Advances in Theoretical and Mathematical Physics

Volume 22 (2018)

Number 3

The quantization of gravity

Pages: 709 – 757



Claus Gerhardt (Ruprecht-Karls-Universität, Institut für Angewandte Mathematik, Heidelberg, Germany)


In a former paper we proposed a model for the quantization of gravity by working in a bundle $E$ where we realized the Hamilton constraint as the Wheeler-DeWitt equation. However, the corresponding operator only acts in the fibers and not in the base space. Therefore, we now discard the Wheeler-DeWitt equation and express the Hamilton constraint differently, either with the help of the Hamilton equations or by employing a geometric evolution equation. There are two possible modifications possible which both are equivalent to the Hamilton constraint and which lead to two new models. In the first model we obtain a hyperbolic operator that acts in the fibers as well as in the base space and we can construct a symplectic vector space and a Weyl system.

In the second model the resulting equation is a wave equation in $\mathcal{S}_0 \times (0, \infty)$ valid in points $(x, t, \xi)$ in $E$ and we look for solutions for each fixed $\xi$. This set of equations contains as a special case the equation of a quantized cosmological Friedmann universe without matter but with a cosmological constant, when we look for solutions which only depend on $t$. Moreover, in case $\mathcal{S}_0$ is compact we prove a spectral resolution of the equation.


unified field theory, quantization of gravity, quantum gravity, gravitational waves, graviton

2010 Mathematics Subject Classification

83C45, 83Cxx, 83-xx

Published 22 October 2018