Boundary conditions in quantum cosmology

The problem of boundary conditions for the cosmological wave function \ensuremath{\psi} is analyzed using a simple model as an example. The model has two degrees of freedom: the scale factor and a homogeneous scalar field. A tunneling wave function is found for this model which describes an ensemble of universes tunneling from ``nothing'' to a de Sitter space, then evolving along the lines of an inflationary scenario and eventually collapsing to a singularity. A boundary condition in superspace is proposed which selects the tunneling solution for \ensuremath{\psi}. This boundary condition can be thought of as a quantum version of the Penrose condition in classical general relativity. The probability current, the concept of time, and the possibility of an experimental test of quantum cosmology are also discussed.