Simple Analytical Solutions of the Wheeler-DeWitt Equation in the Classical Hamilton-Jacobi Limit

TL;DR

This paper derives simple analytical solutions to the Wheeler-DeWitt equation in a cosmological setting, linking potential forms to the classical Hamilton-Jacobi limit and exploring their cosmological implications.

Contribution

It shows how the Wheeler-DeWitt equation reduces to the classical Hamilton-Jacobi equation under a specific constraint, classifies potential forms, and provides analytical solutions for cosmological models.

Findings

Classified potential forms include exponential, quadratic, and cosine types with negative cosmological constant.

Derived analytical solutions for the scale factor and scalar field in the cosine potential case.

Discussed cosmological implications of these solutions in inflation and dark energy contexts.

Abstract

We investigate the Wheeler-DeWitt equation for a flat, homogeneous, and isotropic Universe containing a canonical scalar field with a potential. We show that under the constraint $|\Psi|=1$, where the Wheeler-DeWitt equation exactly becomes the classical Hamilton-Jacobi equation, the form of the potential is completely determined depending on the value of the operator ordering parameter. Furthermore, we demonstrate that the classified potentials admit simple forms, such as the exponential, quadratic with a negative cosmological constant, and cosine-type potential with a negative cosmological constant. Several of these have already been explored in the context of inflation or dark energy. Finally, focusing on the system with the cosine-type potential and a negative cosmological constant in the classified potentials, we derive the analytical solutions for the scale factor and the scalar field and discuss the cosmological implications.