Quantum Field Theory on Noncommutative Space-Times and the Persistence of Ultraviolet Divergences

TL;DR

This paper investigates scalar quantum field theories on various two-dimensional noncommutative space-times, revealing that ultraviolet divergences persist in most cases except on a noncommutative cylinder, highlighting the role of topology.

Contribution

It demonstrates that ultraviolet divergences are not universally removed by noncommutativity and depend on the topology of the space-time, providing insights into quantum field behavior on noncommutative geometries.

Findings

Ultraviolet divergences persist on noncommutative planes with Heisenberg-like relations.

Ultraviolet finiteness occurs on noncommutative cylinders.

Ultraviolet behavior depends on the topology of the space-time.

Abstract

We study properties of a scalar quantum field theory on two-dimensional noncommutative space-times. Contrary to the common belief that noncommutativity of space-time would be a key to remove the ultraviolet divergences, we show that field theories on a noncommutative plane with the most natural Heisenberg-like commutation relations among coordinates or even on a noncommutative quantum plane with $E_q(2)$-symmetry have ultraviolet divergences, while the theory on a noncommutative cylinder is ultraviolet finite. Thus, ultraviolet behaviour of a field theory on noncommutative spaces is sensitive to the topology of the space-time, namely to its compactness. We present general arguments for the case of higher space-time dimensions and as well discuss the symmetry transformations of physical states on noncommutative space-times.