Noncommutative Regularization for the Practical Man

Abstract

It has been proposed that the noncommutative geometry of the "fuzzy" 2-sphere provides a nonperturbative regularization of scalar field theories. This generalizes to compact Kaehler manifolds where simple field theories are regularized by the geometric quantization of the manifold. In order to permit actual calculations and the comparison with other regularizations, I describe the perturbation theory of these regularized models and propose an approximation technique for evaluation of the Feynman diagrams. I present example calculations of the simplest diagrams for the ${\varphi }^4$ model on the 2-sphere, a 2-sphere times a 2-sphere, and 2-dimensional complex projective space. 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.

30 pages. AMS-LaTeX using euler, palatino, graphics, psfrag, and 9 EPS figures. Title explained in ref. [8]

References in corpus (2)

• Gravity coupled with matter and foundation of non-commutative geometry
• D-Particle Dynamics and The Space-Time Uncertainty Relation

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• Noncommutative Gauge Dynamics From The String Worldsheet
• Quantum Field Theories With Compact Noncommutative Extra Dimensions
• Quantisation of θ-expanded non-commutative QED
• The Fuzzy Sphere Star-Product and Spin Networks
• MSc Thesis: Presentation of Certain New Trends in Noncommutative Geometry