The quantization of gravity in globally hyperbolic spacetimes

TL;DR

This paper develops a Hamiltonian formulation of quantum gravity in globally hyperbolic spacetimes by modifying the ADM approach, leading to a hyperbolic Wheeler-DeWitt equation suitable for quantum field theory techniques.

Contribution

It introduces a novel Hamiltonian framework for quantum gravity that eliminates diffeomorphism constraints and treats the Wheeler-DeWitt equation as a hyperbolic PDE in a fiber bundle.

Findings

The Hamiltonian operator is normally hyperbolic in the fiber bundle.

The Wheeler-DeWitt equation becomes a hyperbolic equation with well-posed Cauchy problem.

Standard QFT techniques can be applied to this hyperbolic quantum gravity model.

Abstract

We apply the ADM approach to obtain a Hamiltonian description of the Einstein-Hilbert action. In doing so we add four new ingredients: (i) We eliminate the diffeomorphism constraints. (ii) We replace the densities $\sqrt g$ by a function $\f(x,g_{ij})$ with the help of a fixed metric $\chi$ such that the Lagrangian and hence the Hamiltonian are functions. (iii) We consider the Lagrangian to be defined in a fiber bundle with base space $\so$ and fibers F(x) which can be treated as Lorentzian manifolds equipped with the Wheeler-DeWitt metric. It turns out that the fibers are globally hyperbolic. (iv) The Hamiltonian operator $H$ is a normally hyperbolic operator in the bundle acting only in the fibers and the Wheeler-DeWitt equation $Hu=0$ is a hyperbolic equation in the bundle. Since the corresponding Cauchy problem can be solved for arbitrary smooth data with compact support, we then apply the standard techniques of QFT which can be naturally modified to work in the bundle.