A Novel View on the Physical Origin of E 8 Matej

We consider a straightforward extension of the 4-dimensional spacetime M4 to the space of extended events associated with strings/branes, corresponding to points, lines, areas, 3-volumes, and 4-volumes in M4. All those objects can be elegantly represented by the Clifford numbers X ≡ xγA ≡ x γa1...ar , 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 Cl(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 or (ii) by the left and right action of the Clifford numbers R =exp [αγA] and S =exp [β γA] 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 Cl(1, 3) with a vector space V8,8 whose basis elements are generators of the Clifford algebra Cl(8, 8), which contains the Lie algebra of the exceptional group E8 as a subspace. E8 thus arises from the fact that, just as in the spacetime M4 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.