Generating functional for the gravitational field: Implementation of an evolutionary quantum dynamics

摘要

We provide a generating functional for the gravitational field that is associated with the relaxation of the primary constraints by extending to the quantum sector. This requirement of the theory relies on the assumption that a suitable time variable exists, when taking the T-products of the dynamical variables. More precisely, we start from the gravitational field equations written in the Hamiltonian formalism and expressed via Misner-like variables; hence we construct the equation to which the T-products of the dynamical variables obey and transform this paradigm in terms of the generating functional, as taken on the theory phase-space. We show how the relaxation of the primary constraints (which corresponds to the breakdown of the invariance of the quantum theory under the four-diffeomorphisms) is summarized by a free functional taken on the Lagrangian multipliers, accounting for such constraints in the classical theory. The issue of our analysis is equivalent to a Gupta-Bleuler approach on the quantum implementation of all the gravitational constraints; in fact, in the limit of small h, the quantum dynamics is described by a Schrodinger equation as soon as the mean values of the momenta, associated to the lapse function and the shift vector, are not vanishing. Finally we show how, in the classical limit, the evolutionary quantum gravity reduces to General Relativity in the presence of an Eckart fluid, which corresponds to the classical Counterpart of the physical clock, introduced in the quantum theory.