Stochastic quantization of (λ 4)d scalar theory: Generalized Langevin equation with memory kernel

The method of stochastic quantization for a scalar field theory is reviewed. A brief survey for the case of self-interacting scalar field, implementing the stochastic perturbation theory up to the one-loop level, is presented. Then, it is introduced a colored random noise in the Einsteinʹs relations, a common prescription employed by one of the stochastic regularizations, to control the ultraviolet divergences of the theory. This formalism is extended to the case where a Langevin equation with a memory kernel is used. It is shown that, maintaining the Einsteinʹs relations with a colored noise, there is convergence to a non-regularized theory.