Compactification and signature transition in Kaluza-Klein spinor cosmology

We study the classical and quantum cosmology of a 4 + 1-dimensional space-time with a non-zero cosmological constant coupled to a self-interacting massive spinor field. We consider a spatially flat Robertson-Walker universe with the usual scale factor R (t) and an internal scale factor a (t) associated with the extra dimension. For a free spinor field the resulting equations admit exact solutions, whereas for a self-interacting spinor field one should resort to a numerical method for exhibiting their behavior. These solutions give rise to a degenerate metric and exhibit signature transition from a Euclidean to a Lorentzian domain. Such transitions suggest a compactification mechanism for the internal and external scale factors such that a ∼ R⁻¹ in the Lorentzian region. The corresponding quantum cosmology and the ensuing Wheeler-DeWitt equation have exact solutions in the mini-superspace when the spinor field is free, leading to wavepackets undergoing signature change. The question of stabilization of the extra dimension is also discussed.