  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.
