Poisson brackets with divergence terms in field theories: two examples

TL;DR

This paper investigates how divergence terms in densities affect Poisson brackets in field theories, revealing higher Jacobi identities and illustrating with two integrable system examples.

Contribution

It introduces the concept of divergence terms in Poisson brackets and explores their impact on Jacobi identities through two integrable system examples.

Findings

Divergence terms lead to higher Jacobi identities involving four fields.

The structure of Poisson brackets is affected by boundary terms in field theories.

Two integrable system examples illustrate the theoretical findings.

Abstract

In field theories one often works with the functionals which are integrals of some densities. These densities are defined up to divergence terms (boundary terms). A Poisson bracket of two functionals is also a functional, i.e., an integral of a density. Suppose the divergence term in the density of the Poisson bracket be fixed so that it becomes a bilinear form of densities of two functionals. Then the left-hand side of the Jacobi identity written in terms of densities is not necessarily zero but a divergence of a trilinear form. The question is: what can be said about this trilinear form, what kind of a higher Jacobi identity (involving four fields) it enjoys? Two examples whose origin is the theory of integrable systems are given.