On the Schrodinger Equation for the Minisuperspace Models

TL;DR

This paper derives a time-dependent Schrödinger equation for the FRW cosmological model with scalar matter, introducing an invariant action to address reparametrization issues, and extends the approach to supersymmetric models.

Contribution

It presents a novel method to obtain a time-dependent Schrödinger equation in minisuperspace models by adding a reparametrization-invariant action, including supersymmetric extensions.

Findings

Derived a time-dependent Schrödinger equation for FRW models.

Extended the method to supersymmetric quantum constraints.

Identified a supersymmetric constraint as a square root of the Schrödinger operator.

Abstract

We obtain a time-dependent Schrodinger equation for the Friedmann - Robertson - Walker (FRW) model interacting with a homogeneous scalar matter field. We show that for this purpose it is necesary to include an additional action invariant under the reparametrization of time. The last one does not change the equations of motion of the system, but changes only the constraint which at the quantum level becomes time-dependent Schrodinger equation. The same procedure is applied to the supersymmetric case and the supersymmetric quantum constraints are obtained, one of them is a square root of the Schrodinger operator.