Towards Finite Quantum Field Theory in Non-Commutative Geometry

TL;DR

This work addresses UV regularization in quantum field theory on a curved, non-commutative space by formulating a self-interacting scalar field on the truncated (fuzzy) sphere and quantizing it via the path integral. The non-commutative algebra ${\cal A}_N$ provides a finite set of modes $(l,m)$ and a well-defined integration $I_N$, preserving the sphere's rotational symmetry. The resulting action $S[\Phi]$ yields a finite propagator $1/[l(l+1)+\mu^2]$ and vertex coefficients computed with the Wigner-Eckart theorem, with all diagrams finite for finite $N$; divergences reappear only as $N\to\infty$. The approach offers a non-perturbative UV-regularization tied to the space's short-distance structure, and suggests extensions to spinor and gauge fields as well as links to matrix models and higher-dimensional spheres, motivating further study of large-$N$ behavior and renormalization.

Abstract

We describe the self-interacting scalar field on the truncated sphere and we perform the quantization using the functional (path) integral approach. The theory posseses a full symmetry with respect to the isometries of the sphere. We explicitely show that the model is finite and the UV-regularization automatically takes place.