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VoDou Physics and 130 GeV Truth Quark

My D4-D5-E6-E7-E8 VoDou Physics model is based on Clifford algebras. 
Here is a brief rough outline of how it works 
(ignoring subtleties such as signature etc):

Consider any arbitrary real Clifford algebra Cl(N),
where N is the (arbitrary) dimension of the vector space.
No matter how large is N 
(by going to a larger N, you can assume that N is divisble by 8) 
it can, by the periodicity theorem, be factored into the tensor product

Cl(N) = Cl(8) x...( N/8 times )...x Cl(8)

which is the basis for construction of 
a real hyperfinite II1 von Neumann algebra factor.
Therefore,
I use Cl(8) as building blocks of the physics model.
Since Cl(8) has the structure of the VoDou divination method IFA, 
I call the model VoDou Physics. 

Cl(8) is 2^8 = 16x16 = 256-dimensional, and has graded structure

1   8   28   56   70   56   28   8   1

The first 8 is the vector space;

the first 28 is the bivector Spin(8) Lie algebra;

the spinors are 16-dimensional,
and break down into two mirror-image 8-dim half-spinors.

These things fit together in a natural way 
to form a Lagrangian over 8-dim spacetime:

Integral over vector space of bivector/covariant derivative term
                            and spinor/Dirac operator term

Note that all these parts, including the spinors and Dirac operator,
are natural parts of the Clifford algebra and are NOT ad hoc
additions-put-in-by-hand.

The 8-dim spacetime breaks down into
4-dim Physical SpaceTime that looks topologically like S1 x S3
plus
4-dim Internal Symmetry Space that looks topologically like CP2.

The CP2 = SU(3) / U(2) is a representation space on
which the color SU(3) acts globally
and the electroweak U(2) = SU(2)xU(1) acts locally.

S1 x S3 Physical SpaceTime is 
the Shilov boundary of the bounded domain corresponding to
the symmetric space Spin(2,4)/Spin(1,3)xU(1) = SU(2,2) / SU(2)xU(2)
which has 8 real dimensions and 4 complex dimensions.
Its the symmetry is that
of the 15-dim conformal group Spin(2,4) = SU(2,2)
and you get Gravity and the Higgs mechanism 
by gauging the conformal group as
described by Mohapatra in section 14.6 of his book
Unification and Supersymmetry (2nd edition, Springer-Verlag 1992).

The 28 generators of Spin(8), 
after dimensional reduction of spacetime, are transformed into:

16  generators of U(2,2) = Spin(2,4)xU(1) where Spin(2,4) is
    the conformal group that gives Gravity, the Higgs mechanism, 
    and a comlex phase for particle propagators. 

12 generators of SU(3)xSU(2)xU(1) for the Standard Model, 
whose global group structure is discussed here. 

The gauge boson process is described in more detail on this web page.

More about the Clifford structure, 
including position-momentum duality, is HERE.

The most important calculation in my model, in my opinion,
is my calculation of the Truth quark mass prior to its discovery at Fermilab,
so that calculation of 130 GeV is in fact a prediction.

Fermilab analyses of the data state
that the Truth quark mass is about 170 GeV,
but
I think that their analyses are wrong and that a correct
analysis of the same data is consistent with
a Truth quark mass of about 130 GeV as my model predicted.

Details of my comparison of the data analyses are on
this web page and others to which it is linked. 
A version of that material is on the e-print archive 
as physics/0006041. 

As to the calculations themselves,
they are based on the D4-D5-E6-E7-E8 VoDou Physics model,
which is based on Clifford algebras, symmetric spaces,
and Shilov boundaries of bounded complex homogeneous domains.

A summary of my calculations is on this web page.

More explicit details of the Truth quark mass,
and other quark and lepton mass calculations, are on this web page. 

Here is a more visual outline of the Clifford Algebra Foundation of VoDou Physics:

                   Cl(8N) = 2^(8N) =                                
                                                                    
          = Cl(8)    x ...Ntimes... x   Cl(8)  = (2^8)^N            
                                                                    
2^8 = 256 =   16   x   16  = 1   8    28    56  70  56  28   8   1  
                                                                    
                                 4    16 = U(2,2) = Spin(2,4) x U(1)
   *-*      (8) + (8)                                               
                                 4    12 = SU(3) x SU(2) x U(1)     
*-* or *-* (8,8) + (8,8)                                            
                                                                    
   *-*   (8,8,8) + (8,8,8)                                          
        
D4                   D5              E6           E7      
  Spin(8)             Spin(10)                                
    28                 8 (+1+8)       16 (+1+16)    27 (+1+27)
                                     Generation-1             
  Gravity             SpaceTime       Fermions                
  16-U(4)                4               8          MacroSpace
                                     Generation-1     of      
 Standard Model      Internal Space  AntiFermions  Many-Worlds
12-SU(3)xSU(2)xU(1)      4               8                    

(The Lepton-Quark image to the above left is adapted from a Fermilab web page.)

The 24-cell shown above represents the 24 root vector gauge bosons of the 28-dimensional Spin(8) gauge group. The dual 24-cell

represents:

• First-generation electron; red, green, blue up quarks; red, green, blue down quarks, and neutrino, which are are denoted by

• First-generation positron; anti-red, anti-green, anti-blue up anti-quarks; anti-red, anti-green, anti-blue down anti-quarks, and anti-neutrino, which are denoted by

• 8-dimensional Spacetime that reduces after dimensional reduction to
  • 4-dimensional Physical SpaceTime, whose dimensions are denoted by
  
  • 4-dimensional internal symmetry space, whose dimensions are denoted by
  

Together, the 24-cell plus its dual 24-cell correspond to the 48 root vector vertices of the 52-dimensional exceptional Lie algebra F4., and F4 can be thought of as the Real Part of the 52+26 = 78-dimensional Lie algebra E6 of the D4-D5-E6-E7-E8 VoDou Physics model.

I learned about Clifford Algebras from studying under David Finkelstein at Georgia Tech since around 1981. Both David Finkelstein and I use the the Periodicity-8 tensor product Cl(N) = Cl(8) x ...N/8 times... x Cl(8)) in our physics models. With respect to his model, David Finkelstein says "... An assembly with Clifford statistics we call a squad. The Spinorial Chessboard shows how the dynamics, a large squad of chronons, can spontaneously break down into a Maxwellian assembly of squads of 8 chronons each. Squads of 8 are special this way. ...". Also, see, for example, the paper hep-th/0009086, Clifford Algebra as Quantum Language, by James Baugh, David Ritz Finkelstein, Andrei Galiautdinov, and Heinrich Saller, and the paper by D. Finkelstein and E. Rodriguez, The quantum pentacle, International Journal of Theoretical Physics 23, 887-894 (1984).

Here is the introduction to my May 2002 Cookeville Clifford Algebra talk.

In Phys. Lett. 149B (1984) 117, Green amd Schwarz showed superstring theory to be anomaly-free. Richard Feynman said, in Davies and Brown, Superstrings, Cambridge 1988, pp. 194-195):

I have agreed with Feynman's opinion about superstrings as a model of physics of particles and forces in 4-dimensional spacetime since 1984, when I began to construct a TOE (Theory Of Everything) physics model, the D4-D5-E6-E7-E8 Physics Model, in which I can calculate particle masses and force strengths, including the

Truth Quark mass of 130 GeV.

I sometimes use some other names for my VoDou Physics model:

• D4-D5-E6 physics model, since E6 includes all the Fermion Particles and AntiParticles, the Gauge Bosons, and spacetime of Physical SpaceTime plus Internal Symmetry Space;
• D4-D5-E6-E7 physics model, since E7 includes a representation of the MacroSpace of ManyWorlds;
• D4-D5-E6-E7-E8 physics model, since E8 includes both Geometric and Algebraic Structure of the MacroSpace of ManyWorlds, since the E8 lattice represents unreduced 8-dim Lattice SpaceTime, since E6-E7-E8 can represent the three generations of Fermions, and since the Lie algebra chain D4-D5-E6-E7-E8 terminates with E8;
• HyperDiamond Feynman CheckerBoard model, describing the lattice structure of the model.

According to John C. Baez and S. Jay Olson in their paper at gr-qc/0201030:

"... Ng and van Dam have argued that quantum theory and general relativity give a lower bound delta L > L^(1/3) L_P ^(2/3) on the uncertainty of any distance, where L is the distance to be measured and L_P is the Planck length. Their idea is roughly that to minimize the position uncertainty of a freely falling measuring device one must increase its mass, but if its mass becomes too large it will collapse to form a black hole. ... Amelino-Camelia has gone even further, arguing that delta L > L^(1/2) L_P ^(1/2) ... Here we show that one can go below the Ng-van Dam bound [ and the Amelino-Camelia bound ] by attaching the measuring device to a massive elastic rod. ...

[ while the Ng-van Dam ] result was obtained by multiplying two independent lower bounds on delta L, one from quantum mechanics and the other from general relativity, ours arises from an interplay between competing effects. On the one hand, we wish to make the rod as heavy as possible to minimize the quantum-mechanical spreading of its center of mass. To prevent it from becoming a black hole, we must also make it very long. On the other hand, as it becomes longer, the zero-point fluctuations of its ends increase, due to the relativistic limitations on its rigidity. We achieve the best result by making the rod just a bit longer than its own Schwarzschild radius.

... Relativistic limitations on the rod's rigidity, together with the constraint that its length exceeds its Schwarzschild radius, imply that zero-point fluctuations of the rod give an uncertainty delta L > L_P . ...".

Click here if you don't want any mathematics, but would like to see a Theatrical Version of the Ideas.

Here is a More Detailed Outline

• Start with the Standard Model.
• Now add Gravity.
• Try to Unify the Standard Model with Gravity.
  • Coleman-Mandula
• Supergravity.
• Superstrings.
• The D4-D5-E6-E7-E8 VoDou Physics Model.
  • 3x3 matrices and SL(3,O) = E6
  • Subtle Triality Supersymmetry.
  • Ultraviolet Finiteness and Renormalizability.
• Dimensional Reduction of Spacetime.
• HyperSpace.
• Why is it a D4-D5-E6-E7-E8 Model?
• Particle masses and Force Constants
• What is a Shilov Boundary?
• How do you Visualize the D4-D5-E6-E7-E8 Model?
• MANY-WORLDS, Quantum Computing and the D4-D5-E6-E7-E8 Model.
• D5 and SU(5) Grand Unification
• Weinberg Angle
• Effective IR and UV Cutoffs
• Historical Precedents and Further Description.
  • Here is how my physics model developed from around 1971.
  • Here is how my physics model looked in 1992.
  • Here is SESAPS 2002.
• Building Blocks - Geometry - Lagrangian
• real hyperfinite II1 von Neumann algebra factor

Here is how the mass ratios work:

• the electron / Truth quark mass ratio is fixed, as are the mass ratios of all fundamental fermions;
• the Truth - antiTruth quark pair mass corresponds to the sum of the masses of the weak bosons W+, W-, and W0;
• the W0 / Z0 mass ratio is fixed, and the sum of the masses of the weak bosons W+, W-, and Z0 corresponds to the Vacuum Expectation Value of the Higgs scalar field that transforms the W0 into the Z0;
• the Planck mass is a condensation of fermion-antifermion pairs, somewhat like a pion condensate;
• the SU(5) LeptoQuark X-bosons have a Vacuum Expectation Value corresponding to a decoherence / condensate below the Planck mass.

It is interesting that

Here are links to more details:

Details of that discrete 
HyperDiamond Feynman Checkerboard model, 
including calculation of particle masses 
and charges, or force strength constants, 
are in my 1997 paper 
From Sets to Quarks:
Deriving the Standard Model plus Gravitation 
from Simple Operations on Finite Sets.  
   Table of Contents: 
Chapter 1 - Introduction. 
Chapter 2 - From Sets to Clifford Algebras. 
 
                 MANY-WORLDS QUANTUM THEORY.  
 
Chapter 3 - Octonions and E8 lattices. 
Chapter 4 - E8 spacetime and particles. 
Chapter 5 - HyperDiamond Lattices. 
Chapter 6 - Spacetime and Internal Symmetry Space. 
Chapter 7 - Feynman Checkerboards. 
Chapter 8 - Charge = Amplitude to Emit Gauge Boson. 
Chapter 9 - Mass = Amplitude to Change Direction. 
Chapter 10 - Protons, Pions, and Physical Gravitons. 
Appendix A - Errata for Earlier Papers. 
References.
 
Details of the D4-D5-E6 physics model, 
which is substantially  a continuum spacetime/LaGrangian version 
of the HyperDiamond Feynman Checkerboard model,  
are given in hep-ph/9501252 
Gravity and the Standard Model from D4-D5-E6 Model 
using 3x3 Octonion Matrices. 
 
Another way to look at the continuum version of the D4-D4-E6 model 
is in terms of Octonion Creators and Annihilators. 
 
 

 
This is a more detailed rough outline, 
but without full technical details, of

the model and how I visualize it.

Here, I have not included technical references or 
references to credit who did what and when.  
They can be found in the other pages and papers 
dealing with the D4-D5-E6-E7-E8 model.  
 
In the early 1970s, just before the Standard Model 
was established, Armand Wyler noticed something: 
 
the electromagnetic fine structure constant 
seemed to be related to the ratios of volumes 
of bounded complex homogeneous domains 
and their Shilov boundaries.  
He did not give clear physical reasons for the 
relationship, and his results were dismissed 
by most physicists as mere coincidence.  
However, 
a few have followed his work with interest,  
such as Gustavo R. Gonzlez-Martin, who has written 
papers about the electromagnetic Fine Structure Constant 
and other things from a point of view similar to that of Wyler.  

• Elements of a Discrete Clifford algebra correspond to Basis Elements of a Real Clifford algebra.
• General Elements of a Real Clifford algebra correspond to Superpositions of Basis Elements = Elements of the underlying Discrete Clifford algebra.
  • Volumes of Spaces of Superpositions of some given Sets of Basis Elements correspond to Mass or Charge of Particles or Forces represented by those Basis Elements.
  • Volumes of Spaces of Superpositions of other given Sets of Basis Elements correspond to Volume of Physical SpaceTime and Volume of Internal Symmetry Space represented by those Basis Elements.

Start with the Standard Model.

  
 
The Standard Model has 3 basic parts:
 
1.  a spacetime 
    (You can think of spacetime as being 
     fundamentally a HyperDiamond Lattice, with the 
     continuous manifold being just a convenient 
     way to think of it for doing calculations 
     involving calculus, Lie groups, and such things.
     As a Planck Scale Lattice, it violates Lorentz
     Invariance at the Planck Scale.); 
 
2.  fermion particles and antiparticles, 
    such as electrons and quarks, that 
    roughly speaking are the matter in the model  
    (You can think of them as being things located 
     at points in the spacetime, 
     or vertices of the spacetime lattice); and 
 
3.  gauge bosons, 
    or particles that carry the forces between particles 
    (You can think of them as being "located" on 
     lines in spacetime connecting the particles 
     that are acting on each other, 
     or as being on links of the spacetime lattice).  
 
The gauge group of a given force is the symmetry group 
of the gauge bosons that carry the given force.  
 
The Standard Model is made by taking the total gauge group 
to be the Cartesian product of the 3 gauge groups 
of the 3 forces
(electromagnetic, weak, and color).  
 
In the Standard Model, 
there is NO FUNDAMENTAL RELATION among the three parts.  
 
The spacetime, fermion particles and antiparticles, 
and gauge bosons are just put in by hand, so to speak.  

Paul Langacker, in hep-ph/0304186, says: "... The structure of the standard model is concisely summarized, including the standard model Lagrangian, spontaneous symmetry breaking, the reexpression of the Lagrangian in terms of mass eigenstates after symmetry breaking, and the gauge interactions. The problems of the standard model are described. ...".

Now add Gravity.

 
Since Gravity can be formulated in terms of curvature of spacetime, 
the force of Gravity 
DOES have a FUNDAMENTAL RELATION to spacetime. 
 

 

Try to Unify the Standard Model with Gravity.

 
Physicists tried to combine Gravity 
with the Standard Model by treating Gravity 
like the other forces, that is, 
as a force described by a gauge group.
 
Their choice of gauge group for Gravity was 
the 10-dimensional Poincare group of 
spacetime translations (4-dim), 
spacetime rotations (3-dim), and 
spacetime Lorentz boosts (3-dim).  
 
HOWEVER, the models whose unified simple Lie gauge group 
contained the Cartesian product of the Poincare group 
and the groups of the Standard Model 
did not work well mathematically,
and soon 
there was a theorem (Coleman-Mandula) saying that 
there is NO WAY you can make a Cartesian product of 
the Poincare group and the Standard Model groups 
into a single simple unified Lie gauge group of 
a mathematically consistent gauge field theory 
over a 4-dimensional spacetime.  
 

 

Supergravity.

 
Physicists did not just give up, they tried another way: 
 
SUPERGRAVITY.  
 
It avoided the Coleman-Mandula problem 
by using Lie Superalgebras of Lie Supergroups 
instead of Lie algebras of Lie groups. 
 
Lie superalgebras are not simple Lie algebras, 
but are combinations of two Lie algebras, 
one of which is "bosonic" and the other "fermionic", 
and defining their interactions with each other by using a 
SUPERSYMMETRY between fermions and bosons.  

The bosonic sector of the Lie Superalgebra should contain 
both 
the Lie algebra of the total Standard Model Lie group 
and 
a Lie algebra for gravity.  
 
After playing around some, physicists realized that 
the best Lie algebra for Supergravity theories 
was not the 10-dim Poincare group, 
 
but was the 10-dim deSitter group Spin(5), 
or its noncompact version Spin(2,3).  
 
Spin(5) is the covering group of 5-dim rotations, 
and is 
equal to 10-dim group Sp(2) related 
to 2x2 matrices of quaternions.  
 
Supergravity was very good because now gravity 
was included, and there was a direct relation 
between at leat one force and spacetime.  
 
BUT AGAIN, A MATHEMATICAL PROBLEM CAME UP:
You could classify all the Lie Superalgebras, 
and see which ones 
had Spin(5)=Sp(2) for gravity.  
 
The ones that worked had SO(N) as the other bosonic part. 
SO(N) is the group of N-dim rotations.  
 
Therefore, you could write down the equations 
for the theory with one bosonic part SO(N) 
and another bosonic part Spin(5)=Sp(2). 

Such Lie Superalgebras were called 
OrthoSymplectic osp(N,2).
 
You would get gravity just fine from 
the bosonic part Spin(5)=Sp(2), 
 
BUT you had to decide which N to use for SO(N) 
and how to get the Standard Model groups 
from the other bosonic part SO(N).  
 
The mathematics of the model clearly showed 
that the mathematically nicest N was N=8, 
so that the other bosonic part should be SO(8), 
BECAUSE OF THE OCTONIONIC STRUCTURE OF SO(8).
 
This N=8 Supergravity was also called 
11-dimensional Supergravity, because
it could be formulated in 11-dimensional spacetime.  
Since 11 = 4+7, 
our physical 4-dimensional spacetime 
could be seen as the result of compactifying, 
or curling up into very small things, 
the other 7-dimensions of 11-dim spacetime.  
Since the compactified 7-dim things could 
be thought of as 7-dimensional spheres, 
and 
since 7-dim spheres have a Malcev algebra 
structure that is related to the Lie algebra of SO(8) 
and 
since the 7-dim sphere lives in 8-dim Octonion space 
and is parallelizable by the 7-dim imaginary Octonions, 
the underlying beauty of Supergravity is 
DUE TO OCTONIONIC STRUCTURE.  
 
HOWEVER, 
the Supergravity model was constructed 
using Lie Superalgebras 
in such a way 
that 
SO(8) (interpreted that way) WAS NOT BIG ENOUGH TO 
INCLUDE THE STANDARD MODEL GROUPS SU(3)xSU(2)xU(1).  
 
There was also another problem with Supergravity: 
 
The Supersymmetry of Supergravity was a naive 
1-1 correspondence between fermions and bosons, 
which said 
 
"For every fermion there is a corresponding boson, 
and vice versa."  
 
SUCH NAIVE SUPERSYMMETRY HAS NEVER BEEN OBSERVED: 
 
The fermions - neutrinos, electrons, quarks 
ARE NOT 
in 1-1 correspondence with 
the gauge bosons - photon, weak bosons, gluons, gravitons.  

Burton Richter, of SLAC, says (in hep-ex/0001012): "... The sociologists of science would say that our theories are "socially constructed." That is certainly true initially but our theoretical models are continually tested and the "social constructs" that don't pass are discarded; though sometimes it takes a long time. ... While the standard model has withstood all experimental tests, we know that it is only a low-energy (a few hundred GeV) approximation to a better model. The most popular candidate to be the successor to the plain vanilla standard model is supersymmetry. ... To the experimenters I would say that supersymmetry is a pure "social construct" with no supporting evidence despite many years of effort. It is okay to continue looking for supersymmetry as long as it doesn't seriously interfere with real work (top, Higgs, neutrinos, etc.). ...".

 

Superstrings.

 
Although Supergravity had fatal flaws, 
it had a nice underlying OCTONIONIC unity, 
so the question was 
How to use it as a basis for a better model? 
 
Most physicists decided to:
 
1.  Keep NAIVE 1-1 fermion-boson supersymmetry; 
 
2.  Give up SO(8) as a gauge group; 
 
3.  Generalize point-particle Spin(5)=Sp(2) gravity 
    to gravity based on String theory, the theory of 
    vibrating strings in N-dim spacetime.   
 
Why vibrating strings? 
In the conventional quantum mechanics courses 
of the early education of most physicists, 
they are drilled to think of the example of 
the 1-dim harmonic oscillator - the vibrating string - 
as the fundamental model in terms of which 
physics theories should be visualized.  
 
The result was Superstring theory.  
 
Perturbative superstring theory can be formulated 
in terms of 1-dimensional strings, 
but 
dualities in superstring theory can map perturbative string states 
into non-perturbative states.  Non-perturbative states can 
correspond to p-dimensional membranes, called p-branes, 
rather than 1-dimensional strings, so: 
 
Duality forces superstring theory to include p-branes.   
 
Since the resulting theory contains membranes 
it is now usually termed M-theory.  
 
The most elegant M-theory is 11-dimensional M-theory, 
which is closely related to 11-dimensional supergravity. 
 
As Bergman has pointed out, 
"... the behavior of string theory at high temperature 
and high longitudinal boosts, combined with ... p-branes ... , 
point to ... reformulation of strings, as well as p-branes, 
as composits of bits. ..."
 
If the bits are thought of as pointlike 0-branes, then, 
in view of the "... precise equivalence between 
uncompactified eleven dimensional M-theory and 
the N = infinity limit of the supersymmetric 
matrix quantum mechanics describing D0-branes ..."
conjectured by Banks, Fischer, Shenker, and Susskind, 
11-dimensional M-theory can be taken to mean M(atrix)-theory 
as well as M(embrane)-theory. 
 
Following Bergman, we see that at high energies 
strings and p-branes break down into points, 
so what we have is fundamentally a theory of pointlike bits, 
of which strings and p-branes are only one low-temperature limit. 
 
As Bergman says, the ".. main drawback of the string-bit models 
is that there are so many of them.  ... it is not clear yet 
whether there is a unique superstring-bit model." 
 
To construct a unique low-temperature model from 
the high-temperature bit theory, 
I propose to 
start with a bunch of bits
and then 
use the octonionic symmetries of 11-dimensional M-theory 
as the basis for a theory of interaction among bits, 
with the bits acting somewhat like the nodes of a spin network. 
 
The result: 
the D4-D5-E6-E7-E8 physics model emerges from the bits 
just as it does from the points of 
Simplex Physics above the Planck Energy 
which is similar to its emergence 
from the arrows of quantum set theory 
and from the structure of Metaclifford algebras. 
 
In my opinion:
supergravity in 11 dimensions;   
superstring theory in 10 dimensions; and 
M-theory in 11-dimensions 
all have some nice features (mostly due to octonionic structures) 
but all have the fatal flaw of NAIVE 1-1 fermion-boson supersymmetry 
that has NEVER been observed experimentally, 
while 
the D4-D5-E6-E7-E8 physics model not only has nice octonion structure, 
its subtle triality supersymmetry IS consistent with experiments. 
 
However, 
String Theory in 26 dimensions is useful,
not in its conventional interpretation,
but as a model of dynamics in the MacroSpace of Many-Worlds. 
 
 

 

The D4-D5-E6-E7-E8 VoDou Physics Model

 
I did NOT go from Supergravity to Superstring theory.  
 
I decided to go from Supergravity as follows: 
 
1.  Give up NAIVE 1-1 fermion-boson supersymmetry; 
 
2.  Keep SO(8) as a gauge group with Octonionic structure; 
 
3.  Give up Lie superalgebras and try to put 
    gravity into the SO(8).  
 
To get everything from SO(8), 
recall that the finite (Weyl group) reflection group 
that generates SO(8) is the group of the 
4-dimensional 24-cell.  
 

 
SO(8) has 28 generators:  
      24 for each vertex of the 24-cell
plus   4 for the dimensions of the 4-dim space of the 24-cell.  
 
I let the 28 generators of SO(8) be the 
generators of the gauge bosons:  
 
8+3+1 = 12 of them for the Standard Model, 
whose global group structure is discussed here; 
 
the other 16 for a U(4), 
which contains 15-dim SU(4)=Spin(6),
which contains 10-dim Spin(5)=Sp(2) for gravity.  
The part of the U(4) not used for gravity 
gives the Higgs mechanism and a complex phase for 
quantum propagators.  
 
If the 28-dim adjoint representation of SO(8) 
gives the gauge bosons, then 
 
WHAT GIVES SPACETIME?  
 
SO(8) has an 8-dim vector representation, 
as the 8-dim space on which the rotations act.  
I let that space be an 8-dimensional spacetime.  
 
Next, 
 
WHAT GIVES FERMION PARTICLES AND ANTIPARTICLES? 
 
Fermions should come from spinor representations, 
but SO(8) has no spinor representations, 
 
BUT its 2-fold covering group Spin(8) 
DOES have TWO 8-dim half-spinor representations, 
 
so I let one be 8 fermion particles 
 
and the other be 8 fermion anti-particles.  
 
 
In the D4-D5-E6-E7-E8 model, fermion masses can be calculated. 
 
The tree-level constituent Truth Quark mass is 130 GeV.   


The overall structure of the D4-D5-E6-E7-E8 model 
involves Clifford Algebras and Conformal structures. 

 
 
 

Subtle Triality Supersymmetry:

 
Here is the Dynkin diagram for Spin(8).  
Each vertex represents a representation of Spin(8), 
with the center vertex (Spin(8)) corresponding 
to the 28-dimensional adjoint representation 
that I identified with gauge bosons.  
 

 
The three representations for 
 
spacetime (blue dot), 
 
fermion particles (red dot), and 
 
fermion antiparticles (green dot)
 
are EACH 8-dimensional with Octonionic structure.  
 
They are ALL isomorphic by the Spin(8) Triality Automorphism, 
which can be represented by rotating or interchanging 
the 3 arms of the Dynkin diagram of Spin(8).  
 
The Triality isomorphism between spacetime 
and fermion particles and fermion antiparticles 
constitutes a 
SUBTLE SUPERSYMMETRY between fermions and spacetime.  
 

Ultraviolet Finiteness and Renormalizability

 
Ultraviolet finiteness of the D4-D5-E6-E7-E8 physics model 
may be seen by considering, prior to dimensional reduction, 
the generalized supersymmetry relationship between 
the 28 gauge bosons 
and 
the 8 (first-generation) fermion particles and antiparticles.
 
In the 8-dimensional spacetime,
the dimension of each of the 28 gauge bosons in the Lagrangian is 1, 
and
the dimension of each of the 8 fermion particles is 7/2, 
so that
the total dimension 28x1 = 28 of the gauge bosons 
is equal to 
the total dimension 8x(7/2) = 28 of the fermion particles. 
 
 
After dimensional reduction of spacetime to 4 dimensions,
the 8 fermions get a 3-generation structure 
and
the 28 gauge bosons are decomposed to produce 
the Standard Model of 
U(1) electromagnetism, SU(2) weak force, and SU(3) color force, 
plus a Spin(5) = Sp(2) gauge field
that can produce gravity by the MacDowell-Mansouri mechanism. 

Note that both Gravity and the Standard Model forces are required for the cancellations that produce the ultraviolet finiteness that is useful in the Sakharov Zero Point Fluctuation model of gravity.

The Conformal Structure of the D4-D5-E6-E7-E8 model includes the symmetry of Conformal Group Dilations, which transform the scale of length/energy/mass, which gives the scale transformation properties needed for Renormalization.

Dimensional Reduction of 8-dim Spacetime

 
At this stage my D4-D5-E6-E7-E8 model has a problem: 
Spacetime is 8-dim, not 4-dim.  
 
HOWEVER, Ben Goertzel has noted that, 
given a timestream of spacetime elements abababababababcca, 
you need associativity for spacetime so that you can
mark off any given moment in a timestream
ababab|ababababcca
(by associating everything on either side of the moment)
without the choice of moment making a physical difference.
Since associativity of (ababab)(ababababcca) is required
so that a choice of moment does not change physics,
associativity is required for physical spacetime,
which must therefore be reduced from 8-dim octonion
to 4-dim associative quaternion. 

Another way of seeing the Octonion Non-Associativity 
point of view is to that the basis elements of 
physical spatial dimensions should form Associative Triads 
such as {i,j,k} among the Octonion Imaginaries {i,j,k,E,I,J,K}. 

From yet another point of view, 
dimensional reduction is needed for spacetime to have a 
consistent lightcone structure.  
 
Therefore physical timestreams and consistent lightcones 
require that spacetime be an ASSOCIATIVE QUATERNIONIC SUBSPACE 
of Octonionic 8-dim spacetime, and 
 
the resulting math gives not only a 4-dim spacetime, 
with -+++ signature (1,3), 
 
but also 
 
3 generations of fermions, 
corresponding to the Lie algebras E6, E7, and E8, 
 
and 
 
the decomposition of the 28 gauge bosons into 
the 12 gauge bosons of the Standard Model, 
whose global group structure is discussed here, 
plus 
the 16-dim U(4) that gives Gravity, the Higgs mechanism, 
                           and propagator phase.  

The decomposition of the 28 gauge bosons can be seen from different points of view:

• Conformal Moebius Clifford Transformations
• Clifford Algebras
• Division Algebras
• Weyl Groups and Root Vectors (showing gluon confinement)
  • SU(5) Grand Unification and LeptoQuark X-bosons
  • Casimir operators and questions of signature
• Instantons in 8 and 4 dimensions

 
Where does the associative quaternionic subspace come from? 
Perhaps from timestream or conformal lightcones. 
or perhaps equivalently from MetaClifford Algebra structure 
or from Cohomology and Quadrics. 
 

• Elements of a Discrete Clifford algebra correspond to Basis Elements of a Real Clifford algebra.
• General Elements of a Real Clifford algebra correspond to Superpositions of Basis Elements = Elements of the underlying Discrete Clifford algebra.
  • Volumes of Spaces of Superpositions of some given Sets of Basis Elements correspond to Mass or Charge of Particles or Forces represented by those Basis Elements.
  • Volumes of Spaces of Superpositions of other given Sets of Basis Elements correspond to Volume of Physical SpaceTime and Volume of Internal Symmetry Space represented by those Basis Elements.

In the D4-D5-E6-E7-E8 VoDou Physics model, there are two types of structures that might be regarded as HyperSpace:

Merging Physical SpaceTime with Internal Symmetry Space, to get a HyperSpace of Internal Symmetry Space - this occurs at energies above the quark/hadron phase transition; and

Going from the Shilov Boundary Physical SpaceTime into the Interior of its Bounded Complex Domain, to get a HyperSpace of Complex SpaceTime - this might occur at lower energies for some phenomena, and only at very high energies for other phenomena.

                                  Real            Complex


Physical SpaceTime                4-dim            8-dim 



Internal Symmetry Space           4-dim            8-dim 



Unified Physical+Internal         8-dim           16-dim 

The Basic Building Blocks of the D4-D5-E6-E7-E8 VoDou Physics model are:

 
DIVISION ALGEBRAS - the largest division algebra, 
                    the octonions, are reflexive/recursive 
                    in that the 7 octonion imaginaries 
                    correspond to the 7 associative triples; 
 
MATRIX ALGEBRAS - symmetric JORDAN ALGEBRAS and 
                  antisymmetric LIE ALGEBRAS -  
                  the 248-dimensional E8 Lie Algebra 
                  is the largest exceptional Lie algebra.  
                  It is reflexive/recursive 
                  in that its fundamental representation 
                  is its adjoint representation; 
 
LATTICES - the E8 lattice corresponds to "integral" octonions, 
           and has the same (finite) Coxeter/Dynkin diagram 
           as the E8 Lie algebra, 
           and  
           the 27-dim MacroSpace has a 26-dim subspace that
           corresponds to the Lorentz Leech lattice /\25,1 
           of the Jordan algebra J3(O)o. 
           /\25,1 is reflexive/recursive 
           in that its Coxeter/Dynkin diagram 
           is the (infinite) Leech lattice /\24.  
           /\24 gives the Golay Code, and 
           the 196,560 units of /\24, 
           plus 300 = symmetric part of 24x24, plus 24, 
           give 196,884 
           which is the dimension of a representation space 
           of the largest sporadic finite simple group, 
           the Monster; 
 
and 
 
CLIFFORD ALGEBRAS - from which the other structures can be derived, 
                    and which have the periodicity property 
                    Cl(N+8) = Cl(N) x CL(8). 

• Elements of a Discrete Clifford algebra correspond to Basis Elements of a Real Clifford algebra.
• General Elements of a Real Clifford algebra correspond to Superpositions of Basis Elements = Elements of the underlying Discrete Clifford algebra.
  • Volumes of Spaces of Superpositions of some given Sets of Basis Elements correspond to Mass or Charge of Particles or Forces represented by those Basis Elements.

  Volumes of Spaces of Superpositions of other given Sets of Basis Elements correspond to Volume of Physical SpaceTime and Volume of Internal Symmetry Space represented by those Basis Elements.

 The D4-D5-E6-E7-E8 VoDou Physics model has the Geometry of the A-D-E series:

Symmetric Space      Dimension          Physical Interpretation
 
A0=D1                 1 Real           U(1) Electromagnetism
A1=B1=C1              3 Real              SU(2) Weak Force 
A2                    8 Real              SU(3) Color Force 
A3=D3                15 Real                Gravity+Higgs 
D4                   28 Real        Gravity+Higgs+phase/U(1)xSU(2)xSU(3) 
D5 / D4xU(1)          8 Complex               SpaceTime
E6 / D5xU(1)         16 Complex               Fermions
E7 / E6xU(1)         27 Complex          MacroSpace of ManyWorlds
E8 / E7xSU(2)        28 Quaternionic    ? Correlated Macrospace ?  

These structures are motivated by Saul-Paul Sirag's ideas about Weyl groups, A-D-E, and E7, and lead to Jack Sarfatti's Back-Reaction structures in MacroSpace.

Prior to dimensional reduction of spacetime from 8-dimensional to 4-dimensional,

the Lagrangian for the D4-D5-E6-E7-E8 VoDou Physics model is

the Integral over the Cl(1,7) vector 1+7=8-dimensional SpaceTime of

where

• d is the 8-dim covariant derivative
• P is the scalar field
• F is the adjoint Spin(8) curvature
• S' and S are half-spinor fermion spaces
• D is the 8-dim Dirac operator
• GF is the gauge-fixing term
• GG is the ghost term,

plus a topological Pontrjagin term.

The Pontrjagin term represents Instantons in 8-dimensional spacetime that is locally RP1 x S7. Since, after dimensional reduction of spacetime from 8 to 4 dimensions, the Pontrjagin term goes into the Spin(6) conformal gravity sector of the D4-D5-E6-E7-E8 VoDou physics model, it does not go to the SU(3) color force sector. Therefore, the SU(3) color force Sector has no THETA-term and the D4-D5-E6-E7-E8 Vodou Physics model has no theoretical THETA-CP problem.

After dimensional reduction to 4-dimensional spacetime, the S7 of RP1 x S7 is factored by the Hopf fibration S3 -> S7 -> S4 into 4-dimensional spacetime that is locally RP1 x S3, plus a CP2 part related to 4-dimensional Internal Symmetry Space.

Reduction also produces, for each World of the Many-Worlds, a 4-dimensional lattice Spacetime with MacDowell-Mansouri Gravity, a Higgs Mechanism, and a Complex Propagator Phase;

a 4-dimensional lattice Internal Symmetry Space with 8 Color Force Gluons, 3 Weak Force Bosons, and a Photon that live on the links of the lattice Spacetime; and 3 generations of 8 Fermion Particles and 8 Fermion AntiParticles that live on the vertices of the lattice Spacetime.

In terms of a 5-level grading of the E6 Lie algebra, that is, the Graded Lie Alagebra of type e7, we have

E6(-14) =
              8-dim
                                       + 16-dim
            +  R        + so(1,7)      +  iR
                                       + 16-dim
            + 8-dim

with physical interpretation in the Lagrangian of the D4-D5-E6-E7-E8 VoDou Physics Model as:

• 8-dim of g(-2) plus R of g(0) plus 8-dim of g(2), an 8-Complex-dimensional domain plus an R generator of Complex U(1) symmetry, with an 8-real-dimensional Shilov boundary of the form S1 x S7, corresponds to an 8-dimensional spacetime base manifold over which the Lagrangian integral is integrated;
• so(1,7) of g(0), the double cover of the Lorentz group over the Octonions, corresponds to the 28 generators of gauge bosons in the curvature term of the Lagrangian integrand; and
• 16-dim of g(-1) plus iR of g(0) plus 16-dim of g(1), a 16-Complex-dimensional domain, the Complexified Octonion Plane (CxO)P2, plus an iR generator of Complex U(1) symmetry, with a Shilov boundary ( not entirely real, as the 16-Complex-dimensional domain is not of tube type ) that may be regarded as being a bundle made up of a real fibre S1 x S7 over a base space made up of S1 and CP4 ( note that the CP4 has embedded S7 structure ), so that the real fibre S1 x S7 represents 8 first-generation fermion particles in the Dirac spinor term of the Lagrangian integrand, and an S1 x S7 in the base space represents 8 corresponding antiparticles.

Related Graded Lie Algebra structures give:

• a Bohm Quantum Potential from 26-dimensional String Theory;
• a 27-dimensional M-theory of timelike branes in the MacroSpace of the Many-Worlds; and
• a 28-dimensional F-theory of spacelike branes in the MacroSpace of the Many-Worlds.

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