Quantum cosmology and physics of transitions with a change of the spacetime signature

This paper examines elements of the general theory of transitions with changing spacetime signature in quantum gravity and cosmology as suggested in a pioneer work of A D Sakharov. Unlike the conventional formal method for functional integration, this approach uses as the starting point the Dirac—Wheeler—de Witt operator quantization and its reduction to quantization in Arnowitt—Deser—Misner variables. It has been demonstrated that motivation to consider Euclidean—Lorentzian transitions consists in global ambiguity of physical reduction on the phase space of the theory (a gravitational analog of the problem of Gribov's copies). This ambiguity in particular results in the indefinite sign of the physical inner product of quantum states and leads to the concept of third quantization. An alternative approach is quantization in the York gauge and in special variables of conformal superspace. This quantization is likely to provide a global solution of the ambiguity problem, but the problems of Euclidean—Lorentzian transitions arise in this formalism in the language of complexification of a conformal space of the Wick-rotation type for the time variable in the Klein—Gordon equation. The problem of origin of the early Universe via gravitational tunneling in Hartle—Hawking and Vilenkin quantum states is considered to illustrate applications of the general theory. The mechanism is described by which loop effects generate the normalizable distribution function for the ensemble of chaotic inflationary universes. In a model with a large non-minimal coupling constant for the scalar inflation, this mechanism gives rise to a sharp probability peak at sub-Planckian values for the Hubble constant which is in good agreement with the contemporary observational status of the cosmological inflation theory.