Finite dimesional Hamiltonian formalism for gauge and field theories

TL;DR

This paper develops a finite-dimensional Hamiltonian framework for classical field theories, introducing generalized Poisson brackets and illustrating with scalar fields, string theory, and electromagnetism.

Contribution

It introduces a novel finite-dimensional Hamiltonian formalism for gauge and field theories, emphasizing the role of Legendre correspondence and generalized Poisson brackets.

Findings

Defined generalized Poisson $rak{p}$-brackets for forms

Formulated equations of motion using $rak{p}$-brackets

Applied formalism to scalar fields, string theory, and electromagnetism

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 generalized Poisson $\mathfrak{p}$-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. As illustration of our formalism we present three examples: the interacting scalar fields, conformal string theory and the electromagnetic field.