Novel algebraic structures from the polysymplectic form in field theory

TL;DR

This paper introduces the polysymplectic form in field theory, constructs associated Poisson brackets, and demonstrates their structure as generalized Gerstenhaber algebras, extending classical symplectic concepts.

Contribution

It presents a novel polysymplectic form and develops a new algebraic framework for Poisson brackets in field theory, generalizing classical structures.

Findings

Polysymplectic form acts as an analogue of the symplectic form in field theory.

Poisson brackets on differential forms are constructed.

The algebraic structure is shown to be a generalized Gerstenhaber algebra.

Abstract

The polysymplectic $(n+1)$-form is introduced as an analogue of the symplectic form for the De Donder-Weyl polymomentum Hamiltonian formulation of field theory. The corresponding Poisson brackets on differential forms are constructed. The analogues of the Poisson algebra are shown to be generalized (non-commutative and higher-order) Gerstenhaber algebras defined in the text.