Quantum cosmological Friedman models with a Yang-Mills field and positive energy levels

We prove the existence of a spectral resolution of the Wheeler-DeWitt equation when the matter field is provided by a Yang-Mills field, with or without mass term, if the spatial geometry of the underlying spacetime is homothetic to $\R[3]$. The energy levels of the resulting quantum model, i.e., the eigenvalues of the corresponding self-adjoint Hamiltonian with a pure point spectrum, are strictly positive.Comment: 9 pages, v3: minor corrections to bring it in line with the published versio

Estimates for the volume of a Lorentzian manifold

We prove new estimates for the volume of a Lorentzian manifold and show especially that cosmological spacetimes with crushing singularities have finite volume.Comment: 8 pages, a pdf version of the preprint can also be retrieved from http://www.math.uni-heidelberg.de/studinfo/gerhardt/LM-Volume.pdf v2: A further estimate has been added covering the case when the mean curvature is merely non-negative resp. non-positive (Theorem 1.1

The quantization of gravity in globally hyperbolic spacetimes

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.Comment: 26 pages, v4: some minor typos corrected, to appear in ATMP 201

A unified quantum theory I: gravity interacting with a Yang-Mills field

Using the results and techniques of a previous paper where we proved the quantization of gravity we extend the former result by adding a Yang-Mills functional and a Higgs term to the Einstein-Hilbert action.Comment: v4: minor corrections in some formulas, to appear in ATMP, arXiv admin note: text overlap with arXiv:1205.1427. text overlap with arXiv:1205.142