Decoherence and back reaction in quantum cosmology: Multidimensional minisuperspace examples

We study some aspects of the semiclassical limit of quantum cosmology using multidimensional minisuperspace models with scalar perturbations. Regarding the minisuperspace coordinates as the "relevant" degrees of freedom and the scalar field modes as the "irrelevant" ones, we study the reduced density matrix to analyze the existence of decoherence and correlations. For a Bianchi type-I model we show that the correlations predicted from the peak of the Wigner function associated with a decohered WKB branch are the same as those determined by the semiclassical Einstein equations. We discuss the renormalization of the divergent quantities that appear in the calculation and show that usual dimensional-regularization techniques only allow us to regulate the phase of the reduced density matrix. This illustrates the fundamental difference that exists between the divergences found in the back-reaction equations and the ones that affect the quantum coherence. We analyze a class of models with adiabatic behavior in which there are no problematic divergences affecting the decoherence factor. Finally, we discuss some aspects of the relation between our approach, based on the use of the reduced density matrix, and that formulated in terms of spacetime histories.