Finite Charges from the Bulk Action

TL;DR

This paper presents a method to obtain finite conserved charges in covariant phase space formalism by incorporating corner terms directly from the bulk action, applicable even off-shell, with practical examples in lower dimensions.

Contribution

It shows that corner ambiguities are already present in the bulk action variation and can be used to define finite charges without additional boundary terms.

Findings

Charges become finite when corner terms are included.

The method works in Bondi gauge for 2D and 3D theories.

Actions with well-defined variational principles are derived for boundary conditions with corners.

Abstract

Constructing charges in the covariant phase space formalism often leads to formally divergent expressions, even when the fields satisfy physically acceptable fall-off conditions. These expressions can be rendered finite by corner ambiguities in the definition of the presymplectic potential, which in some cases may be motivated by arguments involving boundary Lagrangians. We show that the necessary corner terms are already present in the variation of the bulk action and can be extracted in a straightforward way. Once these corner terms are included in the presymplectic potential, charges derived from an associated codimension-2 form are automatically finite. We illustrate the procedure with examples in two and three dimensions, working in Bondi gauge and obtaining integrable charges. As a by-product, actions are derived for these theories that admit a well-defined variational principle when the fields satisfy boundary conditions on a timelike surface with corners. An interesting feature of our analysis is that the fields are not required to be fully on-shell.