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arXiv · 0906.2696

Exceptional and Non-crystallographic Root Systems and the Kochen-Specker Theorem

Artur Ruuge
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Abstract

The Kochen-Specker theorem states that a 3-dimensional complex Euclidean space admits a finite configuration of projective lines such that the corresponding quantum observables (the orthogonal projectors) cannot be assigned with 0 and 1 values in a classically consistent way. This paper shows that the irreducible root systems of exceptional and of non-crystallographic types are useful in constructing such configurations in other dimensions. The cases $E_6$ and $E_7$ lead to new examples, while $F_4$, $E_8$, and $H_4$, yield a new interpretation of the known ones. The described configurations have an additional property: they are saturated, i.e. the tuples of mutually orthogonal lines, being partially ordered by inclusion, yield a poset with all maximal elements having the same cardinality (the dimension of space).

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Artur Ruuge. 2009-06-15. Exceptional and Non-crystallographic Root Systems and the Kochen-Specker Theorem. https://doi.org/10.1088/1751-8113/40/11/019

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