Scalar field theories in a Lorentz-invariant three-dimensional noncommutative space-time

TL;DR

This paper investigates scalar quantum field theories in a Lorentz-invariant three-dimensional noncommutative space-time, analyzing one-loop diagrams, infrared singularities, and proposing a method to restore translational symmetry.

Contribution

It introduces a novel approach to preserve translational symmetry in noncommutative scalar field theories by adding an infinite number of tensor fields.

Findings

Non-planar diagrams are finite but exhibit UV/IR mixing.

Violations of momentum conservation persist even in the commutative limit.

An exact translational symmetry can be achieved with tensor fields.

Abstract

We discuss scalar quantum field theories in a Lorentz-invariant three-dimensional noncommutative space-time. We first analyze the one-loop diagrams of the two-point functions, and show that the non-planar diagrams are finite and have infrared singularities from the UV/IR mixing. The scalar quantum field theories have the problem that the violation of the momentum conservation from the non-planar diagrams does not vanish even in the commutative limit. A way to obtain an exact translational symmetry by introducing an infinite number of tensor fields is proposed. The translational symmetry transforms local fields into non-local ones in general. We also discuss an analogue of thermodynamics of free scalar field theory in the noncommutative space-time.