Minkowski space from quantum mechanics

TL;DR

This paper demonstrates how the metric structure of Minkowski space can be derived from quantum mechanical observables in the classical limit, extending Penrose's Spin Geometry Theorem to Poincaré-invariant systems.

Contribution

It extends Penrose's Spin Geometry Theorem to Poincaré-invariant quantum systems, showing Minkowski space geometry emerges from quantum observables in the classical limit.

Findings

Lorentzian distance expressed via quantum observables

Classical Minkowski metric recovered from quantum systems

Quantum-to-classical transition reproduces spacetime geometry

Abstract

Penrose's Spin Geometry Theorem is extended further, from $SU(2)$ and $E(3)$ (Euclidean) to $E(1,3)$ (Poincar\'e) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike straight lines of Minkowski space, considered to be the centre-of-mass world lines of $E(1,3)$-invariant elementary classical mechanical systems with positive rest mass, is expressed in terms of \emph{$E(1,3)$-invariant basic observables}, viz. the 4-momentum and the angular momentum of the systems. An analogous expression for \emph{$E(1,3)$-invariant elementary quantum mechanical systems} in terms of the \emph{basic quantum observables} in an abstract, algebraic formulation of quantum mechanics is given, and it is shown that, in the classical limit, it reproduces the Lorentzian spatial distance between the timelike straight lines of Minkowski space with asymptotically vanishing uncertainty. Thus, the \emph{metric structure} of Minkowski space can be recovered from quantum mechanics in the classical limit using only the observables of abstract quantum mechanical systems.