Dirac decomposition of Wheeler-de Witt equation in the Bianchi class A models (Q2774368)

According to Dirac's method of canonical quantization, the Hamiltonian constraint occurring in the classical Arnowitt-Deser-Misner formalism of general relativity theory (GRT) has to be replaced with a supplementary condition on the wave function which represents the quantum state of space-time. This provides the major equation of canonical quantum GRT called the Wheeler-DeWitt (WD) equation. For Bianchi-type models, it can be reduced to a second-order differential equation which is similar to the Klein-Gordon equation. This brings about the problem that the WD equation provides a probability density which is not positive definite. NEWLINENEWLINENEWLINEAfter reviewing the so-called Dirac square root formalism and some of its applications to cosmological models, the authors propose a new method that factorizes the WD equation into first-order differential equations, using the Pauli matrices. This leads to an equation that resembles the Dirac equation. Assuming that the resulting Hamiltonian is selfadjoint, it is shown that the Dirac factorization works in the case of all Bianchi Class A vacuum models, except for the type-IX model. Afterwards, by an appropriate restriction of the form of the Hamiltonian, the probability density is made positive definite. NEWLINENEWLINENEWLINEFor the Bianchi type-I, II, VI\(_0\), and VII\(_0\) models then the Dirac-like equation is derived and solved. As the time parameter approaches the limit of the singularity of the cosmological model, the Dirac-type equations for the II, VI\(_0\), and VII\(_0\), models are shown to reduce to that one of the type-I model (which is especially interesting in view of earlier discussions on the local character of singularities occurring in GRT).