Generalized Quantization Principle in Canonical Quantum Gravity and Application to Quantum Cosmology

TL;DR

This paper introduces a generalized quantization principle for canonical quantum gravity inspired by the generalized uncertainty principle, leading to modified constraints and a new form of the Wheeler DeWitt equation, with applications to quantum cosmology.

Contribution

It proposes a novel generalized quantization framework for quantum gravity, extending the operator representations and constraints, and applies it to solve a modified Wheeler DeWitt equation in quantum cosmology.

Findings

Derived a generalized Wheeler DeWitt equation for quantum cosmology.

Solved the generalized equation using Sommerfeld's polynomial method.

Showed implications of the generalized quantization on quantum gravitational dynamics.

Abstract

In this paper is considered a generalized quantization principle for the gravitational field in canonical quantum gravity, especially with respect to quantum geometrodynamics. This assumption can be interpreted as a transfer from the generalized uncertainty principle in quantum mechanics, which is postulated as generalization of the Heisenberg algebra to introduce a minimal length, to a corresponding quantization principle concerning the quantities of quantum gravity. According to this presupposition there have to be given generalized representations of the operators referring to the observables in the canonical approach of a quantum description of general relativity. This also leads to generalized constraints for the states and thus to a generalized Wheeler DeWitt equation determining a new dynamical behaviour. As a special manifestation of this modified canonical theory of quantum gravity there is explored quantum cosmology. The generalized cosmological Wheeler DeWitt equation corresponding to the application of the generalized quantization principle to the cosmological degree of freedom is solved by using Sommerfelds polynomial method.