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VoDou Physics and 130GeV Truth Quark

My D4-D5-E6-E7-E8 VoDouPhysics Model meets Einstein'sCriterion:

"... a theorem which at present can notbe based upon anything more than upon a faith in the simplicity,i.e., intelligibility, of nature: there are no arbitraryconstants ... that is to say, nature is so constituted that it ispossible logically to lay down such strongly determined laws thatwithin these laws only rationally completely determined constantsoccur (not constants, therefore, whose numerical value could bechanged without destroying the theory). ...".

Here is an Introduction to VoDouPhysics

Here are SESAPS 2002 and aDecember 2001 overview of D4-D5-E6-E7-E8 VoDou Physics and FermionMasses, including Quark Masses and the Proton/Electron MassRatio.


The Truth Quark, through its stronginteraction with Higgs Vacua, may have two excited energy levels at225 GeV and 173 GeV, above a ground state at 130 GeV. The 173GeV excited state may exist due to appearance of a Planck-energyvaccum with < phi_vac2 > = 10^19 GeV in addition to thelow-energy Standard Model vacuum with < phi_vac1 > = 252GeV.

Here are 26-dim Strings asQuantum Theory, 27-dim M-theory of Timelike Brane-Universes, and28-dim F-theory of Spacelike Brane-Universes.

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 termNote that all these parts, including the spinors and Dirac operator,are natural parts of the Clifford algebra and are NOT ad hocadditions-put-in-by-hand.The 8-dim spacetime breaks down into4-dim Physical SpaceTime that looks topologically like S1 x S3plus4-dim Internal Symmetry Space that looks topologically like CP2.The CP2 = SU(3) / U(2) is a representation space onwhich the color SU(3) acts globallyand the electroweak U(2) = SU(2)xU(1) acts locally.S1 x S3 Physical SpaceTime is the Shilov boundary of the bounded domain corresponding tothe 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 thatof the 15-dim conformal group Spin(2,4) = SU(2,2)and you get Gravity and the Higgs mechanism by gauging the conformal group asdescribed by Mohapatra in section 14.6 of his bookUnification 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 statethat the Truth quark mass is about 170 GeV,butI think that their analyses are wrong and that a correctanalysis of the same data is consistent witha Truth quark mass of about 130 GeV as my model predicted.Details of my comparison of the data analyses are onthis 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 CliffordAlgebra Foundation of VoDouPhysics:

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-Worlds12-SU(3)xSU(2)xU(1) 4 8

(The Lepton-Quark image to the above left isadapted from aFermilab web page.)

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

represents:

Together, the 24-cell plus its dual24-cell correspond to the 48 root vector vertices of the52-dimensional exceptional Lie algebra F4., and F4can be thought of as the Real Part of the 52+26 = 78-dimensional Liealgebra E6 of the D4-D5-E6-E7-E8 VoDou Physics model.


Since my D4-D5-E6-E7-E8 PhysicsModel is based on the Cl(8)Clifford Algebra that is related to the16x16 = 256 Odu of IFA, I think that myphysics model is really in fact VoDouPhysics, so I sometimes call it that.

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

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


In Phys. Lett. 149B (1984) 117, Green amd Schwarz showed superstringtheory to be anomaly-free. RichardFeynman said, in Davies and Brown, Superstrings, Cambridge1988, pp. 194-195):

"... I do feel strongly that this is nonsense! ... I thinkall this superstring stuff iscrazy and is in the wrong direction. ... I don't like it that they'renot calculating anything. ... why are themasses of the various particles such as quarks what theyare? All these numbers ... have no explanations in thesestring theories - absolutely none! ... "

I have agreed with Feynman's opinion about superstrings as a modelof physics of particles and forces in 4-dimensional spacetime since1984, when I began to construct a TOE(Theory Of Everything) physics model, the D4-D5-E6-E7-E8 PhysicsModel, in which I can calculate particlemasses and force strengths, including the

Truth Quark mass of 130GeV.

The model can be formulated as a FeynmanCheckerboard on a discrete HyperDiamondlattice spacetime with a Many-WorldsQuantum Theory.

The Geometry of the SuperImplicate Order, or MacroSpace,can be described at the Nearest Neighbor level by the 27-complex-dimensionalsymmetric space E7 / E6xU(1), where StringTheory in a 26-dimensional subspace can be used to describeSarfatti Back-Reaction and QuantumConsciousness.


I sometimes use some other names for myVoDouPhysics model:

The Planck length is the fundamental lattice link scale inthe D4-D5-E6-E7-E8 VoDou Physicsmodel.

According to JohnC. 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 anymathematics, but would like to see a Theatrical Version of theIdeas.


Here is a More DetailedOutline

( Here is someHistory )
Here is an Introduction to VoDouPhysics.


The Truth Quark, through its stronginteraction with Higgs Vacua, may have two excited energy levels at225 GeV and 173 GeV, above a ground state at 130 GeV. The 173GeV excited state may exist due to appearance of a Planck-energyvaccum with < phi_vac2 > = 10^19 GeV in addition to thelow-energy Standard Model vacuum with < phi_vac1 > = 252GeV.
This Link goes to a Summaryof Results of D4-D5-E6-E7-E8 model calculations of Particle Massesand Force Strength Constants, some of which are shown in thischart:

Here is how the mass ratios work:

It is interesting that

the ratio of the sum of the masses of the weak bosons W+, W-,and W0 to the sum of the masses of the first generation fermions is259.031 GeV / 7.508 GeV = 34.5007

which is very close to

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

 

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 visualizeit.

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.  
The Discrete HyperDiamond GeneralizedFeynman Checkerboard and ContinuousManifolds are related by QuantumSuperposition:
 
 

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 conciselysummarized, including the standard model Lagrangian, spontaneoussymmetry breaking, the reexpression of the Lagrangian in terms ofmass 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, becauseit 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.). ...".

 

The D4-D5-E6-E7-E8 VoDou Physics model has a subtletriality supersymmetry that is consistent with experimentand useful in theory.

 


 

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 bitsand 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 PhysicsModel

 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-cellplus   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 TrialitySupersymmetry:

 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, andthe dimension of each of the 8 fermion particles is 7/2, so thatthe 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 andthe 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 fieldthat can produce gravity by the MacDowell-Mansouri mechanism. 

Note that both Gravity and the Standard Model forces are requiredfor the cancellations that produce the ultraviolet finiteness that isuseful in the Sakharov Zero PointFluctuation model of gravity.

The Conformal Structure of theD4-D5-E6-E7-E8 model includes the symmetry of ConformalGroup Dilations, which transform the scale oflength/energy/mass, which gives the scale transformation propertiesneeded for Renormalization.

   
 

Dimensional Reduction of8-dim Spacetime

into 4-dim Internal Symmetry Spaceand 4-dim Physical 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 canmark off any given moment in a timestreamababab|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 requiredso that a choice of moment does not change physics,associativity is required for physical spacetime,which must therefore be reduced from 8-dim octonionto 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 fromdifferent points of view:

 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.  
The Discrete HyperDiamond GeneralizedFeynman Checkerboard and ContinuousManifolds are related by QuantumSuperposition:
 

In the D4-D5-E6-E7-E8 VoDou Physics model, there are twotypes of structures that might be regarded asHyperSpace:

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 ComplexPhysical 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 physicalconcepts as Momentum Space,Position-MomentumComplementarity, Type IV(2)Domains, Hyperspace,Black Holes, Wavelets,and ConformalSpaceTime.


 

The Basic Building Blocks of the D4-D5-E6-E7-E8VoDou 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). 
The Discrete HyperDiamond GeneralizedFeynman Checkerboard and ContinuousManifolds are related by QuantumSuperposition:

 

 The D4-D5-E6-E7-E8 VoDouPhysics model has the Geometry of the A-D-Eseries:

Symmetric Space      Dimension          Physical Interpretation A0=D1                 1 Real           U(1) ElectromagnetismA1=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               SpaceTimeE6 / D5xU(1)         16 Complex               FermionsE7 / E6xU(1)         27 Complex          MacroSpace of ManyWorldsE8 / E7xSU(2)        28 Quaternionic    ? Correlated Macrospace ?  

These structures are motivated by Saul-PaulSirag's ideas about Weyl groups, A-D-E, and E7, and lead toJack Sarfatti's Back-Reaction structuresin MacroSpace.

Prior to dimensional reduction of spacetime from 8-dimensional to4-dimensional,

the Lagrangianfor the D4-D5-E6-E7-E8 VoDou Physics model is

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

dd P' /\ * dd P + F /\ *F + S' D S + GF + GG

where

plus a topological Pontrjagin term.

The Pontrjagin term represents Instantons in 8-dimensionalspacetime that is locally RP1 x S7. Since, after dimensionalreduction of spacetime from 8 to 4 dimensions, the Pontrjagin termgoes into the Spin(6) conformal gravity sector of the D4-D5-E6-E7-E8VoDou 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 theD4-D5-E6-E7-E8 Vodou Physics model has no theoretical THETA-CPproblem.

After dimensional reduction to 4-dimensionalspacetime, the S7 of RP1 x S7 is factored by the Hopf fibration S3-> S7 -> S4 into 4-dimensional spacetime that is locally RP1 xS3, plus a CP2 part related to 4-dimensional Internal SymmetrySpace.

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

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

In terms of a 5-level grading of the E6 Lie algebra, that is, theGraded 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 Lagrangianof the D4-D5-E6-E7-E8 VoDou Physics Modelas:

 

Related Graded Lie Algebra structuresgive:

 


 

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