Curved spacetimes from quantum mechanics

TL;DR

This paper demonstrates how the local geometry of curved Lorentzian 4-manifolds can be derived from the observables of quantum mechanical systems in the classical limit, linking quantum observables to spacetime metrics.

Contribution

It extends Penrose's Spin Geometry Theorem by showing how classical spacetime geometry emerges from quantum observables in algebraic quantum mechanics.

Findings

Classical spacetime geometry can be reconstructed from quantum observables.

Distance measurements in quantum states asymptotically match spacetime geodesics.

The metric tensor is recoverable from quantum system observables up to third order corrections.

Abstract

The ultimate extension of Penrose's Spin Geometry Theorem is given. It is shown how the \emph{local} geometry of any \emph{curved} Lorentzian 4-manifold (with $C^2$ metric) can be derived in the classical limit using only the observables in the algebraic formulation of abstract Poincar\'e-invariant elementary quantum mechanical systems. In particular, for any point $q$ of the classical spacetime manifold and curvature tensor there, there exists a composite system built from finitely many Poincar\'e-invariant elementary quantum mechanical systems and a sequence of its states, defining the classical limit, such that, in this limit, the value of the distance observables in these states tends with asymptotically vanishing uncertainty to lengths of spacelike geodesic segments in a convex normal neighbourhood $U$ of $q$ that determine the components of the curvature tensor at $q$. Since the curvature at $q$ determines the metric on $U$ up to third order corrections, the metric structure of curved $C^2$ Lorentzian 4-manifolds is recovered from (or, alternatively, can be \emph{defined} by the observables of) abstract Poincar\'e-invariant quantum mechanical systems.