On Noncommutative Geometric Regularisation

TL;DR

This paper explores how noncommutative geometric features of space-time, motivated by string theory and quantum gravity, can lead to regularisation of infrared divergences in quantum field theories.

Contribution

It provides a general proof that noncommutative geometries with minimal momentum uncertainties can IR-regularise field theories within a path integral framework.

Findings

IR regularisation is achievable with noncommutative geometries.

Minimal uncertainties in momenta are key to regularisation.

Path integral approach confirms the regularisation effect.

Abstract

Studies in string theory and in quantum gravity suggest the existence of a finite lower bound to the possible resolution of lengths which, quantum theoretically, takes the form of a minimal uncertainty in positions $\Delta x_0$. A finite minimal uncertainty in momenta $\Delta p_0$ has been motivated from the absence of plane waves on generic curved spaces. Both effects can be described as small noncommutative geometric features of space-time. In a path integral approach to the formulation of field theories on noncommutative geometries, we can now generally prove IR regularisation for the case of noncommutative geometries which imply minimal uncertainties $\Delta p_0$ in momenta.