Covariant hamiltonian formalism for field theory: Hamilton-Jacobi equation on the space G

TL;DR

This paper develops a covariant Hamilton-Jacobi formalism for field theories, including general relativity, providing a background-independent approach that connects classical and quantum gravity through a unified geometric framework.

Contribution

It introduces a covariant Hamilton-Jacobi equation on a space of surfaces, extending the ADM formalism and linking to quantum gravity via the Ashtekar-Wheeler-DeWitt equation.

Findings

Formalism is equivalent to ADM in classical gravity

Yields the Ashtekar-Wheeler-DeWitt equation in quantum domain

Discusses the role of spin networks in classical covariant field theory

Abstract

Hamiltonian mechanics of field theory can be formulated in a generally covariant and background independent manner over a finite dimensional extended configuration space. The physical symplectic structure of the theory can then be defined over a space G of three-dimensional surfaces without boundary, in the extended configuration space. These surfaces provide a preferred over-coordinatization of phase space. I consider the covariant form of the Hamilton-Jacobi equation on G, and a canonical function S on G which is a preferred solution of the Hamilton-Jacobi equation. The application of this formalism to general relativity is equivalent to the ADM formalism, but fully covariant. In the quantum domain, it yields directly the Ashtekar-Wheeler-DeWitt equation. Finally, I apply this formalism to discuss the partial observables of a covariant field theory and the role of the spin networks --basic objects in quantum gravity-- in the classical theory.