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

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