Covariant Extended Phase Space for Fields on Curved Background

TL;DR

This paper develops a covariant Hamiltonian formalism for fields on curved backgrounds, emphasizing a bundle structure of extended phase space, and introduces a new bracket similar to the Peierls bracket.

Contribution

It introduces a covariant, bundle-structured extended phase space for fields and defines a novel bracket akin to the Peierls bracket, advancing the Hamiltonian approach in curved spacetime.

Findings

Established a bundle structure for extended phase space with time as base

Derived Noether currents for symmetry fields in this formalism

Proposed a new bracket analogous to the Peierls bracket

Abstract

It is shown that the nature of physical time requires the extended phase-space in mechanics to have a bundle structure with time as the 1-dimensional base manifold and the phase space as the fiber. This bundle picture of the extended phase space is then applied to fields in a covariant, `directly' Hamiltonian formalism. Variational principle in the covariant phase-space is discussed, Noether currents calculated for symmetry fields and a new bracket analogous to the Peierls bracket is defined.