Abstract
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.
Highlights
The attempts towards the quantization of General Relativity (GR) can be classified as canonical or covariant
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
We solved the puzzle of the noncanonicity firstly indicated in [15] with the concept of gauged canonicity: a transformation is gauged canonical if its fundamental PB are canonical modulo a gauge transformation
Summary
The attempts towards the quantization of GR can be classified as canonical or covariant. Quantum geometrodynamics (see [4] for a recent review) is based on the ADM Hamiltonian formulation [5] This formulation exploits the symmetries of gravity in a 3 + 1 representation, introducing new variables instead of the metric ones, where the most general set of coordinate transformations is reduced to arbitrary 3-dimensional transformations and time reparametrizations. We will outline how the consistency with the quantum framework based on ADM formulation privileges the symmetric operator ordering for the WDW equation, given a specific regularization procedure [8] The former property will be recognized as due to Dirac’s boundary term.
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