The quantization of gravity in globally hyperbolic spacetimes

Abstract

We apply the Arnowitt-Deser-Misner 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 $\varphi(x, g_{ij})$ 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 $\mathcal{S}_0$ 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, and (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 Algebraic Quantum Field Theory (QFT) which can be naturally modified to work in the bundle.