Complex Langevin equations and their applications to quantum statistical and lattice field models

We discuss the calculation of statistical averages of variables lying on $S_1$ or $S_2$ using (complex) Langevin equations. Assuming that the drift term is proportional to the gradient of a possibly complex function S({$x_i$}), $x_i$∈$S_1$ or $S_2$ we give the general form of such Langevin equations. These variables cause unphysical singularities and computational problems; thus we transform them to those of the embedding Euclidean space. We show in several examples that these modified (complex) Langevin equations have good convergence properties using an improved two-stage Runge-Kutta algorithm.