Interacting Stochastic Process and Renormalization Theory

Abstract

A stochastic process with self-interaction as a model of quantum field theory is studied. We consider an Ornstein-Uhlenbeck stochastic process x(t) with interaction of the form x^{(\alpha)}(t)^4, where $\alpha$ indicates the fractional derivative. Using Bogoliubov's R-operation we investigate ultraviolet divergencies for the various parameters $\alpha$. Ultraviolet properties of this one-dimensional model in the case $\alpha=3/4$ are similar to those in the $\phi^4_4$ theory but there are extra counterterms. It is shown that the model is two-loops renormalizable. For $5/8\leq \alpha < 3/4$ the model has a finite number of divergent Feynman diagrams. In the case $\alpha=2/3$ the model is similar to the $\phi^4_3$ theory. If $0 \leq \alpha < 5/8$ then the model does not have ultraviolet divergencies at all. Finally if $\alpha > 3/4$ then the model is nonrenormalizable. This model can be used for a non-perturbative study of ultraviolet divergencies in quantum field theory and also in theory of phase transitions.