  Our attempts to find an explanation for quantum behavior of the Early
Universe appeal, as a rule, to the Wheeler - DeWitt Quantum Geometrodynamics
which relies upon Hamiltonian formulation of General Relativity proposed by
Arnowitt, Deser and Misner (ADM). In spite of the fact that the basic ideas of
this approach were put forward about fifty years ago, even now we do not have
clear understanding what Hamiltonian formulation of General Relativity must be.
An evidence for it gives a recent paper by Kiriushcheva and Kuzmin
[arXiv:0809.0097], where the authors claim that the formulation by ADM and that
by Dirac made in his seminal work of 1958 are not equivalent. If so, we face
the question what formalism should be chosen. Another problem is that we need a
well-grounded procedure of constructing a generator of transformations in phase
space for all gravitational variables including gauge ones. It suggests the
notion of extended phase space. After analyzing the situation, we show that
Hamiltonian formulation in extended phase space is a real alternative to Dirac
and ADM formulations and can be constructed to be equivalent to the original
(Lagrangian) formulation of General Relativity. Quantization in extended phase
space is straightforward and leads to a new description of quantum Universe in
which an essential place is given to gauge degrees of freedom.
