Reparametrization Invariance and the Schr\"odinger Equation

TL;DR

This paper explores systems invariant under time reparametrization, deriving a time-dependent Schr"odinger equation through a two-stage construction, applicable to supersymmetric theories and involving supergravity coupling.

Contribution

It introduces a novel two-stage procedure to construct reparametrization-invariant quantum systems, connecting the Schr"odinger equation with supersymmetric and supergravity frameworks.

Findings

Derived a time-dependent Schr"odinger equation from reparametrization-invariant systems.

Developed a two-stage construction method for such systems.

Extended the approach to supersymmetric quantum mechanics coupled with supergravity.

Abstract

In the present work we consider a time-dependent Schr\"odinger equation for systems invariant under the reparametrization of time. We develop the two-stage procedure of construction such systems from a given initial ones, which is not invariant under the time reparametrization. One of the first-class constraints of the systems in such description becomes the time-dependent Schr\"odinger equation. The procedure is applicable in the supersymmetric theories as well. The $n=2$ supersymmetric quantum mechanics is coupled to world-line supergravity, and the local supersymmetric action is constructed leading to the square root representation of the time-dependent Schr\"odinger equation.