Cartan Connection for Schr\"odinger equation. The nature of vacuum
Abstract
We reinterpret the Schr\"odinger equation as a continuity equation in the space with the Cartan connection given by scaling Lie-B\"acklund group on a specific jet space. In this space, the wave function and their gradient coordinates are treated as independent coordinates. This approach gives a full Cartan connection form a divergence-free condition. Once constructed, the connection makes it possible to investigate the geometry of the space on which this Schr\"odinger-Cartan connection is constructed. This is the idea that generalizes the concepts present in de Broglie-Bohm (pilot wave) theory in a geometric way. We also present this procedure for constructing (non-uniquely) torsion-free Cartan connections for general Partial Differential Equations.