The elementary particles and the root system E8

Abstract

E8 is a famous root system in mathematics. Lisi claims in his paper “An Exceptionally Simple Theory of Everything” that there is a connection between the standard model of the elementary particles and this root system. He claims that standard model has the same structure as the root system E8. If this is true, there must be a mapping from the standard model to E8. This map must identify each elementary particle with an unique root of E8. Also the four charges (weak isospin, weak hypercharge, g3 and g8) of the standard model must have corresponding vectors in R8. These four charge vectors must have the property that taking the inner product of the charge vector and the root identified withan elementary particle, it would result in the charge of that particle. This is needed to guarantee that we have conservation of charge. In this paper we have shown that it is not possible to find a map with four charge vectors. This was done by studying root systems and how different root systems are present in E8 as sub root systems. We also proved that the intersection of E8 with the orthogonal complement of a charge, must be a root system. This limits the possibilities for the charge vectors. For these possible charges it has been checked whether there are enough roots for a given charge to identify the elementary particles. Since this was not the case, we concluded that no mapping with four charge vectors exists.