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

Abstract

Given the non-canonical relationship between variables used in the Hamiltonian formulations of the Einstein-Hilbert action (due to Pirani, Schild, Skinner (PSS) and Dirac) and the Arnowitt-Deser-Misner (ADM) action, and the consequent difference in the gauge transformations generated by the first-class constraints of these two formulations, the assumption that the Lagrangians from which they were derived are equivalent leads to an apparent contradiction that has been called "the non-canonicity puzzle". In this work we shall investigate the group properties of two symmetries derived for the Einstein-Hilbert action: diffeomorphism, which follows from the PSS and Dirac formulations, and the one that arises from the ADM formulation. We demonstrate that unlike the diffeomorphism transformations, the ADM transformations (as well as others, which can be constructed for the Einstein-Hilbert Lagrangian using Noether's identities) do not form a group. This makes diffeomorphism transformations unique (the term "canonical" symmetry might be suggested). If the two Lagrangians are to be called equivalent, canonical symmetry must be preserved. The interplay between general covariance and the canonicity of the variables used is discussed.

On canonical transformations between equivalent hamiltonian formulations of general relativity

“…It has been shown that the full sets of phase-space variables for these two formulations are related to each other by a transformation of (25), which satisfies the condition of canonicity (28) known for the Hamiltonian formulations of non-singular systems. It also preserves the form-invariance of the expressions for the total Hamiltonians (33).…”

“…For singular systems it is also important to preserve the form-invariance of the algebra of constraints, as is the case for the PSS and Dirac formulations (see (30), (38), and (40)). As we have demonstrated in [10], if one is tempted to convert the ADM variables into canonical ones, then, to satisfy (28), all momenta must be involved, which leads to the relationships between "old" and "new" momenta given by Eqs. (163)-(165) of [10]:…”

“…For non-singular systems, the canonicity condition (28) is actually independent of the particular form of the unconstrained Hamiltonian; and the canonical transformations automatically convert the Hamiltonian, written in terms of one set of phasespace variables, into another Hamiltonian. Unlike that, for singular systems the Hamiltonian (that is, the total Hamiltonian) consists of two distinct parts, namely, the "canonical Hamiltonian" and a linear combination of primary constraints, which play different roles.…”

“…The PB of (29) shows that Dirac's modification of L PSS at the Lagrangian level leads to a Hamiltonian formulation in which the phase-space variables are canonically related to those of PSS. Our next goal is to consider the effect of such a canonical change of phase-space variables at all steps of the Dirac procedure.…”

“…Our next goal is to consider the effect of such a canonical change of phase-space variables at all steps of the Dirac procedure. We can utilize the PB of (29), and, by rearranging terms, present the canonical part of the PSS Hamiltonian in a different, but equivalent form by explicitly creating (extracting) combinations that correspond to a canonical change of variables…”