The Birth of E8E_8 out of the Spinors of the Icosahedron

$E_8$ is prominent in mathematics and theoretical physics, and is generally viewed as an exceptional symmetry in an eight-dimensional space very different from the space we inhabit; for instance the Lie group $E_8$ features heavily in ten-dimensional superstring theory. Contrary to that point of view, here we show that the $E_8$ root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of three-dimensional geometry. The $240$ roots of $E_8$ arise in the 8D Clifford algebra of 3D space as a double cover of the $120$ elements of the icosahedral group, generated by the root system $H_3$. As a by-product, by restricting to even products of root vectors (spinors) in the 4D even subalgebra of the Clifford algebra, one can show that each 3D root system induces a root system in 4D, which turn out to also be exactly the exceptional 4D root systems. The spinorial point of view explains their existence as well as their unusual automorphism groups. This spinorial approach thus in fact allows one to construct all exceptional root systems within the geometry of three dimensions, which opens up a novel interpretation of these phenomena in terms of spinorial geometry.Comment: 14 pages, 2 figures, 1 tabl

Clifford quantum computer and the Mathieu groups

One learned from Gottesman-Knill theorem that the Clifford model of quantum computing \cite{Clark07} may be generated from a few quantum gates, the Hadamard, Phase and Controlled-Z gates, and efficiently simulated on a classical computer. We employ the group theoretical package GAP\cite{GAP} for simulating the two qubit Clifford group $\mathcal{C}_2$. We already found that the symmetric group S(6), aka the automorphism group of the generalized quadrangle W(2), controls the geometry of the two-qubit Pauli graph \cite{Pauligraphs}. Now we find that the {\it inner} group ${Inn}(\mathcal{C}_2)=\mathcal{C}_2/{Center}(\mathcal{C}_2)$ exactly contains two normal subgroups, one isomorphic to $\mathcal{Z}_2^{\times 4}$ (of order 16), and the second isomorphic to the parent $A'(6)$ (of order 5760) of the alternating group A(6). The group $A'(6)$ stabilizes an {\it hexad} in the Steiner system $S(3,6,22)$ attached to the Mathieu group M(22). Both groups A(6) and $A'(6)$ have an {\it outer} automorphism group $\mathcal{Z}_2\times \mathcal{Z}_2$, a feature we associate to two-qubit quantum entanglement.Comment: version for the journal Entrop

Unification of Gravity and Yang-Mills-Higgs Gauge Theories

In this letter we show how the action functional of the standard model and of gravity can be derived from a specific Dirac operator. Far from being exotic this particular Dirac operator turns out to be structurally determined by the Yukawa coupling term. The main feature of our approach is that it naturally unifies the action of the standard model with gravity.Comment: 8 pages, late