Beyond Complex Langevin Equations: a Progress Report

After a short review of one of proposals to avoid complex stochastic processes in Complex Langevin studies, the recent progress in the former is reported. In particular, the new developments allow now to construct positive and normalizable representations for gaussian quantum mechanical, as well as field theoretical, path integrals directly in the Minkowski time. A relation to the idea of thimbles is also discussed.Comment: 7 pages, 2 figures, contribution to the Lattice2018 Symposiu

Positive Representations of a Class of Complex Measures

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.Comment: 19 pages, 3 figures, minor changes to bring it into agreement with published versio

Simulations of gaussian systems in Minkowski time

Many research programs aiming to deal with the sign problem were proposed since the advent of lattice field theory. Several of these try to achieve this by exploiting properties of analytic functions. This is also the case for our study. There auxiliary complex variables are introduced and desired weight is obtained after integrating them out. In this note we clarify conceptual difficulties with this procedure encountered in previous works. In the process we observe an exciting connection with thimbles and discover an interesting hidden symmetry present in the problem. Problem of negative eigenvalues of the action will be revisited and considered from a different perspective. As a byproduct we perform simulations of simple quantum systems directly in Minkowski time.Comment: The 36th Annual International Symposium on Lattice Field Theory - LATTICE2018; 22-28 July, 2018; Michigan State University, East Lansing, Michigan, US

Finite confinement from Padé approximants

It is demonstrated that the known perturbation expansion, and its first diagonal Padé approximant, for the Fourier transform of the energy of two classical sources in QCD, agree approximately with the potential used in charmonium phenomenology for 0.5 GeV $\leq$ q $\leq$ 10 GeV. It is argued that the singularity of the exact energy at q = 0 is stronger than g$^{-2}$

Intermittency in the ising systems

It is shown that the Ising model in two dimensions reveals the intermittent behaviour at the critical temperature. We conjecture that intermittency exists generally at the second order phase transition points

Dimensionally reduced SYM4_4 at large-NN: an intriguing Coulomb approximation

We consider the light-cone (LC) gauge and LC quantization of the dimensional reduction of super Yang Mills theory from four to two dimensions. After integrating out all unphysical degrees of freedom, the non-local LC Hamiltonian exhibits an explicit ${\cal N}=(2,2)$ supersymmetry. A further SUSY-preserving compactification of LC-space on a torus of radius $R$, allows for a large-$N$ numerical study where the smooth large-$R$ limit of physical quantities can be checked. As a first step, we consider a simple, yet quite rich, "Coulomb approximation" that maintains an ${\cal N}=(1,1)$ subgroup of the original supersymmetry and leads to a non-trivial generalization of 't Hooft's model with an arbitrary --but conserved-- number of partons. We compute numerically the eigenvalues and eigenvectors both in momentum and in position space. Our results, so far limited to the sectors with 2, 3 and 4 partons, directly and quantitatively confirm a simple physical picture in terms of a string-like interaction with the expected tension among pairs of nearest-neighbours along the single-trace characterizing the large-$N$ limit. Although broken by our approximation, traces of the full ${\cal N}=(2,2)$ supersymmetry are still visible in the low-lying spectrum.Comment: 30 pages, 13 figures, Footnote page 3 replaced, Note Added at the end, 4 References adde