Solution of the noncanonicity puzzle in General Relativity: A new Hamiltonian formulation

We study the transformation leading from Arnowitt, Deser, Misner (ADM) Hamiltonian formulation of General Relativity (GR) to the ΓΓ metric Hamiltonian formulation derived from the Lagrangian density which was firstly proposed by Einstein. We classify this transformation as gauged canonical – i.e. canonical modulo a gauge transformation. In such a study we introduce a new Hamiltonian formulation written in ADM variables which differs from the usual ADM formulation mainly in a boundary term firstly proposed by Dirac. Performing the canonical quantization procedure we introduce a new functional phase which contains an explicit dependence on the fields characterizing the 3+1 splitting. Given a specific regularization procedure our new formulation privileges the symmetric operator ordering in order to: have a consistent quantization procedure, avoid anomalies in constraints algebra, be equivalent to the Wheeler–DeWitt (WDW) quantization. Furthermore we show that this result is consistent with a path-integral approach.