On canonical transformations between equivalent Hamiltonian formulations of General Relativity

TL;DR

This paper analyzes the relationship between two Hamiltonian formulations of General Relativity, demonstrating that they are connected by a canonical transformation that preserves the form of the Hamiltonian equations and diffeomorphism invariance.

Contribution

It establishes that two different Hamiltonian formulations of General Relativity are related through a canonical transformation that maintains the structure of the equations.

Findings

The phase-space variables are related by a canonical transformation.

The transformation converts one total Hamiltonian into the other.

Both formulations preserve four-dimensional diffeomorphism invariance.

Abstract

Two Hamiltonian formulations of General Relativity, due to Pirani, Schild and Skinner (Phys. Rev. 87, 452, 1952) and Dirac (Proc. Roy. Soc. A 246, 333, 1958), are considered. Both formulations, despite having different expressions for constraints, allow one to derive four-dimensional diffeomorphism invariance. The relation between these two formulations at all stages of the Dirac approach to the constrained Hamiltonian systems is analyzed. It is shown that the complete sets of their phase-space variables are related by a transformation which satisfies the ordinary condition of canonicity known for unconstrained Hamiltonians and, in addition, converts one total Hamiltonian into another, thus preserving form-invariance of generalized Hamiltonian equations for constrained systems.