A Novel View on the Physical Origin of E8

Matej Pavsic

doi:10.1088/1751-8113/41/33/332001

paper

A Novel View on the Physical Origin of E8

arXiv:0806.4365 · doi:10.1088/1751-8113/41/33/332001

Abstract

We consider a straightforward extension of the 4-dimensional spacetime $M_{4}$ to the space of extended events associated with strings/branes, corresponding to points, lines, areas, 3-volumes, and 4-volumes in $M_{4}$ . All those objects can be elegantly represented by the Clifford numbers $X \equiv x^{A} γ_{A} \equiv x^{a_{1} ... a_{r}} γ_{a_{1} ... a_{r}}, r = 0, 1, 2, 3, 4$ . This leads to the concept of the so-called Clifford space $C$ , a 16-dimensional manifold whose tangent space at every point is the Clifford algebra $C ℓ (1, 3)$ . The latter space besides an algebra is also a vector space whose elements can be rotated into each other in two ways: (i) either by the action of the rotation matrices of SO(8,8) on the components $x^{A}$ or (ii) by the left and right action of the Clifford numbers $R =$ exp$ [α^A \gam_A]$ and $S =$ exp$ [β^A \gam_A]$ on $X$ . In the latter case, one does not recover all possible rotations of the group SO(8,8). This discrepancy between the transformations (i) and (ii) suggests that one should replace the tangent space $C ℓ (1, 3)$ with a vector space $V_{8, 8}$ whose basis elements are generators of the Clifford algebra $C ℓ (8, 8)$ , which contains the Lie algebra of the exceptional group E $_{8}$ as a subspace. E $_{8}$ thus arises from the fact that, just as in the spacetime $M_{4}$ there are $r$ -volumes generated by the tangent vectors of the spacetime, there are $R$ -volumes, $R = 0, 1, 2, 3, ..., 16$ , in the Clifford space $C$ , generated by the tangent vectors of $C$ .

14 pages

A Novel View on the Physical Origin of E8

Abstract

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