On Finite 4D Quantum Field Theory in Non-Commutative Geometry

TL;DR

The paper addresses the problem of formulating a four-dimensional quantum field theory on a finite, non-commutative geometry that preserves the full $SO(5)$ rotational symmetry of $S^4$. It achieves this by constructing a finite-dimensional, non-commutative (fuzzy) algebra $\mathcal{A}_N$ via a fixed occupation-number subspace and defining coordinates $x_A$ with the $SO(5)$ generators acting through a commutator, yielding a path-integrable, symmetry-preserving scalar theory with a finite number of modes. The key contributions are the explicit non-perturbative UV-regularization mechanism, the exact $SO(5)$ invariance, and well-defined Schwinger functions that satisfy the Osterwalder-Schrader axioms (OS1–OS3 and OS4 under symmetry) while recovering the standard divergences only in the commutative limit $N\to\infty$. This framework provides a concrete, symmetry-preserving route to controlled ultraviolet behavior in 4D QFT via non-commutative geometry, with potential connections to Wilsonian renormalization and a family of fuzzy-sphere models on $S^2$ and $S^4$.

Abstract

The truncated 4-dimensional sphere $S^4$ and the action of the self-interacting scalar field on it are constructed. The path integral quantization is performed while simultaneously keeping the SO(5) symmetry and the finite number of degrees of freedom. The usual field theory UV-divergences are manifestly absent.