Generally covariant quantum mechanics

TL;DR

This paper develops a covariant quantum mechanics framework using operator-valued geodesic equations on curved spacetime, linking quantum geodesics with differential calculus on the algebra of differential operators, and explores applications to black hole bound states.

Contribution

It introduces a covariant operator-valued geodesic formalism on pseudo-Riemannian manifolds and applies it to describe quantum states around black holes, connecting geometry with quantum dynamics.

Findings

Derived covariant quantum geodesic equations on curved spacetime.

Formulated a Schrödinger picture with Klein-Gordon evolution.

Identified gravitationally bound states resembling atomic orbitals near black holes.

Abstract

We obtain generally covariant operator-valued geodesic equations on a pseudo-Riemannian manifold $M$ as part of the construction of quantum geodesics on the algebra $D(M)$ of differential operators. Geodesic motion arises here as an associativity condition for a certain form of first order differential calculus on this algebra in the presence of curvature. The corresponding Schr\"odinger picture has wave functions on spacetime and proper time evolution by the Klein-Gordon operator, with stationary modes being solutions of the Klein-Gordon equation. As an application, we describe gravatom solutions of the Klein-Gordon equations around a Schwarzschild black hole, i.e. gravitationally bound states which far from the event horizon resemble atomic states with the black hole in the role of the nucleus. The spatial eigenfunctions exhibit probability density banding as for higher orbital modes of an ordinary atom, but of a fractal nature approaching the horizon.