Lagrangian symmetries of the ADM action. Do we need a solution to the "non-canonicity puzzle"?

Abstract

We argue that there is nothing puzzling in the fact that the Hamiltonian formulation of a covariant theory, General Relativity, after a non-covariant change of field variables is not canonically related to the formulation based on the original variable, the metric tensor. Were such a puzzle to be "solved" it would lead to the conclusion that a covariant theory can be converted into a non-covariant one in many different ways and without consequence. The non-canonicity of transformations from covariant to non-covariant variables shows the need to work in the original variables so as to be able to restore the covariant gauge transformations in the Hamiltonian approach. Any modification of Dirac's procedure for the constrained Hamiltonian with the aim to prove the legitimacy of non-covariant changes of field variables, or rejection of Dirac's procedure as "not fundamental and undoubted" because it does not allow such changes (as recently suggested by Shestakova [CGQ, 28 (2011) 055009] ), is equivalent to the rejection of the covariance of General Relativity, and to surrender to such temptation is truly puzzling.

Gauge Symmetries and Dirac Conjecture

“…The oldest example, due to Allcock [6], was considered in Section III, and it also leads to a consistent result. Because this example is presented in two parametrisations it allows an explicit demonstration of the parametrisation dependence of the Dirac method, something we have already discussed for some mechanical models [20] and for field theory (different field parametrisations of General Relativity) [21,22]. We have also demonstrated how the natural parametrisation for the Allcock model can be constructed by using, as a criterion, the choice of the simplest properties of the commutator of two transformations.…”

“…The same logic applies to the Dirac conjecture: it is valid for the natural parametrisation of a model, but it is invalid for some "cooked up" parametrisations. For example, the Hamiltonian formulation of EH in natural, metric variables leads to the gauge invariance of full diffeomorphism [24], but in ADM parametrisation it produces a different symmetry [25]; one set of transformations forms a group and the other does not [22]. The same behaviour was observed for the model of Allcock -the property of the commutator of two consecutive transformations is parametrisation dependent, and one can find the parametrisation that has the simplest, zero-valued commutator.…”

Arnowitt–Deser–Misner representation and Hamiltonian analysis of covariant renormalizable gravity

“…Transformation (18) is the same as that given in Eq. ( 41) of [8] for ∂ k N = 0 (projectable case) and ξ 0 = 0. Note: to derive the generator (16) (see [7]) the PB of H k with the total Hamiltonian is needed, or H i , dyN i H i and H i , dyH 0 .…”

“…those for which the conjugate momenta are primary first-class constraints) a different gauge invariance follows. For example, the Hamiltonian formulation of the Einstein-Hilbert action in the original variables, metric, leads to full diffeomorphism in the formulation of either Pirani-Schild-Skinner (PSS) [10,11] or Dirac [12,13], but in ADM variables the transformations are different [13] (for connection with Lagrangian methods see [8,14]).…”

Generalized spherically symmetric gravitational model: Hamiltonian dynamics in extended phase space and the BRST charge

“…If we understand clearly how the Castellani procedure works, it would be strange to discuss seriously the transformation obtained by this procedure for the ADM formulation, as it was done in [19,20], to prove that these transformations do not form a group while the diffeomorphism transformations (1.1) do form. In this connection let us mention that the fact that the transformations (1.1) form a group was known very long ago.…”