NC Geometry and Quantum Fields: Simple Examples

Abstract

These notes contain four stories which are all related somehow to quantum field theory (QFT) and noncommutative geometry (NCG). (I) I sketch how the generalization of Yang-Mills theory to NCG motivates a useful and suggestive computation method to extract Chern-Simons terms from effective fermion actions. (II) Four nonconverging infinite series from a famous letter by Ramanujan are discussed as examples to illustrate a simple technique for regularization. (III) Regularized Hilbert space traces and their relation to the Wodzicki residue are discussed and illustrated for the simple example of matrix valued pseudodifferential operators on ℝⁿ. (IV) To demonstrate the efficiency of the mathematical tools described in (III), the logarithmic divergence of the effective fermion action in four dimensions is computed. It is argued that the result of this computation provides a physical motivation for a particular form of the spectral action principle in NCG.