M. Pavsic: A novel view on the physical origin of E_8


A novel view on the physical origin of E_8

by Matej Pavšiˇc,
Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
E-mail: matej.pavsic@ijs.si
J. Phys. A: Math. Theor. 41 (2008 ) 332001 (10pp)
doi:10.1088/1751-8113/41/33/332001
Received 20 May 2008, in final form 26 June 2008
Published 11 July 2008
Online at stacks.iop.org/JPhysA/41/332001

Abstract
We consider a straightforward extension of the four-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 ≡ x^Aγ_A ≡ x^{a1…ar} γ_{a1…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^A or (ii) by the left and right action of the Clifford numbers R = exp[α^Aγ_A] and S = exp[β^Aγ_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 Cl(1, 3) with a vector space V_{8,8} whose basis elements are generators of the Clifford algebra Cl(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.

Source: M. Pavsic (email of 15 July 2008), http://stacks.iop.org/1751-8121/41/332001

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