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

TL;DR

This paper develops a generating functional for quantum gravity by relaxing primary constraints, leading to an evolutionary quantum dynamics framework that reduces to classical General Relativity with an Eckart fluid in the classical limit.

Contribution

It introduces a novel generating functional approach for quantum gravity that incorporates constraint relaxation and a physical clock, bridging quantum and classical descriptions.

Findings

Quantum dynamics described by a Schrödinger equation in the small ar limit.

Relaxation of primary constraints breaks 4-diffeomorphism invariance.

Classical limit yields General Relativity with an Eckart fluid as a clock.

Abstract

We provide a generating functional for the gravitational field, associated to the relaxation of the primary constraints as extended to the quantum sector. This requirement of the theory, relies on the assumption that a suitable time variable exist, 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 correspond to break down the invariance of the quantum theory under the 4-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 $\hbar$, the quantum dynamics is described by a Schr\"odinger 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.