Quantum Transformations

TL;DR

This paper demonstrates that the stationary quantum Hamilton-Jacobi equation can be reformulated as a classical-like equation with a geometric interpretation, linking quantum potential to space curvature and supporting the equivalence principle.

Contribution

It introduces quantum transformations that relate quantum and classical potentials through projective geometry, providing a geometric perspective on quantum mechanics.

Findings

Quantum Hamilton-Jacobi equation resembles classical form with geometric modifications.

Quantum potential is proportional to a curvature in projective geometry.

Supports the equivalence principle by linking quantum effects to space deformation.

Abstract

We show that the stationary quantum Hamilton-Jacobi equation of non-relativistic 1D systems, underlying Bohmian mechanics, takes the classical form with $\partial_q$ replaced by $\partial_{\hat q}$ where $d\hat q={dq\over \sqrt{1-\beta^2}}$. The $\beta^2$ term essentially coincides with the quantum potential that, like $V-E$, turns out to be proportional to a curvature arising in projective geometry. In agreement with the recently formulated equivalence principle, these ``quantum transformations'' indicate that the classical and quantum potentials deform space geometry.