Complex Probabilities and the Langevin Equation

Complex probabilities arise in quantum systems where the euclidean action is complex, either because the couplings, or the field variables, are complex. Stochastic quantization using the Langevin equation allows one to study such systems, even though the probabilistic interpretation of the euclidean functional integral measure breaks down. The convergence of long time averages computed with the Langevin equation to the analytic continuation of the functional integral averages is investigated for some simple models. The proposed methods have a variety of applications, including nonperturbative studies of chiral gauge theories on the lattice.