On the appearance of time in the classical limit of quantum gravity

Abstract: A possible solution of the problem of time in the Wheeler - DeWitt quantum geometrodynamics is that time appears in semiclassical limit. Following this line of thinking, one can come to the Schrodinger equation for matter fields in curved spacetime with quantum-gravitational corrections. In the present paper, we study the semiclassical limit in the case of a closed isotropic model with a scalar field decomposed into modes. We analyse calculations made within frameworks of three approaches. The first approach was proposed by Kiefer and Singh. Since the Wheeler - DeWitt equation does not contain a time derivative, it is constructed by means of a special mathematical procedure, time variable being a parameter along a classical trajectory of gravitational field. The second method was suggested in the paper of Maniccia and Montani who introduced the Kuchar - Torre reference fluid as an origin of time. And the third is the extended phase space approach to quantization of gravity. In this approach, the temporal Schrodinger equation is argued to be more fundamental than the Wheeler - DeWitt equation, and there is no problem of time. The origin of time is fixing of a reference frame of some observer, who can register macroscopic consequences of quantum gravitational phenomena in the Very Early Universe. To go to the semiclassical limit, the Born - Oppenheimer approximation for gravity is used. In each of the approaches, in the order ${\cal O}(1/M)$, a temporal Schrodinger equation for matter fields in curved spacetime with quantum gravitational corrections is obtained. However, equations and corrections are different in various approaches, and the results depend on additional assumptions made within the scopes of these approaches.