  When classical degrees of freedom and quantum degrees of freedom are
consistently coupled, the former diffuse, while the latter undergo decoherence.
Here, we construct a theory of quantum matter fields and Nordstrom gravity in
which the space-time metric is treated classically. The dynamics is constructed
via the classical-quantum path integral and is completely positive, trace
preserving (CPTP), and respects the classical-quantum split. The weak field
limit of the model matches the Newtonian limit of the full covariant path
integral but it is easier to show that the theory is both diffeomorphism
invariant, CPTP, and has the appropriate classical limit.
