相互作用随机过程与重整化理论
Interacting Stochastic Process and Renormalization Theory
摘要：
研究了一个具有自相互作用的随机过程作为量子场论的模型。考虑一个相互作用为x^{（α）}（t）^4的Ornstein-Uhlenbeck随机过程x（t），其中$α$表示分数阶导数.使用Bogoliubov的R-操作，我们调查紫外发散的各种参数$α$。这种一维模型在$α=3/4$情形下的紫外性质与$ϕ^4_4$理论中的紫外性质相似，但有额外的反项。证明了该模型是双回路可重整化的。对于$5/8\leq α< 3/4$，模型有有限个发散的费曼图。在$α=2/3$的情况下，该模型类似于$ϕ^4_3$理论。如果是$0 \leq α< 5/8$，则模型根本没有紫外发散。最后，如果是$α> 3/4$，则模型不可重正化。该模型可用于量子场论和相变理论中紫外发散的非微扰研究。
英文摘要：
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'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.
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