A SCHR\"Odinger Equation for Mini Universes

TL;DR

This paper develops a gauge fixing method for finite-degree-of-freedom systems with time reparametrization invariance, leading to a Schrödinger equation for mini universes, facilitating their quantization.

Contribution

It introduces a gauge fixing approach in the canonical formulation of reparametrization-invariant systems and derives a Schrödinger equation for quantum mini universes.

Findings

Derived a Schrödinger equation for mini universes.

Applied the method to Robertson--Walker metrics with bosonic fields.

Provided a framework for quantizing finite-dimensional gravitational systems.

Abstract

We discuss how to fix the gauge in the canonical treatment of Lagrangians, with finite number of degrees of freedom, endowed with time reparametrization invariance. The motion can then be described by an effective Hamiltonian acting on the gauge shell canonical space. The system is then suited for quantization. We apply this treatment to the case of a Robertson--Walker metric interacting with zero modes of bosonic fields and write a \S equation for the on--shell wave function.