Exact Results in Quantum Theory

Stochastic quantization is a well known approach which replaces ensemble averages by averaging over a suitable stochastic process. It was designed and proved for positive densities, i.e. real, euclidean actions. However Langevin process can be also defined for complex actions, raising expectations for statistical averaging over complex "distributions". This has attracted a new wave of interest after recent reports of successful study of quantum chromodynamics at finite chemical potential. Nevertheless there is no proof of convergence in the complex case, and indeed the evidence for success is limited.In this talk we will circumvent above problems by introduction of additional variables and direct construction of pairs of corresponding (i.e. complex and positive) weights. As an application, the well known solution for a complex Gaussian distribution will be generalized to arbitrary complex slope. This opens a way to construct positive representations of path integrals directly in Minkowski time and will be done in the second part of the talk.Then some applications to simple, classic quantum systems will be presented. Finally a striking physical interpretation of the new structure will be suggested, albeit with due caution.