On canonical transformations between equivalent hamiltonian formulations of general relativity

Abstract

Two Hamiltonian formulations of general relativity, due to Pirani, Schild and Skinner (Phys. Rev. 87, 452, 1952) and Dirac (Proc. Roy. Soc. A 246, 333, 1958), are considered. Both formulations, despite having different expressions for the constraints, allow one to derive four-dimensional diffeomorphism invariance. The relation between these two formulations at all stages of the Dirac approach to constrained Hamiltonian systems is analyzed. It is shown that the complete sets of their phase-space variables are related by a transformation which satisfies the ordinary condition of canonicity known for unconstrained Hamiltonians and, in addition, converts one total Hamiltonian into another, thus preserving form-invariance of generalized Hamiltonian equations for the constrained systems.

Remarks on the “Non-canonicity Puzzle”: Lagrangian Symmetries of the Einstein-Hilbert Action

“…Because gauge symmetry is an important characteristic of a constrained system, a difference between the symmetries of the PSS (or Dirac) and ADM Hamiltonian formulations indicates that a non-canonical relationship exists between the two formulations. This truism was explicitly confirmed in [4,9] by the calculation of Poisson brackets (PBs) among the phase-space variables. Waxing poetic, the term "the non-canonicity puzzle" was coined in [10] to describe the results of [4,9]; but in [11], using more direct language, it is called "the contradiction that again witnesses about the incompleteness of the theoretical foundations".…”

“…This truism was explicitly confirmed in [4,9] by the calculation of Poisson brackets (PBs) among the phase-space variables. Waxing poetic, the term "the non-canonicity puzzle" was coined in [10] to describe the results of [4,9]; but in [11], using more direct language, it is called "the contradiction that again witnesses about the incompleteness of the theoretical foundations". The source of the "puzzle" or "contradiction" lies in finding how to reconcile the non-equivalence of the two Hamiltonian formulations with their corresponding Lagrangian formulations when "it is supposed" [12] or "it is believed" (as in [11]) "that each of them is equivalent to the Einstein (Lagrangian) formulation" [11,12].…”

“…they are not equivalent), then the corresponding Lagrangians are also not equivalent, contrary to the "belief" which forms the basis of the "puzzle". If two Lagrangians are not equivalent, then the results of [4,9] are fully consistent, there is no puzzle, and the theoretical foundations are sound. The conclusion drawn that PSS and Dirac are not equivalent to the ADM formulation was based on the tenet held by the authors of [4] that Dirac's Hamiltonian method for constrained systems is an unambiguous procedure, applicable to any theory, that leads to a unique symmetry that corresponds exactly to the symmetry present in the Lagrangian.…”

“…2 Such an extension is obtained by replacing the classical EH action by "the effective action including gauge and ghost sectors" [11,12]. For a critical analysis of this approach see [9,13]. 3 In PSS [1] the gamma-gamma part of the EH action is considered; and Dirac in [2] made some additional manipulations in this Lagrangian.…”

“…This outcome contradicts the result of the Arnowitt, Deser, and Misner formulation (ADM, or geometrodynamics) [7,8] where the constraints lead to a different symmetry, one which is known by many names: "spatial diffeomorphism", "special induced diffeomorphism", "field-dependent diffeomorphism", "foliation preserving diffeomorphism", "one-to-one correspondence", "one-to-one mapping" (see [4] and references therein). It was shown [4,9] that the PSS and Dirac Hamiltonians are related by a canonical transformation of the phase-space variables; while the transformation from the Dirac to ADM Hamiltonian is not a canonical change of variables. Canonical transformations must preserve all properties of the Hamiltonian.…”