Decoupling of the re-parametrization degree of freedom and a generalized probability in quantum cosmology

The high degree of symmetry renders the dynamics of cosmological as well as some black hole spacetimes describable by a system of finite degrees of freedom. These systems are generally known as minisuperspace models. One of their important key features is the invariance of the corresponding reduced actions under reparametrizations of the independent variable, a fact that can be seen as the remnant of the general covariance of the full theory. In the case of a system of $n$ degrees of freedom, described by a Lagrangian quadratic in velocities, one can use the lapse by either gauge fixing it or letting it be defined by the constraint and subsequently substitute into the rest of the equations. In the first case, the system is solvable for $n$ accelerations and the constraint becomes a restriction among constants. In the second case, the system can only be solved for $n-1$ accelerations and the "gauge" freedom is transferred to the choice of one of the scalar degrees of freedom. In this paper, we take the second path and express all $n-1$ scalar degrees of freedom in terms of the remaining one, say $q$. By considering these $n-1$ degrees of freedom as arbitrary but given functions of $q$, we manage to extract a two dimensional pure gauge system consisting of the lapse $N$ and the arbitrary $q$: in a way, we decouple the reparametrization invariance from the rest of the equations of motion. The solution of the corresponding quantum two dimensional system is used for the definition of a generalized probability for every configuration $f^i (q)$, be it classical or not. The main result is that, interestingly enough, this probability attains its extrema on the classical solution of the initial $n$-dimensional system.