A Schrodinger Equation for Quantum Universes

TL;DR

This paper develops a gauge fixing method for canonical quantization of finite-degree-of-freedom Lagrangian systems with reparametrization invariance, leading to a Schrödinger-like equation for quantum cosmological models.

Contribution

It introduces a gauge fixing approach that results in an effective Hamiltonian and Schrödinger equation for quantum universes, applied specifically to Robertson--Walker metrics.

Findings

Derived a Schrödinger equation for quantum cosmological models.

Applied the method to Robertson--Walker metrics with bosonic zero modes.

Provided a framework for quantizing reparametrization-invariant 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 (Presented at the International Workshop ``Birth of the Universe and Fundamental Physics'', Rome, May 18-21, 1994).