Geometrical hypotheses underlying wave functions and their emerging dynamics

Classical mechanics for individual physical systems and quantum mechanics of nonrelativistic particles are shown to be exceptional cases of a generalized dynamics described in terms of maps between two manifolds, the source being configuration space. The target space is argued to be two dimensional and flat, and its orthogonal directions are physically interpreted. All terms in the map equation have a geometrical meaning in the target space, and the pullback of its rotational Killing 1-form allows the construction of a velocity field in configuration space. Identification of this velocity field with tangent vectors in the source space leads to the dynamical law of motion. For a specific choice of an arbitrary scalar function present in the map equation, and using Cartesian coordinates in the target space, the map equation becomes linear and can be reduced to the Schrodinger equation. We link the bidimensionality of the target space with the essential nonlocality of quantum mechanics. Many extensions of the framework presented here are immediate, with deep consequences yet to be explored.