Quantum cosmological Friedman models with an initial singularity

Abstract

We consider the Wheeler-DeWitt equation $H\psi=0$ in a suitable Hilbert space. It turns out that this equation has countably many solutions $\psi_i$ which can be considered as eigenfunctions of a Hamilton operator implicitly defined by $H$. We consider two models, a bounded one, $0<r<r_0$, and an unbounded, $0<r<\un$, which represent different eigenvalue problems. In the bounded model we look for eigenvalues $\Lam_i$, where the $\Lam_i$ are the values of the cosmological constant which we used in the Einstein-Hilbert functional, and in the unbounded model the eigenvalues are given by $(-\Lam_i)^{-\frac {n-1}{n}}$, where $\Lam_i<0$. The $\psi_i$ form a basis of the underlying Hilbert space. All solutions have an initial singularity in $r=0$. Under certain circumstances a smooth transition from big crunch to big bang is possible.