Uniqueness of Galilean Conformal Electrodynamics and its Dynamical Structure

TL;DR

This paper explores the action principles and symmetry structures of Galilean Electrodynamics, establishing the uniqueness of the magnetic sector's action, the absence of the electric sector's action, and revealing rich symmetry properties including a state-dependent central charge.

Contribution

It demonstrates the unique existence of an action for the magnetic sector of Galilean Electrodynamics with a scalar field and analyzes its symmetry algebra, including the realization of the full Galilean conformal algebra.

Findings

Unique action exists for magnetic Galilean Electrodynamics with a scalar field.

No action exists for the electric sector of Galilean Electrodynamics.

The theory has a rich symmetry structure with a state-dependent central charge.

Abstract

We investigate the existence of action for both the electric and magnetic sectors of Galilean Electrodynamics using Helmholtz conditions. We prove the existence of unique action in magnetic limit with the addition of a scalar field in the system. The check also implies the non existence of action in the electric sector of Galilean electrodynamics. Dirac constraint analysis of the theory reveals that there are no local degrees of freedom in the system. Further, the theory enjoys a reduced but an infinite dimensional subalgebra of Galilean conformal symmetry algebra as global symmetries. The full Galilean conformal algebra however is realized as canonical symmetries on the phase space. The corresponding algebra of Hamilton functions acquire a state dependent central charge.