  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 Pauli--Lubanski spin vectors 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 systems.

Quantum Physics;General Relativity and Quantum Cosmology