Time in Quantum Gravity

TL;DR

This paper develops a novel approach to derive a time-dependent Schrödinger-Wheeler-DeWitt equation in quantum gravity without relying on the traditional Wheeler-DeWitt equation or WKB approximation, addressing foundational issues in quantum cosmology.

Contribution

It introduces a new method to obtain a time-containing quantum gravity equation using a Gaussian ansatz, consistent with boundary conditions like Hartle-Hawking.

Findings

Derived a time-dependent Schrödinger-Wheeler-DeWitt equation without WKB

Showed Gaussian ansatz aligns with Hartle-Hawking boundary conditions

Provided insights into small scale boundary conditions in quantum cosmology

Abstract

The Wheeler-DeWitt equation in quantum gravity is timeless in character. In order to discuss quantum to classical transition of the universe, one uses a time prescription in quantum gravity to obtain a time contained description starting from Wheeler-DeWitt equation and WKB ansatz for the WD wavefunction. The approach has some drawbacks. In this work, we obtain the time-contained Schroedinger-Wheeler-DeWitt equation without using the WD equation and the WKB ansatz for the wavefunction. We further show that a Gaussian ansatz for SWD wavefunction is consistent with the Hartle-Hawking or wormhole dominance proposal boundary condition. We thus find an answer to the small scale boundary conditions.