Dirac-Bergmann analysis of SW-mapped non-commutative $U(1)$ electrodynamics with external currents

TL;DR

This paper analyzes the constraints and obstructions in non-commutative U(1) electrodynamics with external currents using the Dirac-Bergmann method, clarifying source-compatibility issues at the phase space level.

Contribution

It applies the Dirac-Bergmann algorithm to identify the source-compatibility obstruction directly within the phase space formulation of non-commutative electrodynamics.

Findings

Identifies the source-compatibility obstruction within the Dirac chain.

Shows the secondary constraint's phase-space expression matches the divergence of the mapped Euler-Lagrange equations.

Provides reduced-phase-space results only in a specific first-class subcase.

Abstract

Non-commutative electrodynamics obtained through the Seiberg-Witten map ceases to have equivalent action-level and equation-level realizations once fixed external currents are introduced, and in the action-level construction associated with the Banerjee current map the canonical location of this source-induced obstruction has remained unclear. Working in the full phase space and treating the current as prescribed and non-dynamical, we apply the Dirac-Bergmann algorithm without imposing current conservation as an external condition. The preservation of the Gauss-type secondary constraint produces a third-stage candidate whose phase-space expression is shown to be algebraically identical, at first order in the non-commutativity parameter and for purely space-space non-commutativity, to the canonical pullback of the divergence of the mapped Euler-Lagrange equations. This identity locates the source-compatibility obstruction directly within the Dirac chain. For generic inhomogeneous sources, the next consistency step feeds this object back into the primary multiplier through a source-dependent kernel, so the chain closes by multiplier fixing rather than by the generic appearance of a quaternary constraint. Reduced-phase-space results, including the gauge generator, Dirac brackets and degree-of-freedom count, are obtained only in a restricted sufficient first-class subcase; no broader claim is made for arbitrary source profiles.