An inhomogeneous toy model of the quantum gravity with the explicitly evolvable observables

摘要

An inhomogeneous (1 $+$ 1)-dimensional model of the quantum gravity is considered. It is found, that this model corresponds to a string propagating in some curved background space. The quantization scheme including the Wheeler–DeWitt equation and the “particle on a sphere” type of the gauge condition is suggested. In the quantization scheme considered, the “problem of time” is resolved by building of the quasi-Heisenberg operators acting in a space of solutions of the Wheeler–DeWitt equation, and the normalization of the wave function corresponds to the Klein–Gordon type. To analyze the physical consequences of the scheme, a (1 $+$ 1)-dimensional background space is considered for which a classical solution is found and quantized. The obtained estimations show the way to solution of the cosmological constant problem for this model. Such a solution consists in compensation of the zero-point oscillations of the matter fields by the quantum oscillations of the scale factor. Along with such a compensation, a slow global evolution of a background exists that corresponds to an universe expansion.