The quantization of gravity: Quantization of the Hamilton equations

TL;DR

This paper proposes a novel approach to quantum gravity by quantizing Hamilton equations, leading to a simplified equation involving the Laplacian of the Wheeler-DeWitt metric, and employs Fourier methods on symmetric spaces.

Contribution

It introduces a new quantization method focusing on Hamilton equations and utilizes Fourier analysis on symmetric spaces for quantum gravity.

Findings

Derivation of a simplified quantum gravity equation as a Laplace equation.

Application of Fourier transform techniques to eigenfunctions in symmetric spaces.

Potential framework for analyzing quantum states via tempered distributions.

Abstract

We quantize the Hamilton equations instead of the Hamilton condition. The resulting equation has the simple form $-\D u=0$ in a fiber bundle, where the Laplacian is the Laplacian of the Wheeler-DeWitt metric provided $n\not=4$. Using then separation of variables the solutions $u$ can be expressed as products of temporal and spatial eigenfunctions, where the spatial eigenfunctions are eigenfunctions of the Laplacian in the symmetric space $SL(n,\R[])/SO(n)$. Since one can define a Schwartz space and tempered distributions in $SL(n,\R[])/SO(n)$ as well as a Fourier transform, Fourier quantization can be applied such that the spatial eigenfunctions are transformed to Dirac measures and the spatial Laplacian to a multiplication operator.