On canonical transformations of gravitational variables in extended phase space

TL;DR

This paper demonstrates that different Hamiltonian formulations of General Relativity, using various parametrizations, are related by canonical transformations in an extended phase space that includes gauge degrees of freedom, ensuring their equivalence.

Contribution

It provides a proof that Hamiltonian formulations with different parametrizations are canonically equivalent in extended phase space, including gauge variables.

Findings

Different parametrizations are related by canonical transformations in extended phase space.

The equivalence holds for a wide class of gauges and parametrizations.

This resolves concerns about non-equivalence of Hamiltonian formulations of GR.

Abstract

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