The problem of time in quantum cosmology is formulated in analogy with the same problem in quantum theory of a relativistic particle. The problem is in the interference between the dynamics in future and the dynamics in past in a general solution of the relativistic wave equation. To solve the problem of time we have to modify the relativistic quantum theory and transform the wave equation in a form of the Shroedinger equation. Our suggestion of the solution consists of two steps. First, we formulate a self-consistent modification of quantum rules, in which some degrees of freedom of the dynamical system are not quantized. Such a modification is a variant of the quasi-classical approximation for the original quantum theory. Second, in the case of the closed Universe, a dynamical variable T is proposed, which has the meaning of proper time of the Universe. This dynamical variable is defined as a canonically conjugated coordinate to an eigenvalue of a special Witten-Dirac operator. The latter is a 3D projection of the standard 4D Dirac operator on a spatial slice of the Universe. Witten introduced it in his proof of the positive energy theorem in General Relativity. It is this dynamical variable T which remains classical in our modified theory of quantum Universe. As a result, the modified wave equation of the Universe acquires the Shroedinger equation form.

Page created on February 2, 2006.