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

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

2009-06-15

arxiv.org/abs/0906.2696v1

J. Phys. A 40 (2007), no. 11, 2849--2859

Physics

Quantum Physics

Scientific paper

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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