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Here is an Introduction to VoDou Physics.

Here are SESAPS 2002 and a December 2001 overview of D4-D5-E6-E7-E8 VoDou Physics and Fermion Masses, including Quark Masses and the Proton/Electron Mass Ratio.

The Truth Quark, through its strong interaction with Higgs Vacua, may have two excited energy levels at 225 GeV and 173 GeV, above a

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.

Cl(8N) = 2^(8N) == Cl(8) x...Ntimes...x Cl(8) = (2^8)^N2^8 = 256 =16x 16 = 182856 70 56 28 814 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.

Since my D4-D5-E6-E7-E8 Physics
Model is based on the Cl(8)
Clifford Algebra that is related to the
16x16 = 256 Odu of IFA, I think that my
physics model is really in fact **VoDou****
Physics**, so I sometimes call it that.

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

The Geometry of the Super
Implicate Order, or MacroSpace,
can be described at the Nearest Neighbor level by the 27-complex-dimensional
symmetric space E7 / E6xU(1), where String
Theory in a 26-dimensional subspace can be used to describe
Sarfatti Back-Reaction and **Quantum
Consciousness**.

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 . ...".

- Start with the Standard Model.
- Now add Gravity.
- Try to Unify the Standard Model with Gravity.
- Supergravity.
- Superstrings.
- The D4-D5-E6-E7-E8 VoDou Physics Model.
- 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.
- Building Blocks - Geometry - Lagrangian
- real hyperfinite II1 von Neumann algebra factor

The Truth Quark, through its strong interaction with Higgs Vacua, may have two excited energy levels at 225 GeV and 173 GeV, above a

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

which is very close to

the ratio of the geometric part of the Weak Force Strength to the Electromagnetic Fine Structure Constant is 0.253477 / ( 1 / 137.03608 ) = 34.7355.

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

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.

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. ...".

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

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.

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, ofSLAC, says (in hep-ex/0001012): "... The sociologists of science would say thatour 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 thatsupersymmetry 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.). ...".

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.

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 discussedhere; 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.

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 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**.

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)
- 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.

Merging Physical SpaceTime with Internal Symmetry Space, to get aHyperSpace 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 relevant Complex Structure can be seen in such physical concepts as Momentum Space, Position-Momentum Complementarity, Type IV(2) Domains, Hyperspace, Black Holes, Wavelets, and Conformal SpaceTime.

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.

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 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.

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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