Quantum Oscillations Can Prevent the Big Bang Singularity in an Einstein-Dirac Cosmology

We consider a spatially homogeneous and isotropic system of Dirac particles coupled to classical gravity. The dust and radiation dominated closed Friedmann-Robertson-Walker space-times are recovered as limiting cases. We find a mechanism where quantum oscillations of the Dirac wave functions can prevent the formation of the big bang or big crunch singularity. Thus before the big crunch, the collapse of the universe is stopped by quantum effects and reversed to an expansion, so that the universe opens up entering a new era of classical behavior. Numerical examples of such space-times are given, and the dependence on various parameters is discussed. Generically, one has a collapse after a finite number of cycles. By fine-tuning the parameters we construct an example of a space-time which is time-periodic, thus running through an infinite number of contraction and expansion cycles.Comment: 8 pages, LaTeX, 4 figures, statement on energy conditions correcte

The Dirac Equation and the Normalization of its Solutions in a Closed Friedmann-Robertson-Walker Universe

We set up the Dirac equation in a Friedmann-Robertson-Walker geometry and separate the spatial and time variables. In the case of a closed universe, the spatial dependence is solved explicitly, giving rise to a discrete set of solutions. We compute the probability integral and analyze a space-time normalization integral. This analysis allows us to introduce the fermionic projector in a closed Friedmann-Robertson-Walker geometry and to specify its global normalization as well as its local form.Comment: 22 pages, LaTeX, sign error in equation (3.7) correcte

Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields

Using the Pauli-Villars regularization and arguments from convex analysis, we construct solutions to the classical time-independent Maxwell equations in Dirac's vacuum, in the presence of small external electromagnetic sources. The vacuum is not an empty space, but rather a quantum fluctuating medium which behaves as a nonlinear polarizable material. Its behavior is described by a Dirac equation involving infinitely many particles. The quantum corrections to the usual Maxwell equations are nonlinear and nonlocal. Even if photons are described by a purely classical electromagnetic field, the resulting vacuum polarization coincides to first order with that of full Quantum Electrodynamics.Comment: Final version to appear in Arch. Rat. Mech. Analysi