A description of classical field equations using extensions of graded Poisson brackets

TL;DR

This paper develops an extension of graded Poisson brackets within graded Dirac manifolds to derive classical field equations and evolution of forms, generalizing the role of Poisson brackets in mechanics.

Contribution

It introduces a novel framework for extending graded Poisson brackets in graded Dirac manifolds to formulate field equations and form evolution.

Findings

Extended graded Poisson brackets enable derivation of field equations.

Framework generalizes mechanics' Poisson brackets to field theories.

Provides a new mathematical tool for classical field theory analysis.

Abstract

As it is well-known, Poisson brackets play a fundamental role both in mechanics and in classical field theories. In this paper we develop a theory of extensions of graded Poisson brackets in graded Dirac manifolds. We then show how these extensions can be used to obtain the field equations of a particular theory as well as the evolution of forms of arbitrary order, in a similar way that ordinary Poisson brackets provide in mechanics.