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

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.