Generally covariant quantum mechanics

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 $${\mathcal {D}}(M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 Schrödinger 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.