## 11 Towards a Full Hamiltonian Approach

The Hamilton-Jacobi method is only one possible strategy to define the quasi-local quantities in a large
class of approaches, called the Hamiltonian or canonical approaches. Thus there is a considerable overlap
between the various canonical methods, and hence the cutting of the material into two parts
(Section 10 and Section 11) is, in some sense, artifical. In the previous Section 10 we reviewed
those approaches that are based on the analysis of the action, while in the present we discuss
those that are based primarily of the analysis of the Hamiltonian in the spirit of Regge and
Teitelboim [319].
By a full Hamiltonian analysis we mean a detailed study of the structure of the quasi-local
phase space, including the constraints, the smearing fields, the symplectic structure and the
Hamiltonian itself, according to the standard or some generalized Hamiltonian scenarios, in the
traditional 3 + 1 or in the fully Lorentz-covariant form, or even in the 2 + 2 form, using the metric
or triad/tetrad variables (or even the Weyl or Dirac spinors). In the literature of canonical
general relativity (at least in the asymptotically flat context) there are examples for all these
possibilities, and we report on the quasi-local investigations on the basis of the decomposition they
use. Since the 2 + 2 decomposition of the spacetime is less known, we also summarize its basic
idea.