Operator ordering and the flatness of the universe

Abstract

In order to make predictions in quantum cosmology one has to resolve the problem of the operator ordering in the hamiltonian. This is equivalent to defining a differential operator on superspace, the infinite dimensional manifold of all 3-metrics and matter field configurations on a 3-surface S. We propose that this operator should be the laplacian in a natural metric. There remains a difficulty however because the metric connection on superspace depends on the metric in a nonlinear manner. This nonlinearity would be inconsistent with the interpretation of the lapse function as a Lagrange multiplier. It may cancel out if an equal number of fermion degrees of freedom are included. The probability measure on superspace would be the measure associated with this metric. In situations in which the wave function can be interpreted in terms of classical solutions by the WKB approximation, this choice of measure implies that the probability of finding the 3-metric and matter field configuration in a given region of superspace is proportional to the proper time that the solutions spend in that region. As an application we compute the probability distribution of the density parameter Ω in a minisuperspace model which is in the quantum state defined by a path integral over compact 4-geometries. If we restrict attention to a fixed value of the density, we find that the probability distribution is entirely concentrated at Ω = 1.