Problem of unitarity and quantum corrections in semiclassical quantum gravity

Using both the Born-Oppenheimer idea and the de Broglie–Bohm interpretation of a wave function we represent in a different way the semiclassical quantum gravity from the Wheeler-DeWitt equation in an oscillating regime which can preserve completely the unitary quantum evolution of a matter field at the expense of a nonlinear gravitational field equation, but has the same asymptotic limit as the others. We apply the de Broglie–Bohm interpretation to the nonlinear gravitational field equation to develop a perturbation method to find the quantum corrections of a matter field to the gravity. The semiclassical Einstein equation with the quantum corrections is found for a minimal quantum FRW cosmological model.