  The first mathematically consistent exact equations of quantum gravity in the
Heisenberg representation and Hamilton gauge are obtained. It is shown that the
path integral over the canonical variables in the Hamilton gauge is
mathematically equivalent to the operator equations of quantum theory of
gravity with canonical rules of quantization of the gravitational and ghost
fields. In its operator formulation, the theory can be used to calculate the
graviton S-matrix as well as to describe the quantum evolution of macroscopic
system of gravitons in the non-stationary Universe or in the vicinity of
relativistic objects. In the S-matrix case, the standard results are obtained.
For problems of the second type, the original Heisenberg equations of quantum
gravity are converted to a self-consistent system of equations for the metric
of the macroscopic spacetime and Heisenberg operators of quantum fields. It is
shown that conditions of the compatibility and internal consistency of this
system of equations are performed without restrictions on the amplitude and
wavelength of gravitons and ghosts. The status of ghost fields in the various
formulations of quantum theory of gravity is discussed.
