Quantum Gravity Equation In Schroedinger Form In Minisuperspace Description

TL;DR

This paper derives a Schrödinger-form quantum gravity equation in minisuperspace, establishing a geometric time parameter without relying on the Wheeler-DeWitt equation or WKB expansion, and aligns with previous wavefunction proposals.

Contribution

It introduces a novel approach to quantum gravity in minisuperspace by defining time geometrically, avoiding traditional assumptions, and reproducing Wheeler-DeWitt solutions under specific normalization.

Findings

Derived a Schrödinger-form quantum gravity equation in minisuperspace.

Established a geometric time parameter independent of matter fields.

Reproduces Wheeler-DeWitt wavefunctions with wormhole-based normalization.

Abstract

We start from classical Hamiltonian constraint of general relativity to obtain the Einstein-Hamiltonian-Jacobi equation. We obtain a time parameter prescription demanding that geometry itself determines the time, not the matter field, such that the time so defined being equivalent to the time that enters into the Schroedinger equation. Without any reference to the Wheeler-DeWitt equation and without invoking the expansion of exponent in WKB wavefunction in powers of Planck mass, we obtain an equation for quantum gravity in Schroedinger form containing time. We restrict ourselves to a minisuperspace description. Unlike matter field equation our equation is equivalent to the Wheeler-DeWitt equation in the sense that our solutions reproduce also the wavefunction of the Wheeler-DeWitt equation provided one evaluates the normalization constant according to the wormhole dominance proposal recently proposed by us.