- Start with the Standard Model.
- Now add Gravity.
- Try to Unify the Standard Model with Gravity.
- Supergravity.
- Superstrings.
- The D4-D5-E6-E7-E8 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 stronginteraction with Higgs Vacua, may have two excited energy levels at225 GeV and 173 GeV, above a

D4 is a label for the 28-dim Spin(8) Lie algebra, whose 28-dim adjoint representation gives gauge bosons, which can also be described as pairs of Fermion nearest-neighbors. Here are some reasons why Spin(8) is used. D5 is a label for the 45-dim Spin(1,9) = SL(2,O) Lie algebra, which contains both 28-dim Spin(8) gauge bosons AND Spin(10) / Spin(8) x U(1), whose dimension is 45-28-1=16, the real dimensionality of a bounded complex domain whose Shilov boundary has real dimension 8 has real dimension 8 for 4-dimensional physical spacetime plus 4-dimensional internal symmetry space. E6 is a label for the 78-dim E6 Lie algebra, which contains both the 45-dim Spin(10) Lie algebra AND E6 / Spin(10) x U(1), whose dimension is 78-45-1=32, the real dimensionality of a bounded complex domain whose Shilov boundary has real dimension 16=8+8 for 8 fermion particles and 8 fermion antiparticles.

The "nesting" of D4 = Spin(8) inside D5 = Spin(1,9) = SL(2,O) inside E6 is due to conformal and octonionic structures:

The part of D5 = Spin(1,9) = SL(2,O) that is added to D4 = Spin(8) is the coset space D5 / (D4 x U(1)). It is a complex space corresponding to a bounded complex homogeneous domainwhose Shilov boundary is 8-dimensional. The fermion particle-antiparticle part of E6 that is added to D5 = Spin(1,9) = SL(2,O) is the coset space E6 / (D5 x U(1)). It is a complex space corresponding to a bounded complex homogeneous domainwhose Shilov boundary is 8+8 = 16-dimensional.

If you reduce the original 8-dimensional spacetime into associative 4-dimensional Physical Spacetime and coassociative 4-dimensional Internal Symmetry Space,then, if you look in the original 8-dimensional spacetime at a fermion (First-generation represented by a single octonion)propagating from one vertex to another,there are only 4 possibilities for the same propagationafter dimensional reduction: 1 - the origin and target vertices are bothin the associative 4-dimensional physical spacetime,in which case the propagation is unchanged, and thefermion remains a FIRST generation fermion representedby a single octonion; 2(a) - the origin vertex is in the associative spacetime,and the target vertex in in the Internal Symmetry Space,in which case there must be a new link fromthe original target vertex in the Internal Symmetry Spaceto a new target vertex in the associative spacetime,and a second octonion can be introduced at the originaltarget vertex in connection with the new link,so that the fermion can be regarded after dimensional reductionas a pair of octonions, and therefore as a SECOND generation fermion; 2(b) - the target vertex is in the associative spacetime,and the origin vertex in in the Internal Symmetry Space,in which case there must be a new link tothe original origin vertex in the Internal Symmetry Spacefrom a new origin vertex in the associative spacetime,so that a second octonion can be introduced at the originalorigin vertex in connection with the new link,so that the fermion can be regarded after dimensional reductionas a pair of octonions, and therefore as a SECOND generation fermion;and 3 - both the origin vertex and the target vertexare in the Internal Symmetry Space,in which case there must be a new link tothe original origin vertex in the Internal Symmetry Spacefrom a new origin vertex in the associative spacetime,and a second new link from the original target vertexin the Internal Symmetry Space to a new target vertexin the associative spacetime,so that a second octonion can be introduced at the originalorigin vertex in connection with the first new link,and a third octonion can be introduced at the originaltarget vertex in connection with the second new link,so that the fermion can be regarded after dimensional reductionas a triple of octonions, and therefore as a THIRD generation fermion. As there are no more possibilities, there are no more generations.

E7 is the 133-dim Lie algebra of the coset space E7 / (E6 x U(1)) that describes the MacroSpace of Many-Worlds.

**The D4-D5-E6-E7-E8 VoDou Physics model coset spaces****E7**** / (****E6****x U(1)) and ****E6**** /(****D5**** x U(1)) and****D5**** / (****D4****x U(1)) are ****Conformal****Spaces. ** You can continue the chain to D4/ (D3 x U(1)) where D3 is the15-dimensional Conformal Group whose compact version is Spin(6), andto D3 / (D2x U(1)) where D2 is the 6-dimensional Lorentz Group whose compactversion is Spin(4). Electromagnetism, Gravity,and the ZPF all have in commonthe symmetry of the 15-dimensional D3 Conformal Group whose compactversion is Spin(6), as can be seen by the following structures withD3 Conformal Group symmetry:

- Maxwell's equations of Electromagnetism
- Gravity derived from the Conformal Group using the MacDowell-Mansouri mechanism
- the Quantum Theoretical Hydrogen atom
- the Lie Sphere geometry of SpaceTime Correlations in the Many-Worlds picture

Further, the 12-dimensional Standard Model Lie AlgebraU(1)xSU(2)xSU(3) may be related to the D3 Conformal Group Lie Algebrain the same way that the 12-dimensionalSchrodinger Lie Algebra is related to the D3 Conformal Group LieAlgebra.

In the D4-D5-E6-E7-E8 VoDou Physics model, particle masses and force strengths can be calculated.

For example, the tree-level constituent Truth Quark mass is 130 GeV, and the electromagnetic fine structure constant is 1/137.03608. Click here to seea Summary of Results of D4-D5-E6-E7-E8 model calculationsof Particle Masses and Force Strength Constants.

Particle Masses are calculated byusing:

- ratios of volumes of bounded complex homogeneous domains and their Shilov boundaries; and
- combinatiorial relationships.

Force Strengths are calculated byusing:

- ratios of volumes of bounded complex homogeneous domains and their Shilov boundaries; and
- mass factors for the weak force and gravity, which mass factors have a visualization (arising from e-mail discussion with Dick Andersen): Gauge bosons are visualized as going from a source through a medium to a target:
- The weak force mass factor is related to the Higgs mechanism. The Higgs scalar field absorbs some of the weak bosons as they go through the medium, thus weakening the weak force and producing the weaker effective weak force that is observed by experiments;
- The graviton by itself is long-range and massless, but virtual Planck-mass black holes in spacetime absorb some of the gravitons as they go through the spacetime medium, thus weakening the gravitational force and producing the weaker effective gravitational force that is observed by experiments.

My calculations were motivated by earlierwork of Armand Wyler, and WalterSmilga has, in March 2003, written a paper calculating theelectromagnetic fine structure constant based on the contraction of aspin-1/2 representation of the de Sitter group SO(3,2).

- 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.

WHAT IS A SHILOV BOUNDARY?

Such boundaries have been called Silov and Shilov (two different transliterations from Russian),and are also called characteristic manifolds,Bergman-Shilov boundaries, and (I think) distinguished boundaries, and perhaps other things. They are generally subsets of the topologicalboundaries of bounded complex domains.For example, the bounded complex bidisk, orCartesian product of two disks, each disk in C, is a bounded complex domain in C2. Its Shilov boundary is S1 x S1 , the Cartesian product of two circles.The Shilov boundary can be described as the set (a,b) such that both a and b are on a bounding circle of one of the disks. Its topological boundary is more complicated: the set (a,b) such that either a or b is on a bounding circle of one of the disks. What the Shilov boundary is good for is that it isthe minimal subset of the topological boundary over which you can integrate an analytic functionto reproduce its interior values by the Poisson kernel. Therefore the Shilov boundary is closely related to harmonic functions, Green's functions, etc. There ought to be a good book that discusseslots of examples and is introductory, but if there is, I don't know it. Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains by Hua (Am. Math. Soc., 1979)discusses a lot of examples, but is far from introductory. Also advanced is Geometric Analysis on Symmetric Spaces by Helgason (Am. Math. Soc., 1994).

(There are correspondences betweenSymmetric Spaces and BoundedDomains.)

Harmonic Function Theory by Axler, Bourdon, and Ramey (Springer-Verlag, 1992) is introductory, but does not do a lot of examples. Edward Dunne has some nice WWW pages about related structures, such as Hermitian Symmetric Spaces, E8, F4, and SU(2,2). The Shilov boundaries and related structures are continuum objects.As such, they are not truly fundamental,but they are very useful to enable limited minds like mineto do calculations whose results can be compared with experiments. The Shilov boundaries are used in the D4-D5-E6-E7-E8 model as compact manifolds that represent spacetime, internal symmetry space, and fermion representation space. The volumes of the manifolds are useful in calculations of particle masses and force strength constants. The compact manifold that represents 8-dim spacetime is RP1 x S7, the Shilov boundary of the bounded complex homogeneous domain that corresponds to Spin(1,9) / (Spin(8)xU(1)).

Note that S1/Z2 can be described as an orbifold.

The compact manifold that represents 4-dim spacetime is RP1 x S3, the Shilov boundary of the bounded complex homogeneous domain that corresponds to Spin(6) / (Spin(4)xU(1)). The compact manifold that represents 4-dim internal symmetry spaceis CP2, the Shilov boundary of the bounded complex homogeneous domain that corresponds to SU(3) / (SU(2)xU(1)). The compact manifold that represents the 8-dim fermion representation space is RP1 x S7, the Shilov boundary of the bounded complex homogeneous domain that corresponds to Spin(1,9) / (Spin(8)xU(1)). The manifolds RP1 x S3 and RP1 x S7 are homeomorphic to S1 x S3 and S1 x S7, which are untwisted trivial sphere bundles over S1. The corresponding twisted sphere bundles are the generalized Klein bottles Klein(1,3) Bottle and Klein(1,7) Bottle.

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

Since each of the 3 parts of the model, gauge bosons, spacetime, and fermions, comes from a representation of the 28-dimensional Spin(8) Lie Algebra of the Clifford Algebra Cl(8), you can visualize any part of it as being created from any other part of it, spacetime and fermions being related to each other by Subtle Triality Supersymmetry:

and the (24+4)=28-dim space of gauge bosons being related to the 8 + (8+8) = 24-dim space of spacetime + fermions by the duality between the 24 vertices of a 24-cell and the 24 cells of a 24-cell (or equivalently the 24 vertices of a dual 24-cell):

Therefore everything can be pictured as beingin flux, so to speak, with every part being at the same time both fundamental and derived. If you add to that model-in-flux picture the superpositions of all worlds, or histories, of the Many-Worlds quantum theory, you have the way it should be visualized: a huge superposition of models-in-flux, constrained only by maintaining the relationships among the parts that come from the underlying Octonionic structure. My mind has trouble comprehending that, and it helps my limited mind to have a fixed reference point. How I do that is to FIX (in my mind) the 8-dimensional octonionic spacetime lattice, and therefore the 4-dimensional spacetime lattice that results from dimensional reduction. Then I visualize all the model-in-flux structures as taking place on the spacetime lattice, or really a superposition of as many 8-dim lattices as there are Many-Worlds. Using the FIXED lattice as a reference point lets my limited mind see enough detail so that I can work out calculations of particle masses and force strengths. That is useful and good. HOWEVER, it has lead me sometimes to speak of the FIXED lattice as being THE FIXED SPACETIME LATTICE. That is fundamentally incorrect and bad. The model is better than that.

The natural Clifford algebra structure of the D4-D5-E6-E7-E8 model produces a Many-Worlds Quantum Theory that can also be described by Bohm's Quantum Theory with physical particle structure being described by Compton Radius Vortex structures.

To see how the D4-D5-E6-E7-E8 modelrepresents the Many-Worlds Multiverse MacroSpace, start with theusual configuration space of N copies of spacetime position, in whichthe Multiverse is N copies of all possible spacetime positions. Thengeneralize as follows: Look at each of the N copies separately. Eachcopy in the usual picture is just a position in spacetime, whichposition is in a 4-DIM SPACETIME SPACE. Then note that each positionalso has a 4-dim Internal Symmetry Space in which Standard ModelGauge Bosons are represented, so ADD A 4-DIM INTERNAL SYMMETRY SPACE.Now that we have position and the identity of gauge bosons at theposition, we can generalize further by noting that a fermion particleor antiparticle can be located at each position. Since there are 8first-generation fermion particles, electron; red, blue, green upquark; red, blue, green down quark, neutrino and there are 8corresponding antiparticles, we can generalize by adding an 8-dimrepresentation space for particles and an 8-dim representation spacefor antiparticles, so ADD AN 8-DIM FERMION PARTICLE SPACE and AN8-DIM FERMION ANTIPARTICLE SPACE. (Note that you can representsecond-generation fermions as pairs of first-generation fermions, andyou can represent third-generation fermions as triples offirst-generation fermions, so you don't need more representationspaces for them.) Now, for each of your N particles, you havegeneralized from a 4-dim configuration space to a (4+4) + 8 + 8 =24-DIM GENERALIZED CONFIGURATION SPACE. As jackSarfatti said, "...Deutsch's multiverse really is, is ... a stack of snapshots ... allconnected together by threads ...", so you can let the threadscorrespond to strings, then you have string theory in the space ofpossible outcomes, and you have 24-dim string theory. Note that thisstring theory is in 24-dim, not the critical 26-dim, but it is verynatural to make it a 26-dim string theory, as follows: Each of the(4+4), 8, and 8 dim subspaces of the 24-DIM GENERALIZED CONFIGURATIONSPACE has natural Octonion structure, so it can be considered as the3-dim Octonionic space. The natural algebraic structure for quantumstate spaces is the Jordan algebra structure, so look at the Jordanalgebra of 3-dim Octonionic Space. It is the 27-dimensional space of3x3 Octonionic Hermitian matrices. If you restrict to the traceless3x3 Octonionic Hermitian matrix algebra, you get a26-dimensional space for Unoriented Closed Bosonic String Theory.Closed strings are timelike closed loops, and you can use knownresults from string theory to make a concrete model of the BohmSuperImplicate Order or MacroSpace ofthe Many-Worlds.One concrete result of working out the consequences of the model isthe structure of QuantumConsciousness.

Quantum Information Theory is different from Classical Information Theory. In particular, Cerf and Adami have shown that information theory of quantum computers can give negative conditional entropies for entangled systems. Therefore negative virtual information can be carried by particles, and quantum information processes can be described by particle-antiparticle diagrams much like particle physics diagrams. How does the D4-D5-E6-E7-E8 Model look in terms of Quantum Information? First, look at the Clifford Algebras structure of the D4-D5-E6-E7-E8 model. Then, look at the paper of Calderbank, Rains, Shor, and Sloane, where they say, given the quantum state space C^2^n of n qubits, "... The known quantum codes seemeed to have close connections to a finite group of unitary transformations of C^2^n, known as a Clifford group, ... [containing] all the transformations necessary for encoding and decoding quantum codes. It is also the group generated by fault-tolerant bitwise operations performed on qubits that are encoded by certain quantum codes. ..." Now, look at the example of Steane of the Quantum Reed-Muller code [[ 256, 0, 24 ]], which maps a quantum state space of 256 qubits into 256 qubits, correcting [(24-1)/2] = 11 errors, and detecting 24/2 = 12 errors. Let C(n,t) = n! / t! (n-t)! Then [[ 256, 0, 24 ]] is of the form [[ 2^n, 2^n - C(n,t) - 2 SUM(0 k t-1) C(n,k), 2^t + 2^(t-1) ]] [[ 256, 256 - (1+8+28+56+70+56+28+8+1), 16 + 8 ]] The quantum code [[ 256, 0, 24 ]] can be constructed from the two classical Reed-Muller codes (256, 163, 32) and (256, 93, 16), which are classical Reed-Muller codes of orders 4 and 3, and are dual to each other. Due to the nested structure of Reed-Muller codes, they contain the Reed-Muller codes of orders 2, 1, and 0 : Classical Reed-Muller Codes Order of Length 2^8 = 256 ( 256, 1+8+28+56+70+56+28+8+1, 1 ) 8( 256, 1+8+28+56+70+56+28+8, 2 ) 7 ( 256, 1+8+28+56+70+56+28, 4 ) 6 ( 256, 1+8+28+56+70+56, 8 ) 5 ( 256, 1+8+28+56+70, 16 ) 4( 256, 1+8+28+56, 32 ) 3 ( 256, 1+8+28, 64 ) 2 ( 256, 1+8, 128 ) 1 ( 256, 1, 256 ) 0 In the Lagrangian of the D4-D5-E6 physics model: the Higgs scalar prior to dimensional reduction corresponds to the 0th order classical Reed-Muller code (256, 1, 256), which is the classical repetition code; the 8-dimensional vector spacetime prior to dimensional reduction corresponds to non-0th-order part of the 1st order classical Reed-Muller code (256, 9, 128), which is dual to the 6th order classical Reed-Muller code (256, 247, 4), which is the extended Hamming code, extended from the binary Hamming code (255, 247, 3), which is dual to the simplex code (255, 8, 128) ; the 28-dimensional bivector adjoint gauge boson space prior to dimensional reduction corresponds to the non-1st-order part of the 2nd order classical Reed-Muller code (256, 37, 64) ; and the 8 first generation fermion particles and 8 first generation fermion antiparticles of the 16-dimensional full spinor representation of the 256-dimensional Cl(0,8) Clifford algebra corresponds to the distance of the classical Reed-Muller code (256, 93, 16), as well as to the square root of 256 = 16x16, and to the 16-dimensional Barnes-Wall lattice. The other 8 of the 16+8 = 24 distance of the quantum Reed-Muller code [[ 256, 0, 24 ]]corresponds to the 8-dimensional vector spacetime, and to the 8-dimensional E8 lattice. The total 24 distance of the quantum Reed-Muller code [[ 256, 0, 24 ]]

corresponds to the 24-dimensional Leech lattice,

and to the classical extended Golay code (24, 12, 8) .

The Spin(1,9) D5 of the VoDouD4-D5-E6-E7-E8 physics model is related to

John Baez,in a 2002 post to the sci.physics.research thread Coleman-Mandulatheorem, says:

- 1 - "... The powerful Coleman-Mandula theorem states that within the framework of Lie algebras, there is no way to unify gravity with the gauge symmetries which describe the strong and electroweak interactions. ...".
- 2 - "... supersymmetry defeats "the powerful Coleman-Mandula theorem" ...".

I agree with statement 2, since **supersymmetric Lie algebramodels evade Coleman-Mandula by adding fermionic generators** inthe unified theory, thus expanding the unification algebra from Lieto superLie, and therefore considering the 4-dim structure gravity xstandard model (trivially commuting in a not-very-unified-way) to beembedded (presumably at some higher energy level) in a unifyingsuperLie algebra with respect to 4-dim spacetime,

However, I disagree with statement 1, because it does not takeinto account unification by

combining Gravity( including the 4-dimensional Lorentz Lie algebra Spin(1,3) )and the Standard ModelSU(3)xSU(2)xU(1) acting on a 4-dimensional Physical SpaceTime

into a unified Lie algebra acting on a higher-dimensional SpaceTime, as is done by my D4-D5-E6-E7-E8 VoDou Physics model in whichthe unification SpaceTime is 8-dimensionaland the unified Lie algebra is28-dimensional Spin(1,7), which isthe Lorentz Lie algebra for that 8-dimensional spacetime.

In other words, in my view, Coleman-Mandula does not any more (orless) support supersymmetry than it supports my D4-D5-E6-E7-E8VoDou Physics model.

The best reference that I know for **a proof of theColeman-Mandula theorem** is Steven Weinberg's book The QuantumTheory of Fields, Vol. III (Cambridge 2000). Beginning at page12,

Weinberg gives "... a proof of the celebrated theorem of Coleman and Mandula ... Phys. Rev. 159, 1251 (1967) ... thatthe only possible Lie algebra(as opposed to super-algebra) of symmetry generators consists of

- the generators Pu and Juv of
translations and homogeneous Lorentz transformations... in theories with only massless particles there is also the possibility that in addition to the generators Pu and Juv there are additional generators D and Ku that fill out the Lie algebra ofthe conformal group... ,- together with possible
internal symmetry generators, which commute with Pu and Juvand act on physical states by multiplying them with spin-independent, momentum-independent Hermitian matrices.By 'symmetry generators' here is meant any Hermitian operators:

- that commute with the S-matrix;
- whose commutators are also symmetry generators;
- which take one-particle states into one-article states; and
- whose action on multiparticle states is the direct sum of their action on one-particle states ... .
A further technical requirement will be ... needed ... Coleman and Mandula made the 'ugly technical assumption' that the kernels

Aa(p'p) ...[of] any symmetry generator ... are distributions, which means that each can contain at most a finite number Da of derivatives of delta4(p' - p). To put this another way, each symmetry generator Aa is assumed to act on one-particle states as a polynomial of order Da in the [partial] derivatives d/dpu with matrix coefficients at this point that are allowed to depend on momentum and spin.Apart from the general principles of relativistic quantum mechanics ... the only other assumptions needed in this proof are:

- Assumption 1 For any M there are only a finite number of particle types with mass less than M.
- Assumption 2 Any two-particle state undergoes some reaction at almost all energies (that is, at all energies except perhaps an isolated set.
- Assumption 3 The amplitude for elastic two-body scattering are analytic functions of the scattering angle at almost all energies and angles. ...

- ... Strictly speaking, this assumption is not satisfied in theories with infrared divergences such as quantum electrodynamics, where ... the S-matrix element for any one scattering process involving charged particles actually vanishes, except for elastic forward scattering. In Abelian gauge theories like electrodynamics this problem can be avoided by applying the Comeman-Mandula theorem to the theory with a fictitious gauge boson mass, and then working only with 'infrared-safe' quantities like masses and suitable integrated cross-sections that are finite in the limit of zero gauge boson mass. There is no problem in non-Abelian gauge theories like quantum chromodynamics, in which all massless particle are trapped -- symmetries if unbroken would only govern S-matrix elements for gauge-neutral bound states, like the mesons and baryons in quantum chromodynamics. As far as I know, the Coleman-Mandula theorem has not been proved for non-Abelian gauge theories with untrapped massless particles, like quantum chromodynamics with many quark flavors. ...
... It is not necessary to assume that the S-matrix is governed by a local quantum field theory. ...".

Also, at pages 382-384, Steven Weinberg says:

"... The proof of the Coleman-Mandula theorem ... makes it clear that the list of possible bosonic symmetry generators is essentially the same in d > 2 spacetime dimensions as in four spacetime dimensions: in an S-matrix theory of particles, there are only the momentum d-vector Pu, a Lorentz generator Juv = -Jvu ( with u and v here running over the values 1, 2, ... , d-1, 0 ), and various Lorentz scalar 'charges' ...The anticommutators of the fermionic symmetry generators with each other are bosonic symmetry generators, and therefore must be a linear combination of the Pu, Juv, and various conserved scalars. ...

We will first prove that the general fermionic symmetry generator must transform according to the fundamental spinor representations of the Lorentz group ... and not in higher spinor representations, such as those obtained by adding vector indices to a spinor. ... Here we will use an argument of Nahm, which ... applies in any number of dimensions.

Since the Lorentz transform of any fermionic symmetry generator is another fermionic symmetry generator, the fermionic symmetry generators furnish a representation of the homogeneous Lorentz group ... or, strictly speaking, of its covering group Spin(d-1,1). ... consider the anticommutator {Q,Q*} of any fermionic symmetry generator Q with its Hermitian adjoint. According to the Coleman-Mandula theorem, it is at most a linear combination of Pu, Juv, and scalars. ...[some of Weinberg's details are here omitted by me in this quote]... all the ...[commutation coefficients of Q with the Juv].... are +/- 1/2. The only irreducible representations of the homogeneous Lorentz group with all ...[commutation coefficients of Q with the Juv].... equal to +/- 1/2 are the fundamental spinor representations, so Q must belong to some direct sum of these representations.

We can also use this approach to show that the fermionic generators Q all commute with the d-momentum Puv. ...".

in which the unification SpaceTime is8-dimensional and **the unified Lie algebra is**28-dimensional Spin(1,7), which is **the Lorentz Lie algebra forthat 8-dimensional spacetime**, and

in which **the fundamental first-generation fermions arerepresented by the 8+8 = 16 spinor representations ofSpin(1,7)**.

Further, at pages 2 and 397-398, Steven Weinberg says:

"... the Coleman-Mandula theorem does not apply ...[to]... theories that involve extended objects...... p-Branes ... stable extended objects ... can carry conserved bosonic quantities other than those allowed by the Coleman-Mandula theorem ...".

and that the Coleman-Mandula theorem does not apply at the Plancklength where 4-dimensional PhysicalSpaceTime breaks down into afundamental HyperDiamond lattice structure and combines with4-dimensional Internal Symmetry Spaceto form the unified 8-dimensional SpaceTime and itsfundamental E8 lattice structure.

The mathematical structures used in the D4-D5-E6-E7-E8 model TOE are much like the structures of: IFA - 256-dimensional Cl(0,8) Clifford Algebra; Wei Qi - Spin(8) vector HyperDiamond lattice; I Ching - Spin(8) bivector gauge bosons; Tai Hsuan Ching - Spin(8) spinor fermions; Tarot - 78-dimensional E6.

Maxwell said "... cubic surfaces! Bythrees and nines Draw round his camp your seven-and-twentylines The seal of Solomon inthree dimensions. ... we the form may trace Of himwhose soul, too large for vulgar space, In n dimensions flourishedunrestricted. ...". The symmetry of the 27-lineconfiguration is the same as that of the Weylgroup of the E6 Lie algebra that is fundamental to the D4-D5-E6-E7-E8physics model.

- Our 4-dimensional Physical SpaceTime;
- 4-dimensional Internal Symmetry Space;
- SU(3)xSU(2)xU(1) Gauge Bosons, Gravity, the Higgs Mechanism, and Complex Propagator Phase;
- 3 Generations of Musaka/Ganesha Fundamental Elementary Fermion Particles.

The Musaka/Ganesha Particles are NOTthe Gaja/Ganesha Physical Electrons, Neutrinos, and Quarks, but arethe Musaka/Ganesha Virtual Sea and Valence Particles in theGaja/Ganesha Compton Vortices that are the Physical Electrons,Neutrinos, and Quarks. The Ganeshaterminology is based on works of Sidharth,who described Physical Elementary Particles as ComptonVortices.

Click Here to see my viewof the relationships among

Musaka/Ganesha FundamentalElementary Particles,

Gaja/Ganesha PhysicalCompton Vortex Elementary Particles,

GravitoEM Static RegionGaja/Ganesha ComptonVortex Phenomena, and

GravitoEM Induction RegionGaja/Ganesha ComptonVortex Phenomena.

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; HYPERDIAMOND LATTICE - Lattice structures include the D4 lattice that corresponds to "integral" quaternions, and the E8 lattice that 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).

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 MacroSpace Geometry+Algebra

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

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-dimensionalspacetime that is locally R8, so that the Instantons have as boundarythe 7-sphere S7. Since, after dimensional reduction of spacetime from8 to 4 dimensions, the Pontrjagin term goes into the Spin(6)conformal gravity sector of the D4-D5-E6 physics model, it does notgo to the SU(3) color force sector. Therefore, the SU(3) color forceSector has no THETA-term and the D4-D5-E6 physics model has notheoretical THETA-CP problem.

After dimensional reduction to 4-dimensional spacetime, the S7Instanton boundary is factored by the Hopf fibration S3 -} S7 -} S4into an Instanton with S3 boundary in 4-dimensional spacetime that islocally R4, plus an S4 part related to 4-dimensional InternalSymmetry Space.

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.

The D4-D5-E6-E7-E8 VoDou Physicsmodel has an effective low-energy Infrared cutoff at the scale ofour Universe, and also an effectivehigh-energy HyperDiamond Feynman Checkerboardlattice Ultraviolet cutoff at the Planckenergy scale.

With respect to an **Ultraviolet Cutoff**, Richard Feynmansays, in his book QED (Princeton 1985, 1988), at page 129:

"... perhaps the idea that two points can be infinitely close together is wrong ... If we make the minimum possible distance between two points as small as 10^(-100) centimeters ... the infinities disappear ... but other inconsitencies arise, such as the total probability of an event adds up to slightly more or less than 100%, or we get negative energies in infinitesimal amounts. It has been suggested that these inconsistencies arise because we haven't taken into account the effects of gravity - which ... become important at distances of 10^(-33) cm. ...".

The 10^(-33) cm distance cited by Feynman is equivalent to aPlanck energyscale (10^19 GeV) effective high-energy HyperDiamondFeynman Checkerboard lattice UltraViolet cutoff.

With respect to an **Infrared Cutoff**, Sidney Coleman'sHarvard lectures, plus Robert Brandenburger's homework solutions, ascompiled by Robin Ticciati of Maharishi University says, in his bookQuantum Field Theory for Mathematicians (Cambridge 1999), at pages590, 589, 595:

"... the infrared blowup is effectively renormalized by the bremsstrallung contribution which slips into any experiment due to the finite resolution of the apparatus ... Lorentz invariance is broken by the apparatus, and this is reflected in every aspect of the cut-off theory .. Faddeev-Popov quantization must be supplemented by an infrared regularization which either breaks gauge invariance with a photon mass or breaks Lorentz invariance with a momentum cut-off ... infrared divergences ... are always eliminated when we estimate the finite resolution of the apparatus and compute the inclusive differential cross section. In QED, this means adding to the main process any indistinguishable bremsstrahlung processes. ...".

Some of my discussions with JackSarfatti have involved illustrative examples of Universe-scaleInfrared cutoff phenomenology, such as relating the minimum energy ofa photon or neutrino of any otherwave/particle) to the scale of ourUniverse:

If E = h / L , where L is wavelength, and if you take L to bethe scale of ourUniverse L = 1 / H then,since h = 3.6 x 10^(-21) MeV sec (from PDG2002 Review of Particle Physics) and H = 10^(-28) cm^(-1),then

E = h / L = h H = 3.6 x 10^(-49) MeV sec / cm = 3.6 x 10^(-43) eV sec / cmand, multiplying by c = 3 x 10^10 cm / sec to get rid of the sec / cm ,

h H c = h c / L(now )= 10^(-32) eV is the minimum energy of any particle, including the energy of the lightest boson (the photon) or fermion (the neutrino) at the present stage in the evolution of our universe when L(now) = 10^28 cm.

Even if you go back to the End of Inflationwhen the scale of our universe was (veryroughly) on the order of 10 cm it seems that the scale of ouruniverse L(InflationEnd) = 10 cm = 10^(-27) L(now) we would have

E = h / L(InflationEnd) = 10^(-32) eV / 10^(-27) = 10^(-5) eV so that the minimum neutrino or photon mass, due to the effective Infrared Cutoff of the finite scale of our Universe at the time of the End of Inflation, would be only 10^(-5) eV.

......