  In the earlier works on quantum geometrodynamics in extended phase space it
has been argued that a wave function of the Universe should satisfy a
Schrodinger equation. Its form, as well as a measure in Schrodinger scalar
product, depends on a gauge condition (a chosen reference frame). It is known
that the geometry of an appropriate Hilbert space is determined by introducing
the scalar product, so the Hilbert space structure turns out to be in a large
degree depending on a chosen gauge condition. In the present work we analyse
this issue from the viewpoint of the path integral approach. We consider how
the gauge condition changes as a result of gauge transformations. In this
respect, three kinds of gauge transformations can be singled out: Firstly,
there are residual gauge transformations, which do not change the gauge
condition. The second kind is the transformations whose parameters can be
related by homotopy. Then the change of gauge condition could be described by
smoothly changing function. In particular, in this context time dependent
gauges could be discussed. We also suggest that this kind of gauge
transformations leads to a smooth changing of solutions to the Schrodinger
equation. The third kind of the transformations includes those whose parameters
belong to different homotopy classes. They are of the most interest from the
viewpoint of changing the Hilbert space structure. In this case the gauge
condition and the very form of the Schrodinger equation would change in
discrete steps when we pass from a spacetime region with one gauge condition to
another region with another gauge condition. In conclusion we discuss the
relation between quantum gravity and fundamental problems of ordinary quantum
mechanics.
