Noncommutativity from the symplectic point of view

TL;DR

This paper introduces a novel symplectic approach to incorporate noncommutative geometry into commutative field theories, providing a systematic method that extends beyond the traditional Moyal product, with applications to known physical models.

Contribution

It proposes a new systematic symplectic method to embed noncommutative geometry into commutative theories, expanding the tools available for noncommutative field theory formulation.

Findings

A new symplectic formalism for NC geometry in commutative systems.

Explicit derivation of NC Lagrangians for known theories.

Demonstration of the method's effectiveness in reproducing established models.

Abstract

The great deal in noncommutative (NC) field theories started when it was noted that NC spaces naturally arise in string theory with a constant background magnetic field in the presence of $D$-branes. Besides their origin in string theories and branes, NC field theories have been studied extensively in many branches of physics. In this work we explore how NC geometry can be introduced into a commutative field theory besides the usual introduction of the Moyal product. We propose a systematic new way to introduce NC geometry into commutative systems, based mainly on the symplectic approach. Further, as example, this formalism describes precisely how to obtain a Lagrangian description for the NC version of some systems reproducing well known theories.