Noncommutative fields and actions of twisted Poincare algebra

TL;DR

This paper explores the challenges of defining noncommutative fields under twisted Poincaré symmetry and proposes a tentative framework for noncommutative classical fields on Moyal space, aiming to facilitate noncommutative field theories.

Contribution

It introduces a tentative definition for noncommutative classical fields with desired properties and suggests approaches to identify suitable noncommutative Minkowski spaces.

Findings

Proposes a framework for noncommutative classical fields of any spin.

Identifies difficulties in extending classical field concepts to noncommutative settings.

Suggests methods to find noncommutative Minkowski spaces compatible with deformed Poincaré symmetries.

Abstract

Within the context of the twisted Poincar\'e algebra, there exists no noncommutative analogue of the Minkowski space interpreted as the homogeneous space of the Poincar\'e group quotiented by the Lorentz group. The usual definition of commutative classical fields as sections of associated vector bundles on the homogeneous space does not generalise to the noncommutative setting, and the twisted Poincar\'e algebra does not act on noncommutative fields in a canonical way. We make a tentative proposal for the definition of noncommutative classical fields of any spin over the Moyal space, which has the desired representation theoretical properties. We also suggest a way to search for noncommutative Minkowski spaces suitable for studying noncommutative field theory with deformed Poincar\'e symmetries.