Modified Hamilton formalism for fields

TL;DR

This paper introduces a modified Hamilton formalism for fields using a worldsheet-targetspace approach, replacing null-vectors with null-polyvectors and generalizing the Legendre transformation to preserve action extremality.

Contribution

It develops a new Hamiltonian framework for fields based on higher-rank forms and null-polyvectors, extending the classical formalism to a finite-dimensional setting.

Findings

Formulation of a Hamiltonian formalism using null-polyvectors

Generalization of Legendre transformation for fields

Preservation of action extremality in the new formalism

Abstract

In Hamiltonian mechanics the equations of motion may be considered as a condition on the tangent vectors to the solution; they should be null-vectors of the symplictic structure. Usually the formalism for the field case is done by replacing the finite dimensional configuration space by an infinite dimensional one. In the present paper we work in worldsheet-targetspace formalism. The null-vectors of symplectic 2-form are replaced by null-polyvectors of a higher rank form on a finite dimensional manifold. The action in this case is an integral of a differential form over a surface in phase space. The method to obtain such a description from the Lagrange formalism generalizes the Legendre transformation. The requirement for this transformation to preserve the value of the action and its extremality leads to a natural definition of this procedure.