Flat space/spacetime from Quantum Mechanics

In 1966, R. Penrose suggested the so-called spin network model to illustrate how the structure of quantum spacetime could look like, and, using combinatorial methods, he showed how the conformal structure of the Euclidean 3-space can be recovered from this model in the classical, large spin limit [1].

In the present talk, after recalling the original ideas, we report on two extensions of this result [2-4]: how the metric structure of the Euclidean 3-space and of the flat Minkowski space can be recovered in the classical limit from quantum mechanics using only the observables of abstract (algebraic form of) quantum mechanics.

[1] R. Penrose, Combinatorial quantum theory and quantized directions, in {\it Advances in Twistor Theory}, Eds. L. P. Houghston, R. S. Ward,
    Pitman Publishing Ltd, London 1979
[2] L. B. Szabados, A note on Penrose's Spin-Geometry Theorem and the geometry of 'empirical quantum angles', Found. Phys. vol. 52, 96 (2022),
    arXiv: 2112.14538 [gr-qc] 
[3] L. B. Szabados, Three-space from quantum mechanics, Found. Phys. vol 52, 102 (2022), arXiv: 2203.04827 [quant-ph] 
[4] L. B. Szabados, Minkowski space from quantum mechanics, Found. Phys. vol. 54, 24 (2024), arXiv: 2309.06150 [quant-ph]