Two-Component Formulation of the Wheeler-DeWitt Equation

TL;DR

This paper reformulates the Wheeler-DeWitt equation for a cosmological model as a two-component Schrödinger equation, simplifying its solution to a harmonic oscillator eigenvalue problem and revealing superpositions of parity solutions.

Contribution

It introduces a novel two-component formulation of the Wheeler-DeWitt equation, enabling easier solution methods and analysis of solution parity and mode expansion properties.

Findings

Solutions are superpositions of two definite-parity states.

Mode expansion is always infinite for both parities.

Explicit example satisfies DeWitt's boundary condition.

Abstract

The Wheeler-DeWitt equation for the minimally coupled FRW-massive-scalar-field minisuperspace is written as a two-component Schr\"odinger equation with an explicitly `time'-dependent Hamiltonian. This reduces the solution of the Wheeler-DeWitt equation to the eigenvalue problem for a non-relativistic one-dimensional harmonic oscillator and an infinite series of trivial algebraic equations whose iterative solution is easily found. The solution of these equations yields a mode expansion of the solution of the original Wheeler-DeWitt equation. Further analysis of the mode expansion shows that in general the solutions of the Wheeler-DeWitt equation for this model are doubly graded, i.e., every solution is a superposition of two definite-parity solutions. Moreover, it is shown that the mode expansion of both even and odd-parity solutions is always infinite. It may be terminated artificially to construct approximate solutions. This is demonstrated by working out an explicit example which turns out to satisfy DeWitt's boundary condition at initial singularity.