Shape Invariant Potentials in Supersymmetric Quantum Cosmology

TL;DR

This paper explores the application of shape invariant potentials in supersymmetric quantum cosmology, demonstrating how they simplify solving the Schrödinger-Wheeler-DeWitt equation for a flat FRW model with a scalar field.

Contribution

It introduces the use of shape invariance in supersymmetric quantum cosmology, providing a case study that connects factor ordering choices with supersymmetric partner Hamiltonians.

Findings

Shape invariance helps relate different factor orderings.

Ground and excited states are explicitly constructed.

Supersymmetric Hamiltonians are explicitly formulated.

Abstract

In this brief review, we comment on the concept of shape invariant potentials, which is an essential feature in many settings of $N=2$ supersymmetric quantum mechanics. To motivate its application within supersymmetric quantum cosmology, we present a case study to illustrate the value of this promising tool. Concretely, we take a spatially flat FRW model in the presence of a single scalar field, minimally coupled to gravity. Then, we extract the associated Schr\"odinger-Wheeler-DeWitt equation, allowing for a particular scope of factor ordering. Subsequently, we compute the corresponding supersymmetric partner Hamiltonians, $H_1$ and $H_2$. Moreover, we point out how the shape invariance property can be employed to bring a relation among several factor orderings choices for our Schr\"odinger-Wheeler-DeWitt equation. The ground state is retrieved, and the excited states easily written. Finally, the Hamiltonians, $H_1$ and $H_2$ are explicitly presented within a $N=2$ supersymmetric quantum mechanics framework.