Here are some excerpts from some e-mail messages I sent
to my friend Carlos Castro discussing my VoDou Physics Model in
September 2004:
With respect to Cl(2,6), you say
"... now I have a way to connect my work with yours ...".
The way I see that connection is described on my web page at
... clfpq2.html#physicspaths
where I say that
"... dimensional reduction
changes Real M(16,R) of Cl(0,8), Cl(1,7), and Cl(4,4)
to the Quaternionic M(8,Q) of Cl(2,6) or Cl(3,5) ...".
In other words,
in my view,
at high energies where spacetime is really 8-dimensional
the relevant Clifford algebra is
Cl(1,7) = Cl(0,8) = the 16x16 real matrix algebra
which has real/octonionic structure
but
when you to lower energies (where our experiments are done)
the cooling forces a quaternionic structure to freeze out
(or, in other language, real/octonionic symmetry is broken
by choosing a particular quaternionic substructure)
and
the result is that at our lower energies things look
like Cl(2,6) = the 8x8 quaternion matrix algebra,
which
has a Cl(2,4)= the 4x4 quaternion matrix algebra =
= the conformal Lie algebra for gravity + Higgs
and
if you factor the 15-dim Spin(2,4) conformal Lie algebra
out of the 28-dim Spin(2,6) Lie algebra of high-energy gauge bosons
you see
that the remaining 13 generators correspond to:
1 is a U(1) to be added to Spin(2,4) to get U(2,2) = U(1)xSU(2,2),
and physically represents propagator complex phases;
1 is an electromagnetic U(1);
3 are weak SU(2); and
8 are color SU(3).
The way those generators form those groups is based on
the Weyl group root vector symmetries, and not as
conventional group/subgroup structures. One reason
that conventional physicists/mathematicians have a hard time
understanding what I have done is that such Weyl/root vector
stuff is not found in existing textbooks or papers.
A description is on my web page at
... Sets2Quarks4a.html#WEYLdimredGB
I should also note that the quaternionic structure of Cl(2,4) is
inherited by its subalgebra Cl(1,3) = 2x2 quaternionic matrices
which is consistent with the physical interpretation of
fermions in terms of quaternionic half-spinors and with
John Baez's statement
"... fermions are quaternionic and bosons are real ..."
on which I have elaborated on my web page at
... clfpq2.html#13vs31
Also,
your comment that: "... Machian dream ...express[ing] the
maximal force = F = m( Planck ) c^2 / L(Planck)
as F = M ( Universe ) c^2 / R ( Hubble )
then we get one of the Dirac-Eddington's large number
coincidences :
M ( Universe ) / m ( Planck ) =
R ( Hubble ) / L ( Planck ) = of the order of 10^{ 61 } .
Since the Planck mass = 10^{ 19 } m ( proton )
you get the famous M ( Universe ) = 10^{ 80} m ( proton )
which is the Dirac-Eddington number. ..."
is
somewhat related to some comments that I put on my web page at
... LARGEsmall.html
with respect to infinite dimensional Clifford algebras,
my view is to
consider the conventional Hyperfinite II1 factor
which is roughly an infinite-dimensional version of
the spinor representation of Complex Clifford algebras,
which have periodicity 2 and so are like an infinite limit
of what John Baez calls "... the fermionic Fock space over C^(2n) ..."
and
then generalize it to the case of Real Clifford Algebras
with periodicity 8 so that what you get is an infinite
limit of a tensor product of a lot of copies of 256-dim Cl(8),
as I have described on my web pages, including
... II1vNfactor.html
Each Cl(8) would describe physics locally in the neighborhood
of a given spacetime point, as described in my physics model
which is summarized in papers and web pages such as
... 2002SESAPS.html
All the Cl(8) things in the generalized Hyperfinite II1 factor
(roughly an infinite tensor product) would be linked together
to form (at the next higher energy level above our quaternionic
4-dim physical spacetime plus 4-dim CP2 internal symmetry space)
a higher-energy real/octonionic 8-dim spacetime
as described on my web pages, including
... ClifTensorGeom.html
When you take quantum superpositions in the many-worlds quantum theory,
you get things such as quantum loops/graphs of higher and higher order,
whose description involves the prime numbers
as described on my web pages, including
... Rzetazeta.html#qsohprime
and which may be closely related to the p-adic geometry
that you use, Matti. However, I have a hard time visualizing
p-adic stuff, so I have not written much about it.
I had been trying to read about it and understand it recently,
but I have become depressed and discouraged about doing further work.
I should note that my model also gives two significant sets of
calculations not given on that web page. They are:
1 - WMAP ratio calculation at
... coscongraviton.html
and
2 - neutrino mass and mixing angle calculations at
... snucalc.html#asno
... I see it as:
The 256-dimensional Cl ( 8 ) has the graded structure :
1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1
the 1-dim ( Higgs ) ,
the 8-dim ( spacetime that breaks to 4-dim physical spacetime plus CP2 )
and the 28-dim ( gravity plus gauge bosons )
and
it also has spinor structure in addition to graded structure.
At high energies before dimensional reduction
the Clifford algebra is Cl(0,8) = Cl(1,7) = 16x16 real matrices
and
the full spinor space is 8-dim which reduces to
8-dim half-spinor space for 8 fundamental fermion particles
plus
a mirror image 8-dim half-spinor space for 8 fundamental antiparticles.
After dimensional reduction
the Clifford algebra is Cl(2,6) = 8x8 quaternionic matrices
and
the full spinor space is 8x1 quaternionic left ideal
which reduces to
a half-spinor 4x1 quaternions which are 16-dim and
so correspond to 8 Dirac fermion particles with two helicities
(the neutrino right helicity is suppressed due to inherited
structure from real/octonionic Cl(1,7))
plus
a mirror image half-spinor 4x1 quaternions which are 16-dim and
so correspond to 8 Dirac fermion antiparticles with two helicities
(the antineutrino left elicity is suppressed due to inherited
structure from real/octonionic Cl(1,7)).
This structure is consistent with the
quaternionic structure of conformal Cl(2,4) = 4x4 quaternionic matrices
and with the
quaternionic structure of Cl(1,3) = 2x2 quaternionic matrices,
so
the 4-dim physical spacetime has signature (1,3)
and not signature (3,1) of Cl(3,1) = 4x4 real matrices.
--------------------------------------------
...[all Cl(8) multivectors]...
were not completely encoded [by spinors]
(and neither were grades 0 (Higgs) 1 (spacetime plus CP2)
or 2 (gauge bosons) )
because
the spinor fermions are only the left ideals,
sort of the square root of the 16x16 = 256,
which
leaves the right ideals free to represent the gammas
(this was the basic insight of David Hestenes with respect
to 4-dim Clifford algebra in his spacetime algebra program).
Of course, the gammas appear in the structure of the Lagrangian.
From my point of view, there are only 3 generations, as follows:
The 16 8-dim spinors correspond to
8 first-generation fermion particles
and
8 first-generation fermion antiparticles
and
they propagate in 8-dim spacetime,
where there is only one generation (the first).
The 8-dim propagation looks like this:
origin * ------ > destination lying in 8-dim spacetime.
However,
after dimensional reduction,
the 8-dim spacetime is no longer one layer
----------------------------------------------
but splits into two layers:
------------------------------------ 4-dim Internal Symmetry Space
------------------------------------ 4-dim physical spacetime
and
If the spinor propagator origin * and destination >
both lie in the 4-dim physical spacetime,
then the low-dim propagator looks like this:
origin * ------ > destination lying in 4-dim physical spacetime
and the spinor fermion particle (or antiparticle) looks like
a 4-dim first-generation fermion,
and is represented by a single octonion basis element
corresponding to the origin * .
However,
if the origin * lies in the 4-dim Internal Symmetry Space
then the low-dim propagator looks like this:
-- origin * ------------------- 4-dim Internal Symmetry Space
|
|
-- virtual origin (*) ------ > destination -- 4-dim physical spacetime
and the spinor fermion particle (or antiparticle) looks like
a 4-dim second-generation fermion,
and is represented by a pair of octonion basis elements
corresponding to the origin * plus
the virtual origin (*) .
and
if the > destination lies in the 4-dim Internal Symmetry Space
then the low-dim propagator looks like this:
--------- (>*) intermediate virtual orgin/destination -- 4-dim Internal
/ \ Symmetry Space
/ \
-- origin * > destination -------------------------- 4-dim physical
spacetime
and the spinor fermion particle (or antiparticle) looks like
a 4-dim second-generation fermion,
and is represented by a pair of octonion basis elements
corresponding to the origin * plus
the intermediate virtual orgin/destination (>*) .
The only other possible case is that both the
origin and the > destination lie in the 4-dim Internal Symmetry Space
in which case the low-dim propagator looks like this:
----------- origin * (>*) intermediate virtual orgin/destination -- 4-dim
| / \ Internal
| / \ Symmetry Space
-- virtual origin (*) > destination -------------------------- 4-dim
physical spacetime
and the spinor fermion particle (or antiparticle) looks like
a 4-dim third-generation fermion,
and is represented by a triple of octonion basis elements
corresponding to the origin * plus
the virtual origin (*) plus
the intermediate virtual orgin/destination (>*) .
What this means is that physically there can only be 3 generations,
and
the first generation is represented by octonion basis elements;
the second generation is represented by pairs of octonion basis elements;
and
the third generation is represented by triples of octonion basis elements.
The single - pair - triple relationship gives
combinatorial relationships among the masses of the various
generations of fermions,
which produces
constituent masses that are roughly consistent with experiment,
which
is a reason that I think that my point of view is correct.
It is (as far as I know) the only point of view that
gives roughly equal constituent masses in the first generation quarks,
but
in the second generation charm is somewhat heavier than strange
and
in the third generation truth is a lot heavier than beauty,
and
using these calculated masses lets me calculate
realistic Kobayashi-Maskawa parameters,
also something that nobody else has done AFAIK.
All this is in my web pages and even in my papers on arXiv
(they are probably there because they were put up prior to arXiv
blacklisting me or during the brief interval that arXiv allowed
its endorsement system to work without interference from the
blacklisting moderators).
Also,
there is experimental evidence that there are only 3 generations,
such as the Z width experiments.
According to a Particle Data Group Particle Properties review
http://pdg.lbl.gov/ (quote from the August 2001 revision by D. Karlen):
"... The most precise measurements of the number of light neutrino types,
Nv, come from studies of Z production in e+e- collisions.
The invisible partial width ... is determined by
subtracting the measured visible partial widths,
corresponding to Z decays into quarks and charged leptons,
from the total Z width.
The invisible width is assumed to be due to Nv light neutrino species
each contributing the neutrino partial width ...
as given by the Standard Model. ...
The combined result from the four LEP experiments
is Nv = 2.984 +/- 0.008 ...".
What I am doing is taking the Cl(8) Clifford algebra
and making a Lagrangian by using different parts
of Cl(8) in different ways:
grade 0 is a Higgs term;
grade 1 is a vector spacetime term (the manifold over
which the Lagrangian density is integrated);
grade 2 is a bivector gauge boson term (like a YM term);
+half-spinors are fermion particles; and
-half-spinors are fermion antiparticles.
As I understand it, you want to ask
WHY those parts should play those roles,
and
in order to do that you want to start with
a supergravity in a Clifford superspace
and see whether its various terms fall into the places
where I put them.
A way to look at superspace in terms of Clifford algebras
is described in a paper by Doran, Lasenby and Gull at
http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/grass_mech.html
in which they say (their term for "Clifford algebra" is "geometric algebra"):
"... Given a set of n Grassmann generators ... we can map these into
geometric algebra by introducing a set of n independent vectors ...
and replacing the product of Grassmann variables by the exterior product
... In this way any combination of Grassmann variables can be replaced
by a multivector ...
... Berezin calculus ... can be handled entirely within the algebra
... by introducing the reciprocal frame ...[and]... Integration is defined
to be equivalent to right differentiation ...
... Thus we see that Grassmann calculus amounts to no more than
Clifford contraction and the results of "Grassmann analysis" ...can all
be expressed as simple algebraic identities for multivectors ...
Furthermore these results are now given a firm geometric significance
through the identification of Clifford elements with directed line,
plane segments etc ...".
I guess you want to apply the Doran-Lasenby-Gull viewpoint
to high-dimensional supergravity. There is a paper cached in
html form at
http://64.233.161.104/search?q=cache:U7Cy3y87nb4J:www.cgtp.duke.edu/~drm/PCMI2001/notes.pdf+supergravity+geometric+algebra&hl=en&ie=UTF-8
by Savdeep Sethi that says:
"... The aim of this lecture is to overview the general structure
and features of supergravity ... Why 11 dimensions? ...
... restrict to D = 11 ... This representation, as in all these cases,
must have an equal number of bosons and fermions.
Each application ... changes a boson to a fermion, and vice-versa.
In this case, we get 128 bosons and 128 fermions. ...
... a gravitino ... is often called a spin 3/2 particle
(a vanilla spinor is typically called a spin 1/2 particle).
There are no fermions beyond the gravitino in 11 dimensions ...
...
For D > 11, if you impose SUSY, and follow the same construction,
you will find higher spin fields.
Massless fields with spin greater than 2 cannot
(in a way that we currently understand)
consistently be coupled to gravity.
There is a strong belief that such theories do not make sense.
In this respect, 11-dimensional supergravity is quite unique.
It is the minimal and maximal supergravity possible in 11 dimensions,
and we believe that it gives a good description of M theory on large
smooth spaces ...".
Therefore, it seems to me that if you
describe 11-dim supergravity in Clifford terms,
you will NOT end up with my model,
because
my model does NOT have spin 3/2 gravitinos
and
my model does NOT have a 1-1 supersymmetric correspondence
of 128 bosons with 128 fermions.
What I have is a more subtle supersymmetry in the 8-dim structure,
whereby
the 28 gauge bosons are the 8/\8 = 28 bivectors
and
the there are 8 +half-spinor fermion particles
and 8 -half-spinor fermion antiparticles.
Due to triality,
the vector 8 = the +half-spinor 8 = the -half-spinor 8
so that
you can identify the 28 bosons with
the /\ product of the 8 fermions with the 8 antifermions.
If you look carefully at the spinor fermion term in
the 8-dim Lagrangian, you see that the dimensionality
of each half-spinor factor is 7/2 (compare the dimensionality
of 3/2 in conventional 4-dim Lagrangians).
Therefore,
since the dimensionality of a gauge boson is 1,
the total dimensionality of the bosons = 28 x 1 = 28
which is exactly balanced by
the total dimensionality of the fermions = 8 x 7/2 = 28,
and
you should get nice cancellations in my model at the 8-dim level.
In other words,
my subtle supersymmetry does the nice things of 11-dim supergravity
but
does not have its unrealistic particle content of 128 bosons
and 128 fermions including spin 3/2 gravitinos.
Here is how my model contains parts of the graded structure of Cl(8),
1 8 28 56 35+35 56 28 8 1
where I have written the middle 70 as 35+35 because it is self-dual
under Hodge duality.
By Hodge duality, the
1 8 28 56 35
is dual to the
35 56 28 8 1
As I have discussed on my web site at
... clfpq2.html#clifstructure
Dennis Marks has shown that the
1 8 28 56 35
correspond to physical stuff in the coordinate representation
while the
35 56 28 8 1
correspond to physical stuff in the momentum representation.
As he says
"... Complementarity between space-time and momentum-energy
is achieved by bit inversion, which interconverts between position
representation and momentum representation ...".
That means that the physical interpretation of Cl(8) graded stuff is:
coordinate rep of Higgs scalar 1 1 momentum rep of Higgs scalar
coordinate rep of spacetime 8 8 momentum rep of spacetime
coordinate rep of gauge bosons 28 28 momentum rep of gauge bosons
coordinate rep of trivectors 56 56 momentum rep of trivectors
coordinate rep of 4-vectors 35 35 momentum rep of 4-vectors
In my Lagrangian,
I usually do not write out the trivector and 4-vector terms,
because they are not physically effective at low energies after
dimensional reduction.
However, here is what they are (in my model) at high energies:
In the full 8-dimensional theory,
the 56 trivectors are related to the structure of
1+3=4-dimensional subspaces of 1+7=8-dimensional spacetime
that are connected with the E8 HyperDiamond lattice
links that are (normalized) sums of 4 of the basis octonions.
To reduce the dimension of spacetime to 1+3=4 dimensions,
an associative 3-form is used to pick a particular quaternionic
substructure of the 8-dim octonionic structure.
This effectively fixes a particular trivector,
so the 56 trivectors do not play a dynamical role
in my model at low energies after dimensional reduction.
Here and now, we do not have the technology
to do experiments that could test
the structure of the full 56-dimensional trivector sector.
The 70 4-vectors in the 4-grade subspace are reducible to
two sets of 35 4-vectors each.
They are the symmetric parts of 8x8 = 8/\8 + 8sym8 = 28 + 36
after taking out the 1-dimensional scalar 0-vector Higgs scalar.
The 1-dimensional scalar 0-vector representing the Higgs scalar
can be thought of as the trace of the full symmetric
1+35=36-dimensional space of symmetric 8x8 real matrices.
The 35 4-vectors are the traceless symmetric 8x8 matrices.
They are related to the coassociative 4-form that is fixed
in the dimensional reduction process to determine the
internal symmetry space. At our low energy levels,
below the Planck-scale at which dimensional reduction occurs
in the D4-D5-E6 model, the 35 4-vectors do not play a
dynamical role that we can test experimentally here and now.
However, they show that the Higgs mechanism is related
fundamentally to BOTH
the particles and fields of the internal symmetry space
and
the spacetime of conformal MacDowell-Mansouri gravity.
Therefore, my model does use all the graded parts of Cl(8),
and also the spinor structure of Cl(8),
but the 56 and 35 parts are not physically effective at
low energies after dimensional reduction,
so usually I don't write them explicitly in my Lagrangian,
which I usually use to calculate force strengths,
particle masses, etc in the energy region where we do experiments now.
it might be useful for me to say how I see that spinors are
related to the graded structure of 256-dim Cl(8),
based on the material on my web site including
... 8idempotents.html
As Pertti Lounesto says in his book
Spinor Valued Regular Functions in Hypercomplex Analysis
(Report-HTKK-MAT-A154 (1979) Helsinki University of Technology)
(in this quote I have changed his notation for a Clifford
algebra from R_(p,q) to Cl(p,q)) at pages 40-42:
"... To fix a minimal left ideal V of Cl(p,q)
we can choose a primitive idempotent f of Cl(p,q) so that V = Cl(p,q) f .
By means of an orthonormal basis { e_1 , e_2 , ... , e_n }
for [the grade-1 vector part of Cl(p,q)] Cl^1(p,q)
we can construct a primitive idempotent f as follows:
Recall that the 2^n elements
e_A = e_a_1 e_a_2 ... e_a_k , 1 < a_1 < a_2 < ... < s_k < n
constitute a basis for Cl(p,q). ...
dim_R V = 2^X ,
where X = h or X = h + 1 according as p - q = 0, 1, 2 mod 8
or p - q = 3, 4, 5, 6, 7 mod 8 and h = [ n / 2 ] .
Select n - X elements e_A, e_A^2 = 1 ,
so they are pairwise commuting and generate a group of order 2^( n - X ) .
then the idempotent ...
f = (1/2)( 1 + e_A_1) (1/2)( 1 + e_A_2 ) ... (1/2)( 1 + e_A_( n - X ) )
is primitive ...
... Cl(0,8) has a primitive idempotent
f =
(1/2)( 1 + e_1248 )(1/2)( 1 + e_2358 )(1/2)( 1 + e_3468 )(1/2)( 1 + e_4578 ) =
=(1/16)( 1 +
+ e_1248 + e_2358 + e_3468 + e_4578 + e_5618 + e_6728 + e_7138 -
- e_3567 - e_4671 - e_5712 - e_6123 - e_7234 - e_1345 - e_2456 +
+ e_J )
with four factors [and where J = 12345678 ] ...".
Physically (refer to my recent earlier messages for details),
in my model I identify the first scalar and 7 4-vectors with
the Higgs and the 7 diagonal elements of traceless symmetric 35
in their coordinate representations
and
I identify the last pseudoscalar and 7 4-vectors of the dual 35 with
them in their momentum-space representations.
In other words, 8 of the symmetric 36
of 8x8 = 8/\8 + 8sym8 = 28 + 36 = 28 + 35 + 1
correspond to 8 elements of the Primitive Idempotent,
1 for scalar Higgs and 7 for 4-vector 35.
The other 8 terms of the 16-term Primitive Idempotent
correspond to the 1 pseudoscalar and 7 of the 4-vector dual 35.
Since Primitive Idempotent produces Spinors,
the Higgs and Spinors are connected,
which gives a Cl(8) structure interpretation for Yukawa couplings,
something that is not clearly motivated in the usual Standard Model.
Of course, the Higgs/electroweak coupling in my model,
based on electroweak gauge boson connection with Higgs scalar,
is the same as that of the usual Standard Model.
In short,
the spinors in my model come from a Primitive Idempotent living in
the scalar, the 35+35 4-vector, and pseudoscalar
parts of the Cl(8) graded structure.
Carlos, you are correct that I have been imprecise in saying
things like "... a 35 of the 4-vector is like the 35 in
8x8 = 8/\8 + 8sym8 = 28 + 36 = 28 + 35 + 1 ...".
However,
there is a structural sense in which it is a valid way to
visualize the elemets of Cl(8).
Consider ...[this]... image of a 16x16 array of 256 elements.
The two diagonal 8x8 parts represent the even subalgebra.
The 1+1 red dots are the scalar and its dual pseudoscalar.
The 28+28 blue dots are the bivectors and their dual 6-vectors.
The 35+35 white dots are the 4-vectors.
As you can see, the 64+64 elements of the even subalgebra
can be thought of as:
64 = 28 bivectors (8/\8) plus 1 scalar and 35 4-vectors (8sym8),
with
the other 64 being interpreted similarly in dual grades.
The scalar and 7+7 of the 35+35 and the pseudoscalar
have yellow dots, and they correspond to the 16 terms
of the Primitive Idempotent.
As you can see,
they lie on the diagonals of the two 8x8s.
The odd parts (the two off-diagonal 64-element 8x8s) of Cl(8) are:
the 8+8 green dots are vectors and pseudovectors
and
the 28+28+28+28 = 56+56 gold dots are the trivectors and their dual 5-vectors.
I hope this clarifies to some degree the way that I visualize Cl(8).
Please note that details of this nice visualization are peculiar to Cl(8),
which has exceptional and unusual symmetries and structure,
and is also a fundamental building block of all real Clifford algebras,
due to real 8-periodicity.
For instance,
if you were to look at Cl(16) = Cl(2x8) = Cl(8) (x) Cl(8)
you would see a 2-level tensor-nesting of the 16x16 structure,
and the Cl(16) diagonal would be made up of 16 dots each of which
looked like an entire Cl(8) 16x16 with its 16-element diagonal,
so that
Cl(16) spinors would look like 16x16 = 256 elements,
with 128 +half-spinors and 128 -half-spinors,
which is indeed the case.
...[this is from part of]... my web page at
... Jordan.html#3x3trless4x4
"... The 28-real-dimensional degree-4 quaternionic Jordan algebra J4(Q)
of 4x4 Hermitian matrices over the Quaternions
p D B A
D* q E C
B* E* r F
A* C* F* t
where * denotes conjugate and p,q,r,t are in the reals R and A,B,C,D,E,F
are in the quaternions Q ...
The 4x28 = 112-real dimensional Quaternification of J4(Q) can be
represented as the Symmetric Space E8 / E7 x SU(2)
...
J4(Q) contains the traceless 28-1 = 27-dimensional subalgebra J4(Q)o that
"has the unique structure of" the 27-dimensional exceptional Jordan
algebra J3(O) of 3x3 Hermitian matrices over the Octonions
p B A
B* q C
A* C* r
where * denotes conjugage and p,q,r are in the reals R and A,B,C are
in the Octonions O.
The 2x27 = 54-real dimensional Complexification of J3(O) = J4(Q)o can
be represented as the Symmetric Space E7 / E6 x U(1)
...
J3(O) contains a traceless 27-1 = 26-dimensional subalgebra J3(O)o that
can be represented as the Symmetric Space E6 / F4 ...".
In other words, the chain 26, 27, 28
gives you Jordan algebra structures J3(O)o, J3(O) = J4(Q)o, J4(Q)
which in turn
give you the Lie algebra structures E6, E7, E8
which Lie algebras are all either E8 or E8 subgroups.
To connect with (real) Clifford algebras, here are some more parts of
that same web page:
"... how to fit E8 into Clifford structure ... use
248-dim E8 = 120-dim bivector adjoint of D8 + 128-dim D8 half-spinor
and so embed E8 in the Clifford algebra Cl(16),
with graded structure
1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1
and total dimension 2^16 = 65,536 = (128+128)(128+128)
Since Cl(16) = Cl(2x8) = Cl(8)xCl(8)
( For example, 120 = 1x28 + 8x8 + 28x1 and 128 = 8x8 + 8x8. )
E8 can be represented in a tensor product of Cl(8) algebras ...".
Frank Dodd (Tony) Smith, Jr.