Evolutionary Reformulation of Quantum Gravity

General Relativity is a background independent theory which identify the gravitational interaction into the metric properties of the space-time and this peculiar nature makes very subtle even simple questions about its quantization. To deal with a canonical method for the fields dynamics necessarily involves the notion of a physical time variable, whose conjugate momentum fixes the Hamiltonian function. Already in the context of classical General Relativity, the task of recovering a physical clock acquires non-trivial character, depending on the local properties of the spacetime. However, a well grounded algorithm devoted to this end was settled down by Arnowitt-Deser-Misner (ADM) in It consists of a space-time slicing based on a one parameter family of spacelike hypersurfaces Σ3t , defined via the parametric representation t = t(t, x) (μ = 0, 1, 2, 3 and i = 1, 2, 3). In what follows, we denote the set of coordinates {t, xi} by x, in order to emphasize that the slicing procedure can be recast as a 4-diffeomorphism, i.e. ds = gμν(t )dtdt = ḡμ̄ν̄(x )dxdx . The main issue of adopting the coordinates x is that they allow to separate the 4-metric tensor into six evolutionary components, which determine the induced 3-metric tensor hij of the hypersurfaces and four Lagrangian multipliers, corresponding to the lapse function N and to the shift vector N . These non-evolutionary variables have a precise geometrical meaning, given by the relation ∂tt μ = Nn + N ∂it μ (where gμνn n = 1 and gμνn ∂it ν = 0), n(t) denoting the orthonormal vector to the family Σt . The classical dynamics of the 3-metric hij is governed, in vacuum, by the following set of equations