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

TL;DR

This paper explores how exceptional and non-crystallographic root systems can be used to construct new quantum configurations related to the Kochen-Specker theorem, providing new examples and interpretations in various dimensions.

Contribution

It demonstrates the application of irreducible root systems of exceptional and non-crystallographic types in constructing Kochen-Specker configurations, including new examples for $E_6$ and $E_7$.

Findings

New Kochen-Specker configurations from $E_6$ and $E_7$

Reinterpretation of known configurations from $F_4$, $E_8$, and $H_4$

Configurations are saturated with uniform maximal elements

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).