Three-space from quantum mechanics

• URL: http://arxiv.org/abs/2203.04827v4
• Date: Tue, 6 Sep 2022 21:22:51 GMT
• Title: Three-space from quantum mechanics
• Authors: L\'aszl\'o B. Szabados
• Abstract summary: The spin geometry of Penrose is extended from $SU(2)$ to $E(3)$ (Euclidean) invariant elementary quantum mechanical systems. The emphdistance between the centre-of-mass lines of the elementary subsystems of a classical composite system can be recovered from their emphrelative orbital angular momenta by $E(3)$-invariant classical observables.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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