Dirac decomposition of Wheeler-DeWitt equation in the Bianchi Class A models

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