equations are derived by the well -known order by order expansion in $M^{-1}$, where $M$ acts as a parameter analogous to the mass of the heavy modes (in our case the minisuperspace ones) in the Born-Oppenheimer approximation. The Hamilton-Jacobi equation satisfied by $S_{(n)}$ will have a $D-1$ parameter family of solutions, the subindex $(n)$ labeling such parameters characterizes a particular solution and thus a specific WKB branch.

The reduced density matrix associated with the wave function $(3.4)$ is then given by: $$\rho_{red}(r_2,r_1) =\sum\limits_{n,n'} e^{iM[S_{(n)}(r_1) - S_{(n')}(r_2)]}
    C_{(n)}(r_1)C_{(n')}(r_2) I_{n,n'}(r_2,r_1)
                                                             $$ where we call $$I_{n,n'}(r_2,r_1)= \int \chi^{*}_{(n')}(r_2,\Phi)\chi_{(n)}(r_1,\Phi) d\Phi
                                                              $$ the influence functional. The justification for this name comes from the fact that it has been shown (Paz and Sinha 1991, Kiefer 1991) that it is exactly analogous to the Feynman-Vernon (1963) influence functional in the case in which the environment is initially in a pure state. For models with $r >1$ it can be shown that it is indeed a functional of two histories, which is in turn related to the Gell-Mann Hartle decoherence functional (Gell-Mann and Hartle 1990, Griffith 1984, Omnes 1988). It is the central object of our consideration.

Decoherence and Correlation
---------------------------

We will now study the emergence of 