Relativistically covariant formulation of the canonical theory of classical fields I

TL;DR

This paper develops a Lorentz covariant canonical theory for classical fields on space-like hypersurfaces, defining Hamilton's equations and Poisson brackets that satisfy Poincaré algebra, aligning with covariant symplectic structures.

Contribution

It introduces a covariant formulation of the canonical theory for classical fields, including Hamilton's equations and Poisson brackets on space-like hypersurfaces, consistent with Poincaré symmetry.

Findings

Poisson brackets satisfy Poincaré algebra

Hamilton's equations are covariantly formulated

Poisson structure matches covariant symplectic approach

Abstract

An explicit Lorentz covariant formulation of the canonical theory for classical fields is established on a space-like hypersurface. Hamilton's equations and a Poisson bracket are defined on the space-like hypersurface. The Poisson bracket relations between total momentum and total angular momentum satisfies the Poincar{\'e} algebra. It is shown that our Poisson bracket has the same symplectic structure that was brought in the covariant symplectic approach.