A note on the implementation of Poincar\'e symmetry in noncommutative field theory

TL;DR

This paper proposes a method to implement Poincaré symmetry in noncommutative field theories by allowing the noncommutative parameter to transform as a tensor, ensuring consistency with the star product and coordinate definitions.

Contribution

It introduces a universal correction to Poincaré transformations in NCFT by incorporating the variation of the noncommutative parameter, enabling symmetry implementation.

Findings

Poincaré symmetry can be realized with a variable noncommutative parameter.

The correction involves a term proportional to the variation of ta^{} in the transformation.

Obstructions exist for implementing generalized space-time symmetries like conformal or diffeomorphisms.

Abstract

We argue that Poincar\'e symmetry can be implemented in NCFT if we allow the parameter of noncommutitive deformation $\theta^{\mu\nu}$ to change as a two-tensor under the corresponding space-time symmetry. The implementation is consistent with the definition of $\theta^{\mu\nu}$ in terms of space-time coordinates and with the Moyal star product. Inspired from the standard definition of a variational symmetry we found a universal way to correct the implementation of the Poincar\'e symmetry by a term proportional to the variation of $\theta^{\mu\nu}$ in such a way that the new transformation define a symmetry of the theory. Finally we present as an example the case of NCYM theory and comment about the obstructions to implement generalized space-time symmetries in NCFT like conformal or diffeomorphism transformations.