A unified field theory I: The quantization of gravity

Abstract

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 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. \nd In the second model the resulting equation is a wave equation in $\so \times (0,\infty)$ 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 $\so$ is compact we prove a spectral resolution of the equation.

This paper has been published in ATMP 22 (2018) under the title "The quantization of gravity". The title of the arXiv version, however, will remain unchanged since it had been referred to under the original title in a number of other papers

References in corpus (3)

• Wave Equations on Lorentzian Manifolds and Quantization
• The quantization of a black hole
• The quantum development of an asymptotically Euclidean Cauchy hypersurface

Cited by in corpus (8)

• The quantization of a Kerr-AdS black hole
• The quantization of a black hole
• A unified field theory II: Gravity interacting with a Yang-Mills and Higgs field
• Deriving a complete set of eigendistributions for a gravitational wave equation describing the quantized interaction of gravity with a Yang-Mills field in case the Cauchy hypersurface is non-compact
• The quantization of gravity: Quantization of the Hamilton equations
• The quantization of gravity: The quantization of the full Einstein equations
• Applications of canonical quantum gravity to cosmology
• A partition function for Schwarzschild-AdS and Kerr-AdS black holes and for quantized globally hyperbolic spacetimes with a negative cosmological constant