On Covariant Poisson Brackets in Classical Field Theory

TL;DR

This paper connects the multisymplectic Poisson bracket with the Peierls-De Witt bracket in classical field theory, clarifying longstanding issues and ambiguities in defining covariant Poisson brackets geometrically.

Contribution

It demonstrates how the multisymplectic Poisson bracket can be derived from the Peierls-De Witt bracket for a specific class of functionals, resolving previous problems.

Findings

Derived the multisymplectic Poisson bracket from the Peierls-De Witt bracket.

Identified ambiguities due to non-unique relations between forms and functionals.

Showed that the class of functionals does not form a Poisson subalgebra.

Abstract

How to give a natural geometric definition of a covariant Poisson bracket in classical field theory has for a long time been an open problem - as testified by the extensive literature on "multisymplectic Poisson brackets", together with the fact that all these proposals suffer from serious defects. On the other hand, the functional approach does provide a good candidate which has come to be known as the Peierls - De Witt bracket and whose construction in a geometrical setting is now well understood. Here, we show how the basic "multisymplectic Poisson bracket" already proposed in the 1970s can be derived from the Peierls - De Witt bracket, applied to a special class of functionals. This relation allows to trace back most (if not all) of the problems encountered in the past to ambiguities (the relation between differential forms on multiphase space and the functionals they define is not one-to-one) and also to the fact that this class of functionals does not form a Poisson subalgebra.