Divergences in QFT on the Noncommutative Minkowski Space with Grosse-Wulkenhaar potential

TL;DR

This paper investigates quantum field theory on a two-dimensional noncommutative Minkowski space with a Grosse-Wulkenhaar potential, revealing fundamental issues with defining certain distributions and products, especially at and above the self-dual point.

Contribution

It explicitly constructs the retarded propagator in this setting and demonstrates its non-tempered nature, highlighting non-standard divergences in the theory.

Findings

Retarded propagator is not a tempered distribution.

Planar products of distributions cannot be defined at or above the self-dual point.

Problems are not due to ordinary ultraviolet divergences.

Abstract

We study quantum field theory on the two-dimensional Noncommutative Minkoswki space with a Grosse-Wulkenhaar potential. We explicitly construct the retarded propagator and show that it is not a tempered distribution. This leads to problems when trying to define planar products of such distributions, as they appear in the Yang-Feldman series. At and above the self-dual point, these can no longer be defined, not even at different points. This shows that we do not deal with an ordinary ultraviolet divergence.