  We consider an ``integral'' extension of the classical notion of affine
connection providing a correspondence between paths in the manifold and
diffeomorphisms of the manifold. These path-diffeomorphisms are a
generalization of parallel translations along paths via the connection. In this
way, one can translate nonanalytic functions and distributions rather than
tangent vectors. We describe the integral holonomy and the integral curvature.
  On the symplectic or quantum level this construction makes up the symplectic
or quantum paths, as well quantum connection, quantum curvature and quantum
holonomy.
  The construction of path-diffeomorphisms, being applied to trajectories of a
given dynamical system produces a transformation of the system to a new one
which has an a priori chosen equilibrium point. In the symplectic (quantum)
case, this dynamic localization provides a coherent-type representation of the
quantum flow.
