A starting point for the present work was the statement recently discussed in the literature that two Hamiltonian formulations for the theory of gravity, the one proposed by Dirac and the other by Arnowitt -Deser -Misner, may not be related by a canonical transformation. In its turn, it raises a question about the equivalence of these two Hamiltonian formulations and their equivalence to the original formulation of General Relativity. We argue that, since the transformation from components of metric tensor to the ADM variables touches gauge degrees of freedom, which are non-canonical from the point of view of Dirac, the problem cannot be resolved in the limits of the Dirac approach. The proposed solution requires the extension of phase space by treating gauge degrees of freedom on an equal footing with other variables and introducing missing velocities into the Lagrangian by means of gauge conditions in differential form. We illustrate with a simple cosmological model the features of Hamiltonian dynamics in extended phase space. Then, we give a clear proof for the full gravitational theory that the ADM-like transformation is canonical in extended phase space in a wide enough class of possible parametrizations.

A way of constructing mathematically correct quantum geometrodynamics of a closed universe is presented. The resulting theory appears to be gauge-noninvariant and thus consistent with the observation conditions of a closed universe, by that being considerably distinguished from the traditional Wheeler -DeWitt one. For the Bianchi-IX cosmological model it is shown that a normalizable wave function of the Universe depends on time, allows the standard probability interpretation and satisfies a gauge-noninvariant dynamical Schrödinger equation. The Wheeler -DeWitt quantum geometrodynamics is represented by a singular, BRST-invariant solution to the Schrödinger equation having no property of normalizability.

Quantum geometrodynamics in extended phase space describes phenomenologically the integrated system "a physical object + observation means (a gravitational vacuum condensate)". The central place in this version of QGD belongs to the Schrödinger equation for a wave function of the Universe. An exact solution to the "conditionally-classical" set of equations in extended phase space for the Bianchi-IX model and the appropriate solution to the Schrödinger equation are considered. The physical adequacy of the obtained solutions to existing concepts about possible cosmological scenarios is demonstrated. The gravitational vacuum condensate is shown to be a cosmological evolution factor.

The Wheeler -DeWitt equation was proposed 50 years ago and until now it is the cornerstone of most approaches to quantization of gravity. One can find in the literature the opinion that the Wheeler -DeWitt equation is even more fundamental than the basic equation of quantum theory, the Schrödinger equation. We still should remember that we are in the situation when no observational data can confirm or reject the fundamental status of the Wheeler -DeWitt equation, so we can give just indirect arguments in favor of or against it, grounded on mathematical consistency and physical relevance. I shall present the analysis of the situation and comparison of the standard Wheeler -DeWitt approach with the extended phase space approach to quantization of gravity. In my analysis I suppose, firstly, that a future quantum theory of gravity must be applicable to all phenomena from the early Universe to quantum effects in strong gravitational fields, in the latter case the state of the observer (the choice of a reference frame) may appear to be significant. Secondly, I suppose that the equation for the wave function of the Universe must not be postulated but derived by means of a mathematically consistent procedure, which exists in path integral quantization. When applying this procedure to any gravitating system, one should take into account features of gravity, namely, non-trivial spacetime topology and possible absence of asymptotic states. The Schrödinger equation has been derived early for cosmological models with a finite number of degrees of freedom, and just recently it has been found for the spherically symmetric model which is a simplest model with an infinite number of degrees of freedom. The structure of the Schrödinger equation and its general solution appears to be very similar in these cases. The obtained results give grounds to say that the Schrödinger equation retains its fundamental meaning in constructing quantum theory of gravity.

Last years a certain attention was attracted to the statement that Hamiltonian formulations of General Relativity, in which different parametrizations of gravitational variables were used, may not be related by a canonical transformation. The example was given by the Hamiltonian formulation of Dirac and that of Arnowitt -Deser -Misner. It might witness for non-equivalence of these formulations and the original (Lagrangian) formulation of General Relativity. The problem is believed to be of importance since many authors make use of various representations of gravitational field as a starting point in searching a way to reconcile the theory of gravity with quantum principles. It can be shown that the mentioned above conclusion about non-equivalence of different Hamiltonian formulations is based on the consideration of canonical transformations in phase space of physical degrees of freedom only, while the transformations also involve gauge degrees of freedom. We shall give a clear proof that Hamiltonian formulations corresponding to different parametrizations of gravitational variables are related by canonical transformations in extended phase space embracing gauge degrees of freedom on an equal footing with physical ones. It will be demonstrated for the full gravitational theory in a wide enough class of parametrizations and gauge conditions.