Three-space from quantum mechanics

The spin geometry theorem of Penrose is extended from $SU(2)$ to $E(3)$ (Euclidean) invariant elementary quantum mechanical systems. Using the natural decomposition of the total angular momentum into its spin and orbital parts, the \emph{distance} between the centre-of-mass lines of the elementary subsystems of a classical composite system can be recovered from their \emph{relative orbital angular momenta} by $E(3)$-invariant classical observables. Motivated by this observation, an expression for the `empirical distance' between the elementary subsystems of a \emph{composite quantum mechanical system}, given in terms of $E(3)$-invariant quantum observables, is suggested. It is shown that, in the classical limit, this expression reproduces the \emph{a priori} Euclidean distance between the subsystems, though at the quantum level it has a discrete character. `Empirical' angles and 3-volume elements are also considered.