Perturbative noncommutative regularization
Eli Hawkins · CERN Document Server (European Organization for Nuclear Research) · 1999
I propose a nonperturbative regularization of quantum field theories with contact interactions (primarily, scalar field theories). This is given by the geometric quantization of compact Kähler manifolds and generalizes what has already been proposed by Madore, Grosse, Klimčík, and Prešnajder for the two-sphere. I discuss the perturbation theory derived from this regularized model and propose an approximation technique for evaluating the Feynman diagrams. This amounts to a momentum cutoff combined with phase factors at vertices. To illustrate the exact and approximate calculations, I present, as examples, the simplest diagrams for the l f 4 model on the spaces S 2 , S 2 ×S 2 , and C P 2 . This regularization fails for noncompact spaces. I give a brief dimensional analysis argument as to why this is so. I also discuss the relevance of the topology of Feynman diagrams to their ultra-violet and infra-red divergence behavior in this model.