Three-space from quantum mechanics

TL;DR

This paper extends Penrose's spin geometry theorem to Euclidean invariant quantum systems, proposing a quantum definition of distance and angles that aligns with classical Euclidean geometry in the classical limit.

Contribution

It generalizes the spin geometry theorem to E(3) invariance and introduces quantum observables for empirical distances and angles between subsystems.

Findings

Quantum distance reduces to Euclidean distance classically.

Discrete quantum character of the empirical distance.

Proposed quantum observables for angles and volume elements.

Abstract

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.