On the energy-momentum tensor in Moyal space

TL;DR

This paper investigates the properties of the energy-momentum tensor in non-commutative Moyal space, revealing conflicts between gauge invariance and conservation when matter fields are coupled.

Contribution

It demonstrates the incompatibility of two methods for achieving gauge invariance and conservation of the energy-momentum tensor in non-commutative gauge theories with matter.

Findings

Gauge invariance via Wilson line conflicts with conservation redefinition.

Pure gauge theories can have gauge-invariant energy-momentum tensors.

Coupling matter fields introduces fundamental incompatibilities.

Abstract

We study the properties of the energy-momentum tensor of gauge fields coupled to matter in non-commutative (Moyal) space. In general, the non-commutativity affects the usual conservation law of the tensor as well as its transformation properties (gauge covariance instead of gauge invariance). It is known that the conservation of the energy-momentum tensor can be achieved by a redefinition involving another star-product. Furthermore, for a pure gauge theory it is always possible to define a gauge invariant energy-momentum tensor by means of a Wilson line. We show that the latter two procedures are incompatible with each other if couplings of gauge fields to matter fields (scalars or fermions) are considered: The gauge invariant tensor (constructed via Wilson line) does not allow for a redefinition assuring its conservation, and vice-versa the introduction of another star-product does not allow for gauge invariance by means of a Wilson line.