Coherent State Representation of Semiclassical Quantum Gravity

We elaborate the recently introduced asymptotically exact semiclassical quantum gravity derived from the Wheeler-DeWitt equation by finding a particular coherent state representation of a quantum scalar field in which the back-reaction of the scalar field Hamiltonian exactly gives rise to the classical one. In this coherent state representation classical spacetime emerges naturally from semiclassical quantum gravity. Electronic address: [email protected] 1 Canonical quantum gravity based on the Wheeler-DeWitt (WDW) equation has been used as one of its applications to provide a self-consistent theory of quantum fields in a curved spacetime (for review and references, see [1]). Most of the methods in this direction rely on the Born-Oppenheimer idea that the larger mass scale out of several mass scales of a quantum system becomes first semiclassical and the relatively smaller mass scales retain the quantum mechanical nature. In this approach to quantum field theory in the curved spacetime the classical spacetime emerges from the quantum domain but the matter fields (typically scalar fields) keep their quantum mechanical properties satisfying the timedependent functional Schrödinger equation. The advantage of this approach is that one may treat semiclassical theory relatively simply, including some parts of quantum gravitational corrections to the matter fields and replacing the energy-momentum tensor by its quantum mechanical expectation values. The disadvantage is that the quantum corrections of gravity to matter fields may not be achieved fully and the renormalization problem of wave functions may not be resolved in this approach to semiclassical quantum gravity. This approach to semiclassical quantum gravity, despite its shortcomings, proves quite useful when one considers the quantum gravity effects semiclassically and especially in the context of cosmology. However, there has been one step not completely resolved in deriving classical gravity from the WDW equation. One usually assumes that classical gravity could be directly obtained from the WDW equation as the Hamilton-Jacobi equation. It could, of course, presumably sound physical in a certain domain, but the logical steps are not fulfilled, since considering the different mass scales of the gravitational fields and matter fields it would be more correct for the gravity to emerge first from the quantum domain and then the larger mass scaled matter fields. The scheme of deriving semiclassical quantum gravity 1 We distinguish this approach so-called semiclassical quantum gravity from conventional quantum field theory in curved spacetimes. We refer to [2] for review and references of the latter approach.