I would like to add some comments to Tony Smith's interesting essay.
1-
In page 4, in the second line, a notation less likely to be
misunderstood would have been something like
120 = 28 + 28 + ( 8 x 8) = 28 + 28 + 64
2-
It is known that the E_8 algebra admits the Larsson
7- grading GL ( 8, R ) decomposition of E_8 :
8 + 28 + 56 + 64 + 56 + 28 + 8
with E_8 grades
-3, -2, -1, 0, +1, +2, +3
However, as Smith argues, the E_8 graded structure does not look exactly like the simplex-polytope
decompostion graded structure corresponding to the 256-dimensional
Cl (8) Clifford algebra with Clifford graded structure
1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1
with Clifford grades
0, 1, 2, 3, 4, 5, 6, 7, 8
which looks like the decomposition of a simplex-polytope in 7-dim space
which has 2^(7+1) = 2^8 = 256 elements (edges, faces, etc).
The 256 - 248 = 8 things missing from E_8 itself are:
the 1 element of Clifford grade 0
6 elements of the 70 of Clifford grade 4
the 1 element of Clifford grade 8.
Therefore, the most natural embedding of E_8 is not inside Cl (8),
but inside Cl(16) which has 120-dim SO (16) bivectors and 128-dim SO ( 16)
chiral-spinors
that combine to make E_8.
However, since real 8-periodicity gives Cl(16) = Cl(8) x Cl(8) (tensor product),
one can use the Cl(16) embedding to write E_8 as a *subalgebra* of the
tensor product of two copies Cl(8) x Cl ( 8).
The 64 in the zero grade part is the dim of the GL(8,R )
algebra with 8 x 8 = 64 generators. Despite that Gl ( 8, R ) group is not compact,
from the point of view of compact Lie groups the
64-dimensional Lie algebra U ( 8 ) can naturally be embedded in Spin(16). In general,
U ( n ) can be embedded in SO ( 2n ) and admits a realization in terms of the
Clifford ( 2n ) algebra generators.
The GL(8,R) algebra is used to construct Metric Affine theories of Gravity in an 8-dimensional
spacetime and the 64 root vectors of E_8 are shown in blue in Tony Smith's images.
The other even-grade parts of E_8 are the 28 of grade -2 and the 28 of
grade +2, which correspond to two D4 Lie algebras with 24 root
vectors each,
and they are the yellow (24 dark and 24 bright) root vectors in the
images of Tony Smith's essay.
The odd-grade parts are 8 + 56 of grades -3 and -1
and 56 + 8 of grades +1 and +3
and they correspond to the 64 red and 64 green fermionic root vectors
shown in those images.
3-
Others prefer to look at 248 as
28 + 28 + 3 x ( 8 x 8 ) = 28 + 28 + 3 x (64) = 248 =
SO(8) + SO(8) + 3 x (O x O)
where O = Octonions.
28 + 28 = 28 bivectors in D = 8 plus their 28 "mirrors"
dual momentum conjugates = 56 in total.
Tony Smith sees the bivectors 28 = 16 + 12, as
16 corresponding to the dimensions of the algebra
U ( 2, 2 ) = SU ( 2, 2 ) x U ( 1 ) associated to the
Macdowell-Mansouri-Chamseddine-West Conformal Gravity approach to 4D gravity.
And, 12 corresponding to the dimensions of SU ( 3 ) x SU ( 2 ) x U ( 1) =
8 gluons + 3 Weak bosons + 1 photon of the Standard Model.
4-
One of the reasons why Supersymmetric E_8 Grand Unification is very appealing is because supersymmetric gauge theories ( "superconnections")
allows one to work with bosons and fermions at once.
So the leptons and quarks associated to the SUSY E_8 theory are the *gluinos*, the super-partners, of the E_8 gauge bosons because the adjoint and fundamental 248-dim representations of E_8 happens to coincide in this "exceptional" case.
Usually when one works with a bosonic E_8 gauge theory, the fermions, and the scalar matter fields, are assigned to the sections of a spinor (vector) bundle.
Despite this technical subtlety, why supersymmetry is important,
it is true that the Octonions O in the factor
3 x (8 x 8) = 3 x (O x O)
permits to account for 3 copies = 3 fermion generations as follows :
The electron, neutrino, up and down quark (
with 3 colors each ) gives 8 different degrees of freedom of the Octet
= 1 + 1 + 3 + 3 = 8.
When one multiplies 8 times the number of
spinorial components of a Weyl spinor ( a chiral spinor) in D = 8,
given by ( 1/2 ) 2^{ 4 } = 8, one then gets 8 x 8 = 64 = the total number of spinorial degrees of freedom of the first generation Octet. Thus the 3 generations yields a net factor of 3 x 64 which is appealing if one works in
D = 8.
5-
As I've argued in one of the sections of my article on the E_8 Geometry of Clifford (16) Superspace Conformal Gravity and Yang-Mills Grand Unification
http://www.scribd.com/doc/3870934/E8-Geometry-of-
Clifford-16-Superspace-GravityYangMills-GrandUnification
One of the noncompact forms of E_8 contains SO ( 8, 2 ) as a subgroup,
and such that a E_8 gauge theory in D = 8
contains the SO ( 8, 2 ) Conformal gauge theory of Gravity in D = 8,
and which upon compactification ( from 8 to 4 dim ) on a CP^2 manifold
as shown by Batakis, with Torsion, furnishes Conformal gravity and the Standard Model in D = 4.
CP^2 = SU ( 3 ) / U ( 2 ) = SU ( 3 ) / SU ( 2 ) x U ( 1)
The CP^2 has SU (3), SU (2), U (1) built in. Thus, D = 8 is essential.
6-
Tony Smith agrees that his way of getting 3 generations does NOT assign them all in the fundamental E_8, but the second and third generations
emerge as composites from the process of converting octonionic 8-dim
spacetime into Kaluza-Klein 4+4 spacetime with 4-dim physical
spacetime and 4-dim CP^2 internal symmetry space.
This model differs from the SO(10) GUT models, emerging from the E_8 GUT models of the late 70's and early 80's, with four and three generations (plus their mirror fermions) with a massive neutrino,
in that Smith's first-generation, the fundamental neutrino remains massless (the second and third generation neutrinos get small mass by processes beyond tree level) and has no right-handed component.
7-
Finally, pertaining tp the Murray-von Neumann hyperfinite type factor II_1, defined as the complex Clifford algebra of an infinite dim Euclidean space,
it is very reasonable to argue that an infinite d
im Clifford algebra can be written as an infinite tensor product of Cl ( 8 ) factors ( 16 x 16 matrices ) or as an infinite tensor product of
2 x 2 complex matrices, like the Pauli spin matrices,
furnishing an an infinite dim Fermionic Fock space.
Carlos Castro Perelman, September 7, 2007
