  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.
