Improvement of a conserved current density versus adding a total derivative to a Lagrangian density

TL;DR

This paper explores the relationship between adding superpotential terms to conserved currents and adding total derivatives to Lagrangians, showing their equivalence in certain classical relativistic field theories and improving properties of conserved quantities.

Contribution

It demonstrates the equivalence of superpotential additions and total derivative modifications in classical field theories through specific examples and theoretical analysis.

Findings

Superpotential terms do not alter conservation laws or charges.

Adding total derivatives to Lagrangians leaves equations of motion unchanged.

The two operations are related and potentially equivalent for global symmetries.

Abstract

For classical relativistic field theory in Minkowski space-time, the addition of a superpotential term to a conserved current density is trivial in the sense that it does not modify the local conservation law nor change the conserved charge, though it may allow us to obtain a current density with some improved properties. The addition of a total derivative term to a Lagrangian density is also trivial in the sense that it does not modify the equations of motion of the theory. These facts suggest that both operations are related and possibly equivalent to each other for any global symmetry of an action functional. We address this question following the study of two quite different (and well known) instances: the Callan-Coleman-Jackiw improvement of the canonical energy-momentum tensor for scalar and vector fields (providing an on-shell traceless energy-momentum tensor) and the construction of a current density satisfying a zero curvature condition for two-dimensional sigma models on deformed spaces (notably the squashed three-sphere and warped AdS spaces). These instances correspond to fairly different implementations of the general results. An appendix addresses the precise relationship between the approaches to local conservation laws based on active and passive symmetry transformations, respectively.