Canonical Structure of Classical Field Theory in the Polymomentum Phase Space

TL;DR

This paper develops a space-time symmetric Hamiltonian formalism for classical field theory using the De Donder-Weyl approach, introducing a graded Poisson structure and applying it to various physical models.

Contribution

It introduces a generalized Poisson structure and graded Lie algebra framework for the De Donder-Weyl formalism in field theory, extending the canonical structure.

Findings

Defines a graded Poisson bracket on forms in field theory.

Formulates field equations using the graded Poisson bracket and DW Hamiltonian.

Applies the formalism to scalar fields, electrodynamics, and strings.

Abstract

Canonical structure of the space-time symmetric analogue of the Hamiltonian formalism in field theory based on the De Donder-Weyl (DW) theory is studied. In $n$ space-time dimensions the set of $n$ polymomenta is associated to the space-time derivatives of field variables. The polysymplectic $(n+1)$-form generalizes the simplectic form and gives rise to a map between horizontal forms playing the role of dynamical variables and vertical multivectors generalizing Hamiltonian vector fields. Graded Poisson bracket is defined on forms and leads to the structure of a Z-graded Lie algebra on the subspace of the so-called Hamiltonian forms for which the map above exists. A generalized Poisson structure arises in the form of what we call a ``higher-order'' and a right Gerstenhaber algebra. Field euations and the equations of motion of forms are formulated in terms of the graded Poisson bracket with the DW Hamiltonian $n$-form $H\vol$ ($\vol$ is the space-time volume form and $H$ is the DW Hamiltonian function). A few applications to scalar fields, electrodynamics and the Nambu-Goto string, and a relation to the standard Hamiltonian formalism in field theory are briefly discussed. This is a detailed and improved account of our earlier concise communications (hep-th/9312162, hep-th/9410238, and hep-th/9511039).