The case against asymptotic freedom

In this talk I give an overview of the work done during the last 15 years in collaboration with the late Adrian Patrascioiu. In this work we accumulated evidence against the commonly accepted view that theories with nonabelian symmetry -- either two dimensional nonlinear $\sigma$ models or four dimensional Yang-Mills theories -- have the property of asymptotic freedom (AF) usually ascribed to them.Comment: 18 pages, 2 figure

Status of Complex Langevin

I review the status of the Complex Langevin method, which was invented to make simulations of models with complex action feasible. I discuss the mathematical justification of the procedure, as well as its limitations and open questions. Various pragmatic measures for dealing with the existing problems are described. Finally I report on the progress in the application of the method to QCD, with the goal of determining the phase diagram of QCD as a function of temperature and baryonic chemical potential.Comment: Plenary talk given at Lattice 2017, Granada; 21 pages, 14 figures; a reference added, some more typos correcte

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

New Universality Classes in One--Dimensional O(N)O(N)--Invariant Spin--Models with an nn--Parametric Action

An action with $n$ parameters, which generalizes the $O(N) - R P^{N-1}$ -model, is considered in one dimension for general $N$. We use asymptotic expansion techniques to determine where the model becomes critical and show that for the actions considered there exists a family of hypersurfaces whose asymptotic behaviour determines a one-parameter family of new universality classes. They interpolate between the $O(N)$-vector-model-class and the $R P^{N-1}$-model-class. Furthermore continuum limits are discussed, including the exceptional case $N=2$.Comment: 13 page