11 Towards a Full Hamiltonian Approach

The Hamilton–Jacobi method is only one possible strategy for defining 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, artificial. In Section 10 we reviewed those approaches that are based on the analysis of the action, while in this section we discuss those that are based primarily on the analysis of the Hamiltonian in the spirit of Regge and Teitelboim [433].20

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.

 11.1 The 3 + 1 approaches
  11.1.1 The quasi-local constraint algebra and the basic Hamiltonian
  11.1.2 The two-surface observables
 11.2 Approaches based on the double-null foliations
  11.2.1 The 2 + 2 decomposition
  11.2.2 The 2 + 2 quasi-localization of the Bondi–Sachs mass-loss
 11.3 The covariant approach
  11.3.1 The covariant phase space methods
  11.3.2 The general expressions of Chen, Nester and Tung: Covariant quasi-local Hamiltonians with explicit reference configurations
  11.3.3 The reference configuration of Nester, Chen, Liu and Sun
  11.3.4 Covariant quasi-local Hamiltonians with general reference terms
  11.3.5 Pseudotensors and quasi-local quantities

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