The Schr\"odinger Equation Describes a Particular Quantum Geometry

TL;DR

This paper proposes a novel interpretation of quantum mechanics as quantum geometry by changing variables to relate the Schrödinger equation to a geometric framework, reproducing known atomic spectra and linking to gravity.

Contribution

It introduces a variable change that interprets quantum mechanics as quantum geometry, connecting the Schrödinger equation with a geometric metric and relating the coupling constant to Newton's gravitational constant.

Findings

Reproduces hydrogen atom energy spectrum using quantum geometry

Introduces a coupling constant related to Newton's gravitational constant

Suggests massive objects couple to space-time geometry, massless ones do not

Abstract

This paper posits the existence of, and finds a candidate for, a variable change that allows quantum mechanics to be interpreted as quantum geometry. The Bohr model of the Hydrogen atom is thought of in terms of an indeterministic electron position and a deterministic metric and the motivation for this paper is to try to change variables to have a deterministic position and momentum for the electron and nucleus but with an indeterministic (quantum) metric that reproduces the physics of the Bohr model. This mapping is achieved by allowing the metric in the Hamiltonian to be different to the metric in the space-time distance element and then representing the two metrics with vierbeins and assuming they are canonically conjugate variables. Effectively, the usual Schr\"odinger space-time variables have been re-interpreted as four of the potentially sixteen parameters of the metric tensor vierbein in the distance element while the metric tensor vierbein in the Hamiltonian is an operator expressible as first-order derivatives in these variables or vice versa. I then argue that this reproduces observed quantum physics at the sub-atomic level by demonstrating the energy spectrum of electron orbitals is exactly the same as the usual relativistic Bohr model for the Hydrogen atom in a certain limit. Next, by introducing a single dimensionless running coupling that shows up in the analogous place as, but in addition to, Planck's constant in the commutator definition I argue that this allows massive objects to couple to the physical space-time geometry but not massless ones - no matter coupling value. This claim is based on a fit to the Schwarzschild metric with a few simple assumptions and thus obtaining an effective theory of how the quantum geometries at nearby space-time points couple to one another. This demonstrates that this coupling constant is related to Newton's gravitational constant.