatment, as required in Monte Carlo calculations. This is the well-known sign problem. The complex Langevin approach amounts to effectively construct a posi\-tive distribution on the complexified manifold reproducing the expectation values of the observables through their analytical extension. Here we discuss the direct construction of such positive distributions paying attention to their localization on the complexified manifold. Explicit localized repre\-sentations are obtained for complex probabilities defined on Abelian and non Abelian groups. The viability and performance of a complex version of the heat bath method, based on such representations, is analyzed.Comment: Proceedings of Lattice 2017 (The 35th International Symposium on Lattice field Theory). 8 pages, 4 figure

SU(6)⊃\supsetSU(3)xSU(2) and SU(8)⊃\supsetSU(4)xSU(2) Clebsch-Gordan coefficients

Tables of scalar factors are presented for 63x63 and 120x63 in SU(8)$\supset$SU(4)xSU(2), and for 35x35 and 56x35 in SU(6)$\supset$SU(3)xSU(2). Related tables for SU(4)$\supset$SU(3)xU(1) and SU(3)$\supset$SU(2)xU(1) are also provided so that the Clebsch-Gordan coefficients can be completely reconstructed. These are suitable to study meson-meson and baryon-meson within a spin-flavor symmetric scheme.Comment: 30 pages, mostly table