  The physical phase space in gauge systems is studied. Effects caused by a
non-Euclidean geometry of the physical phase space in quantum gauge models are
described in the operator and path integral formalisms. The projection on the
Dirac gauge invariant states is used to derive a necessary modification of the
Hamiltonian path integral in gauge theories of the Yang-Mills type with
fermions that takes into account the non-Euclidean geometry of the physical
phase space. The new path integral is applied to resolve the Gribov
obstruction. Applications to the Kogut-Susskind lattice gauge theory are given.
The basic ideas are illustrated with examples accessible for non-specialists.
