Symplectic Groupoids and Poisson Electrodynamics

TL;DR

This paper introduces a geometric framework for Poisson electrodynamics using symplectic groupoids, revealing new insights into gauge fields, transformations, and emergent gravity in noncommutative spacetime.

Contribution

It develops a novel geometric approach to Poisson electrodynamics based on symplectic groupoids, connecting gauge fields to bisections and providing a gauge-invariant action functional.

Findings

Explicit examples show curved and compact momentum spaces.

Gauge transformations relate to spacetime diffeomorphisms.

Emergent gravity phenomena are demonstrated.

Abstract

We develop a geometric approach to Poisson electrodynamics, that is, the semi-classical limit of noncommutative $U(1)$ gauge theory. Our framework is based on an integrating symplectic groupoid for the underlying Poisson brackets, which we interpret as the classical phase space of a point particle on noncommutative spacetime. In this picture gauge fields arise as bisections of the symplectic groupoid while gauge transformations are parameterized by Lagrangian bisections. We provide a geometric construction of a gauge invariant action functional which minimally couples a dynamical charged particle to a background electromagnetic field. Our constructions are elucidated by several explicit examples, demonstrating the appearances of curved and even compact momentum spaces, the interplay between gauge transformations and spacetime diffeomorphisms, as well as emergent gravity phenomena.