On canonical transformations of gravitational variables in extended phase space

Abstract

In the recent years, a certain attention was attracted to the statement that Hamiltonian formulations of general relativity (GR), in which different parametrizations of gravitational variables were used, may not be related by a canonical transformation. An example was given by Dirac’s Hamiltonian formulation and that of Arnowitt-Deser-Misner. It might be indicative of a non-equivalence of these formulations and the original (Lagrangian) formulation of GR. The problem is believed to be of importance since many authors make use of various representations of the gravitational field as a starting point in searching for a way to reconcile the theory of gravity with quantum principles. It can be shown that the above-mentioned conclusion on non-equivalence of different Hamiltonian formulations is based on a consideration of canonical transformations in the phase space of physical degrees of freedom only, while the transformations also involve the gauge degrees of freedom. We shall give a clear proof that the Hamiltonian formulations corresponding to different parametrizations of the gravitational variables are related by canonical transformations in extended phase space embracing the gauge degrees of freedom on equal footing with the physical ones. It will be demonstrated for the full gravitational theory in a wide enough class of parametrizations and gauge conditions.