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

摘要

Hamiltonian mechanics of field theory can be formulated in a generally covariant and background independent manner over a finite dimensional extended configuration space. I study the physical symplectic structure of the theory in this framework. This structure can 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 fully covariant and yields directly the Ashtekar-Wheeler-DeWitt equation, the basic equation of canonical quantum gravity. 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.