Bering's proposal for boundary contribution to the Poisson bracket

TL;DR

This paper demonstrates that Bering's boundary term Poisson bracket can be derived from a previous formalism by omitting certain terms, extending the formula to more general cases, but notes issues with invariance under variable changes.

Contribution

It shows how Bering's boundary contribution to the Poisson bracket relates to earlier formalisms and extends the formula to non-ultralocal brackets with field-dependent coefficients.

Findings

Bering's boundary Poisson bracket can be derived from earlier formalisms.

The formula is extended to non-ultralocal brackets with field-dependent coefficients.

The approach lacks invariance under field redefinitions.

Abstract

It is shown that the Poisson bracket with boundary terms recently proposed by Bering (hep-th/9806249) can be deduced from the Poisson bracket proposed by the present author (hep-th/9305133) if one omits terms free of Euler-Lagrange derivatives ("annihilation principle"). This corresponds to another definition of the formal product of distributions (or, saying it in other words, to another definition of the pairing between 1-forms and 1-vectors in the formal variational calculus). We extend the formula (initially suggested by Bering only for the ultralocal case with constant coefficients) onto the general non-ultralocal brackets with coefficients depending on fields and their spatial derivatives. The lack of invariance under changes of dependent variables (field redefinitions) seems a drawback of this proposal.