Hamiltonian formalism with several variables and quantum field theory, PartI

TL;DR

This paper develops a Hamiltonian formalism for classical field theory using a pataplectic approach, introducing new brackets and equations of motion for forms, with applications to scalar fields and string theory.

Contribution

It introduces a novel Hamiltonian framework for field theories with multiple variables, defining new brackets and formulating equations of motion in this context.

Findings

Defined Poisson p-brackets and omega-brackets for forms.

Formulated equations of motion using these brackets.

Applied formalism to scalar fields and conformal string theory.

Abstract

We discuss in this paper the canonical structure of classical field theory in finite dimensions within the {\it{pataplectic}} hamiltonian formulation, where we put forward the role of Legendre correspondance. We define the Poisson $\mathfrak{p}$-brackets and $\mathfrak{\omega}$-brackets which are the analogues of the Poisson bracket on forms. We formulate the equations of motion of forms in terms of $\mathfrak{p}$-brackets and $\mathfrak{\omega}$-brackets with the $n$-form ${\cal H}\omega $. As illustration of our formalism we present two examples: the interacting scalar fields and conformal string theory.