On Field Theoretic Generalizations of a Poisson Algebra

TL;DR

This paper explores generalizations of Poisson algebras within field theory, introducing graded brackets and Nambu-Poisson structures that extend classical mechanics concepts to multiple variables and higher-order relations.

Contribution

It introduces new algebraic structures, including graded Poisson brackets on forms and an $(n+1)$-ary Nambu bracket, extending Poisson algebra concepts to field theory.

Findings

Development of a graded Poisson bracket on differential forms

Introduction of an $(n+1)$-ary Nambu bracket satisfying the fundamental identity

Application of Nambu brackets to joint evolution of multiple dynamical variables

Abstract

A few generalizations of a Poisson algebra to field theory canonically formulated in terms of the polymomentum variables are discussed. A graded Poisson bracket on differential forms and an $(n+1)$-ary bracket on functions are considered. The Poisson bracket on differential forms gives rise to various generalizations of a Gerstenhaber algebra: the noncommutative (in the sense of Loday) and the higher-order (in the sense of the higher order graded Leibniz rule). The $(n+1)$-ary bracket fulfills the properties of the Nambu bracket including the ``fundamental identity'', thus leading to the Nambu-Poisson algebra. We point out that in the field theory context the Nambu bracket with a properly defined covariant analogue of Hamilton's function determines a joint evolution of several dynamical variables.