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^{(α)}(t)^4, where $α$ indicates the fractional derivative. Using Bogoliubov&#39;s R-operation we investigate ultraviolet divergencies for the various parameters $α$. Ultraviolet properties of this one-dimensional model in the case $α=3/4$ are similar to those in the $ϕ^4_4$ theory but there are extra counterterms. It is shown that the model is two-loops renormalizable. For $5/8\leq α< 3/4$ the model has a finite number of divergent Feynman diagrams. In the case $α=2/3$ the model is similar to the $ϕ^4_3$ theory. If $0 \leq α< 5/8$ then the model does not have ultraviolet divergencies at all. Finally if $α> 3/4$ 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.