On the Canonical Structure of the De Donder-Weyl Covariant Hamiltonian Formulation of Field Theory I. Graded Poisson brackets and equations of motion

TL;DR

This paper introduces a Poisson bracket analogue for the De Donder-Weyl covariant Hamiltonian formulation of field theory, establishing a graded algebra structure and deriving equations of motion with illustrative examples.

Contribution

It defines a new Poisson bracket on differential forms in the DW formulation, linking it to the Schouten-Nijenhuis bracket and generalizing symplectic structures to field theory.

Findings

The bracket forms a Gerstenhaber graded algebra.

The Poisson bracket generates the exterior differential.

Derived Hamiltonian field equations using the bracket.

Abstract

The analogue of the Poisson bracket for the De Donder-Weyl (DW) Hamiltonian formulation of field theory is proposed. We start from the Hamilton- Poincar\'{e}-Cartan (HPC) form of the multidimensional variational calculus and define the bracket on the differential forms over the space-time (=horizontal forms). This bracket is related to the Schouten-Nijenhuis bracket of the multivector fields which are associated with the horizontal forms by means of the "polysymplectic form". The latter is given by the HPC form and generalizes the symplectic form to field theory. We point out that the algebra of forms with respect to our Poisson bracket and the exterior product has the structure of the Gerstenhaber graded algebra. It is shown that the Poisson bracket with the DW Hamiltonian function generates the exterior differential thus leading to the bracket representation of the DW Hamiltonian field equations. Few illustrative examples are also presented.