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

Carlo Rovelli (1, 2)

1 CPT - Centre de Physique Théorique - UMR 6207
2 CPT - Centre de Physique Théorique - UMR 7332

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.

Domaines: Relativité Générale et Cosmologie Quantique [gr-qc]

Carlo Rovelli. Covariant hamiltonian formalism for field theory: Hamilton-Jacobi equation on the space G. Thomas-H Elze. Decoherence and Entropy in Complex Systems: Selected Lectures from DICE 2002, Springer, pp.36-62, 2003, Lecture Notes in Physics. ⟨hal-00126306⟩