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Chapitre D'ouvrage Année : 2022

From Electrons to Elephants and Elections

Résumé

In quantum mechanics, the measurement of an observable does not reveal a pre-existing value/reality but depends on the measurement contexts: the geometry of mutually commuting observables. This has led to many ‘proofs’ that the algebra of observables and of eigenvalues is counter intuitive compared to the Boolean logic. Paraphrasing Niels Bohr: The world “reality” is also a word, a word which we must learn to use correctly. In this essay, words built from a two-generator free group G are used to investigate contextuality in quantum mechanics. Our language has two important features related to geometry and topology. Group theoretical commutators correspond to quantum commutation of observables and a ‘geometric contextuality’ corresponds to quantum contextuality. Then, selecting the free group G as the fundamental group π1 of an appropriate 4-manifold M – a 4-dimensional exotic space-time–, quantum measurements in a d-dimensional Hilbert space can be seen as taking place on homeomorphic but not diffeomorphic manifolds Md corresponding to subgroups of finite index d of π1. We use the word ‘exotic contextuality’ to feature the geometries with non Boolean logic occuring from those manifolds. Some Brieskorn 3-manifolds M = Σ(p, q, r) (with p, q and r mutually coprimes) play a special role in the aforementioned phenomenon of ‘exotic contextuality’.
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Dates et versions

hal-03813033 , version 1 (21-10-2022)

Identifiants

  • HAL Id : hal-03813033 , version 1

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Michel Planat. From Electrons to Elephants and Elections. The Frontiers Collection, Springer, pp.469 - 488, 2022. ⟨hal-03813033⟩
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