  Determining the physical Hilbert space is often considered the most difficult
but crucial part of completing the quantization of a constrained system. In
such a situation it can be more economical to use effective constraint methods,
which are extended here to relativistic systems as they arise for instance in
quantum cosmology. By side-stepping explicit constructions of states, such
tools allow one to arrive much more feasibly at results for physical
observables at least in semiclassical regimes. Several questions discussed
recently regarding effective equations and state properties in quantum
cosmology, including the spreading of states and quantum back-reaction, are
addressed by the examples studied here.
