Noether currents and charges for Maxwell-like Lagrangians

TL;DR

This paper analyzes Noether currents for Maxwell-like Lagrangians, demonstrating that proper energy-momentum currents emerge when considering background variations or gravitational dynamics, with applications to gauge fields and electrodynamics.

Contribution

It clarifies the Noether procedure for diffeomorphism-invariant p-form field models, showing consistent currents without metric variation in certain cases.

Findings

Noether currents coincide on variable and fixed backgrounds.

Proper energy-momentum currents require gravitational dynamics.

The procedure yields correct results even without metric variation in electrodynamics.

Abstract

Hilbert-Noether theorem states that a current associated to diffeomorphism invariance of a Lagrangian vanishes on shell modulo a divergence of an arbitrary superpotential. Application of the Noether procedure to physical Lagrangians yields, however, meaningful (and measurable) currents. The well known solution to this ``paradox'' is to involve the variation of the metric tensor. Such procedure, for the field considered on a fixed (flat) background, is sophisticated logically (one need to introduce the variation of a fixed field) and formal. We analyze the Noether procedure for a generic diffeomorphism invariant $p$-form field model. We show that the Noether current of the field considered on a variable background coincides with the current treated in a fixed geometry. Consistent description of the canonical energy-momentum current is possible only if the dynamics of the geometry (gravitation) is taken into account. However, even the ``truncated'' consideration yields the proper expression. We examine the examples of the free $p$-form gauge field theory, the GR in the coframe representation and the metric-free electrodynamics. Although, the variation of a metric tensor is not acceptable in the latter case, the Noether procedure yields the proper result.