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 $\Gamma\Gamma$ 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.