The quantum development of an asymptotically Euclidean Cauchy hypersurface

TL;DR

This paper develops a quantum gravity model where the quantum evolution of an asymptotically Euclidean Cauchy hypersurface is described by a wave equation, with solutions characterized via separation of variables and eigenvalue problems.

Contribution

It establishes a method to find physically relevant solutions of the quantum wave equation for asymptotically Euclidean hypersurfaces using eigenvalue analysis.

Findings

Eigenvalues are both temporal and spatial and coincide under asymptotic Euclidean conditions.

Physically interesting solutions are smooth functions with polynomial growth.

The approach links eigenfunctions to the quantum development of the hypersurface.

Abstract

In our model of quantum gravity the quantum development of a Cauchy hypersurface is governed by a wave equation derived as the result of a canonical quantization process. To find physically interesting solutions of the wave equation we employ the separation of variables by considering a temporal eigenvalue problem which has a complete countable set of eigenfunctions with positive eigenvalues and also a spatial eigenvalue problem which has a complete set of eigendistributions. Assuming that the Cauchy hypersurface is asymtotically Euclidean we prove that the temporal eigenvalues are also spatial eigenvalues and the product of corresponding eigenfunctions and eigendistributions, which will be smooth functions with polynomial growth, are the physically interesting solutions of the wave equation. We consider these solutions to describe the quantum development of the Cauchy hypersurface.