The Koszul-Tate Cohomology in Covariant Hamiltonian Formalism

TL;DR

This paper develops a covariant Hamiltonian formalism for degenerate quadratic Lagrangians, providing a complete set of Hamiltonian forms and explicit Koszul-Tate resolutions to connect Hamiltonian and Euler-Lagrange solutions.

Contribution

It introduces a method to construct non-degenerate Hamiltonian forms and explicit Koszul-Tate resolutions for almost regular quadratic Lagrangians in covariant field theory.

Findings

Constructed complete non-degenerate Hamiltonian forms.

Derived characteristic splittings of phase bundles.

Explicit Koszul-Tate resolution for Lagrangian constraints.

Abstract

We show that, in the framework of covariant Hamiltonian field theory, a degenerate almost regular quadratic Lagrangian $L$ admits a complete set of non-degenerate Hamiltonian forms such that solutions of the corresponding Hamilton equations, which live in the Lagrangian constraint space, exhaust solutions of the Euler--Lagrange equations for $L$. We obtain the characteristic splittings of the configuration and momentum phase bundles. Due to the corresponding projection operators, the Koszul-Tate resolution of the Lagrangian constraints for a generic almost regular quadratic Lagrangian is constructed in an explicit form.