Quantum Field Theory on the Noncommutative Plane with $E_q(2)$ Symmetry

TL;DR

This paper develops a scalar quantum field theory on a noncommutative plane with $E_q(2)$ symmetry, revealing increased ultraviolet divergences and exploring symmetry transformations in this noncommutative setting.

Contribution

It constructs a quantum field theory on the noncommutative plane with $E_q(2)$ symmetry and analyzes its properties, including divergence behavior and symmetry transformations.

Findings

Quantum field theory exhibits more severe UV divergences than in commutative case.

Field theory can be viewed as second quantization of particle theory on noncommutative space.

Detailed analysis of symmetry transformations for $E_q(2)$ quantum group.

Abstract

We study properties of a scalar quantum field theory on the two-dimensional noncommutative plane with $E_q(2)$ quantum symmetry. We start from the consideration of a firstly quantized quantum particle on the noncommutative plane. Then we define quantum fields depending on noncommutative coordinates and construct a field theoretical action using the $E_q(2)$-invariant measure on the noncommutative plane. With the help of the partial wave decomposition we show that this quantum field theory can be considered as a second quantization of the particle theory on the noncommutative plane and that this field theory has (contrary to the common belief) even more severe ultraviolet divergences than its counterpart on the usual commutative plane. Finally, we introduce the symmetry transformations of physical states on noncommutative spaces and discuss them in detail for the case of the $E_q(2)$ quantum group.