Intrinsic Dynamics of Manifolds: Quantum Paths, Holonomy, and Trajectory Localization

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.