Quantum Einstein-Dirac Bianchi universes

We study the minisuperspace quantization of spatially homogeneous (Bianchi) cosmological universes sourced by a Dirac-spinor field. The quantization of the homogeneous spinor leads to a finite-dimensional fermionic Hilbert space and, thereby, to a multicomponent Wheeler-DeWitt equation whose main features are (i) the presence of spin-dependent Morse-type potentials and (ii) the appearance of a $q$-number squared-mass term, which is of order $\mathcal{O}({\hslash }^2)$ and which is affected by ordering ambiguities. We give the exact quantum solution of the Bianchi type-II system (which contains both scattering states and bound states) and discuss the main qualitative features of the quantum dynamics of the (classically chaotic) Bianchi type-IX system. We compare the exact quantum dynamics of fermionic cosmological billiards to previous works that described the spinor field as being either classical or Grassmann-valued.