  The effort to discover a quantum theory of gravity is motivated by the need
to reconcile the incompatibility between quantum theory and general relativity.
Here, we present an alternative approach by constructing a consistent theory of
classical gravity coupled to quantum field theory. The dynamics is linear in
the density matrix, completely positive and trace preserving, and reduces to
Einstein's theory of general relativity in the classical limit. Consequently,
the dynamics doesn't suffer from the pathologies of the semiclassical theory
based on expectation values. The assumption that general relativity is
classical necessarily modifies the dynamical laws of quantum mechanics -- the
theory must be fundamentally stochastic in both the metric degrees of freedom
and in the quantum matter fields. This allows it to evade several no-go
theorems purporting to forbid classical-quantum interactions. The measurement
postulate of quantum mechanics is not needed -- the interaction of the quantum
degrees of freedom with classical space-time necessarily causes decoherence in
the quantum system. We first derive the general form of classical-quantum
dynamics and consider realisations which have as its limit deterministic
classical Hamiltonian evolution. The formalism is then applied to quantum field
theory interacting with the classical space-time metric. One can view the
classical-quantum theory as fundamental or as an effective theory useful for
computing the back-reaction of quantum fields on geometry. We discuss a number
of open questions from the perspective of both viewpoints.
