Dynamics without Time for Quantum Gravity: Covariant Hamiltonian Formalism and Hamilton-Jacobi Equation on the Space G

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.

The Weiss variation of the gravitational action

“…(A7) 23 One may show this using the formula for acceleration (3.17) and the requirement that n µ have unit norm.…”

“…where we have made a substitution in (4.56) using (4.55). It is common to identify (4.56) as the Hamilton-Jacobi equation for (vacuum) GR, as is often done in the literature [21,23,29], and it may be shown that (4.56) define the dynamics for (vacuum) GR [22]. We note that equations (4.56) do not form the Hamilton-Jacobi equation for GR in the same sense as the Hamilton-Jacobi equation in mechanics; H gf is a Hamiltonian density, not a Hamiltonian, so (4.56) should be viewed as a set of local constraints.…”

Characteristics, bicharacteristics and geometric singularities of solutions of PDEs

“…In particular, there is a Hamiltonian system on boundary data (see [32]). Rovelli [38] showed that this considerations can be generalized (in a covariant way) to any classical field theory. In particular he was able to write down a Hamilton-Jacobi equation on the space of boundary data of the field equations and show that the action functional provide a canonical solution.…”

Principal symbol of Euler–Lagrange operators

“…Unfortunately, there is no consensus on a covariant Hamiltonian framework for field theories and different approaches often differ on details; see [2], [3], [4], [5], [6], [7], [8], [9].…”