# On the geometrical hypotheses underlying wave functions and their emerging dynamics

## TL;DR

This paper presents a geometric framework unifying classical and quantum mechanics through maps between manifolds, revealing the underlying geometry of wave functions and deriving the Schrödinger equation from these principles.

## Contribution

It introduces a novel geometric approach that generalizes mechanics using maps between manifolds, linking quantum non-locality to the bi-dimensionality of a target space.

## Key findings

- Derivation of Schrödinger equation from geometric principles
- Identification of quantum non-locality with bi-dimensional target space
- Unified geometric framework for classical and quantum mechanics

## Abstract

Classical mechanics for individual physical systems and quantum mechanics of non-relativistic 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 2-dimensional and flat, and their orthogonal directions are physically interpreted. All terms in the map equation have a geometrical meaning in the target space, and the pull-back of its rotational Killing one-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 Schr\"odinger equation. We link the bi-dimensionality of the target space with the essential non-locality of quantum mechanics. Many extensions of the framework here presented are immediate, with deep consequences yet to be explored.