Fermions in quantum cosmology

This paper considers the extension of the ideas of quantum cosmology and, in particular, the proposal of Hartle and Hawking for the boundary conditions of the Universe, to models which incorporate fermions in a realistic manner. We consider inhomogeneous fermionic perturbations about a homogeneous, isotropic minisuperspace background model, by expanding the fermion fields in spinor harmonics on the spatial sections, taken to be three-spheres. The Dirac action is thus found to take the form of an infinite sum of terms, each describing a time-dependent Fermi oscillator. On quantization, we find that the Wheeler-DeWitt equation for the wave function of the Universe may be decomposed into a set of time-dependent Schroumldinger equations, one for each fermion mode, and a background minisuperspace Wheeler-DeWitt equation, which includes a term in its potential describing the back reaction of the fermionic perturbations on the homogeneous modes. Our quantization procedure employs the holomorphic representation for the fermion modes, which permits them to be treated in a manner very similar to the case of bosonic perturbations considered by Halliwell and Hawking. We set initial conditions for the Schrödinger equations by applying the proposal of Hartle and Hawking that the quantum state of the Universe is defined by a path integral over compact four-metrics and regular matter fields. We find this to imply that the fermion modes start out in their ground state. Particles are created in the subsequent (inflationary) evolution, and their number, defined with respect to instantaneous Hamiltonian diagonalization, is calculated and is found to be finite. We calculate the back-reaction term and find, after regularization, that its effect is negligible. We construct a model of a fermionic particle detector and, in the case of an exact de Sitter background, examine its response to the state picked out by the Hartle-Hawking proposal. We show that it experiences a thermal spectrum at the de Sitter temperature, with a distribution of the Fermi-Dirac form, although the distribution does not have the correct density-of-states factor to be precisely Planckian.