About the proper time and the mass of the universe

For a closed universe, a modification of the quantum gravity where the dynamics is reduced to the motion in the orbit of a general covariance groups has been proposed. To connect these motion parameters, namely, proper time and spatial shifts, to observations, classical equations of motion were introduced into the quantum theory as additional conditions. The equations account for differential conservation laws for additional dynamical variables, which form the spatial density of distribution and motion of the universe's proper mass in the representation of Arnovitt, Deser and Misner (ADM). This made it possible to determine the average values of the parameters of proper time and spatial shifts in the evolutionary history of the universe. In order to preserve the homogeneity and isotropy of space, the proper mass of the universe should next be set equal to zero. Nonzero values of its proper mass (mass spectrum) were allowed in the operator canonical representation of the quantum gravity, which was also introduced instead of the ADM representation.