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

TL;DR

The paper clarifies that non-canonical transformations in the Hamiltonian formulation of General Relativity do not undermine covariance, emphasizing the importance of working with original variables to preserve gauge symmetries.

Contribution

It demonstrates that non-covariant variable changes in Hamiltonian GR are not problematic and defends the use of Dirac's procedure for constrained systems.

Findings

Non-canonicity does not threaten covariance in Hamiltonian GR

Working in original variables restores covariant gauge transformations

Rejection of Dirac's procedure implies abandoning covariance

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.