Finite Field Theory on Noncommutative Geometries

TL;DR

This paper explores quantum field propagators on noncommutative geometries, demonstrating finite behavior at coincident points in noncommutative spaces and contrasting it with divergent commutative cases, including flat and curved geometries.

Contribution

It provides explicit calculations of propagators on noncommutative flat and curved geometries, highlighting finite limits at coincident points and extending noncommutative field theory understanding.

Findings

Finite propagator limits in noncommutative geometries

Divergence in the commutative limit

Extension to curved noncommutative spaces

Abstract

The propagator is calculated on a noncommutative version of the flat plane and the Lobachevsky plane with and without an extra (euclidean) time parameter. In agreement with the general idea of noncommutative geometry it is found that the limit when the two `points' coincide is finite and diverges only when the geometry becomes commutative. The flat 4-dimensional case is also considered. This is at the moment less interesting since there has been no curved case developed with which it can be compared.