Dirac Decomposition of Wheeler-DeWitt Equation on the Bianchi Class A Models

TL;DR

This paper introduces a Dirac-like factorization of the Wheeler-DeWitt equation for Bianchi Class A models, resulting in a positive-definite probability density and revealing a spin-like degree of freedom influencing early universe quantum behavior.

Contribution

It extends the Dirac-Square-Root formalism to quantum cosmology, providing a new first-order differential equation approach with spinor solutions for Bianchi models.

Findings

Positive-definite probability density achieved

Conserved current established

Universe exhibits anisotropy-oscillation like Zitterbewegung in early quantum stage

Abstract

The Wheeler-DeWitt equation for the Bianchi Class A cosmological models is expressed generally in terms of the second-order differential equation like the Klein-Gordon equation. To obtain the positive-definite probability density, a new method extending the Dirac-Square-Root formalism, which factorizes the Wheeler-DeWitt equation into the first-order differential equation using the Pauli matrices, is investigated. The solutions to the Dirac type equation thus obtained are expressed in terms of two-component spinor form. The probability density defined by the solution is positive-definite and there is a conserved current. The newly found spin-like degree of freedom causes the universe to go through an early quantum stage of evolution with agitated anisotropy-oscillation like Zitterbewegung.