Hamiltonian formulation of General Relativity 50 years after the Dirac celebrated paper: do unsolved problems still exist?

TL;DR

This paper reviews the Hamiltonian formulations of General Relativity developed over the past 50 years, examining their equivalence, unresolved issues, and the benefits of the extended phase space approach through a simplified model.

Contribution

It critically analyzes longstanding problems in Hamiltonian formulations of gravity and demonstrates the advantages of the extended phase space method with a simple model.

Findings

Questions about the equivalence of Dirac and ADM formulations remain open

The extended phase space approach offers clear advantages in handling gauge degrees of freedom

The paper clarifies the construction rules for generators of phase space transformations

Abstract

About 50 years ago, in 1958, Dirac published his formulation of generalized Hamiltonian dynamics for gravitation. Several years later Arnowitt, Deser and Misner (ADM) proposed their description of the dynamics of General Relativity which became a basis of the Wheeler - DeWitt Quantum Geometrodynamics. There exist also other works where the Hamiltonian formulation of gravitational theory was discussed. In spite of decades passed from the famous papers by Dirac and ADM, there are unsolved problems. Namely, are the Dirac and ADM formulations equivalent to each other? Are these formulations equivalent to the original (Lagrangian) Einstein theory? Is the group of transformation in phase space the same as the group of gauge transformation of the Einstein theory? What are rules according to which a generator of transformations in phase space should be constructed? Let us mention also another approach based on extended phase space where gauge degrees of freedom are treated on the equal ground with physical degrees of freedom. Our purpose is to review the above questions and to demonstrate advantages of the extended phase space approach by the example of a simple model with finite number degrees of freedom.