M ay 2 00 3 A nonlinear quantum model of the Friedmann universe

A discussion is given of the quantisation of a physical system with finite degrees of freedom subject to a Hamiltonian constraint by treating time as a constrained classical variable interacting with an unconstrained quantum state. This leads to a quantisation scheme that yields a Schrödinger-type equation which is in general nonlinear in evolution. Nevertheless it is compatible with the probabilistic interpretation of quantum mechanics and in particular the construction of a Hilbert space with a Euclidean norm is possible. The new scheme is applied to the quantisation of a Friedmann universe with a massive scalar field whose dynamical behaviour is investigated numerically. PACS numbers: 04.60.Ds, 04.60.Kz