Quantum Einstein equations

We derive the quantum Einstein equations (which are the quantum generalisation of the Einstein equations of classical gravity) from Bohmian quantum gravity. Bohmian quantum gravity is a non-classical geometrodynamics (in the ADM formalism) which describes the time evolution of a 3-geometry and of a matter field (or other matter degrees of freedom) on a three manifold. The evolution is determined by a velocity law which is defined by the wave function. The wave function itself satisfies the Wheeler-DeWitt equation. We cast the Bohmian dynamics into the form of the Einstein field equations, where the interesting novelty is a contribution to the energy-momentum tensor that depends on the quantum potential.