Galilean symmetry in noncommutative field theory

TL;DR

This paper explores how noncommutative field theories can exhibit two types of Galilean symmetries, including an exotic extension where boosts do not commute, highlighting symmetry structures affected by interactions.

Contribution

It demonstrates the existence of both conventional and exotic Galilean symmetries in noncommutative field theories, especially after reordering interactions, and analyzes symmetry breaking due to interactions.

Findings

Existence of two Galilean symmetries in noncommutative field theory

Identification of an exotic two-parameter central extension of the Schrödinger group

Interaction breaks conformal symmetry

Abstract

When the interaction potential is suitably reordered, the Moyal field theory admits two types of Galilean symmetries, namely the conventional mass-parameter-centrally-extended one with commuting boosts, but also the two-fold centrally extended ``exotic'' Galilean symmetry, where the commutator of the boosts yields the noncommutative parameter. In the free case, one gets an ``exotic'' two-parameter central extension of the Schroedinger group. The conformal symmetry is, however, broken by the interaction.