Clifford Algebras and theD4-D5-E6-E7-E8 Vodou Physics Model

Dirac Operator

Cl(1,3) from Cl(1,2)and Cl(1,1)

Minkowski Spacetime

is an important structure in many physics theories, includingthe D4-D5-E6-E7-E8 VoDou Physics Model.There are two Clifford Algebras for Minkowski Spacetime:

Cl(3,1) = M(4,R) = 4x4 Matrix Algebra over the Real Numbers for signature (3,1); and

Cl(1,3) = M(2,Q) = 2x2 Matrix Algebra over the Quaternions for signature (1,3).

They are not isomorphic. The key to understanding how they work isthe isomorphism

Q x Q = M(4,R)

of the 4x4 Matrix Algebra over the Real Numbers with the tensorproduct of two Quaternionic spaces.

To see how that isomorphism works, let { 1, i, j, k } be a basisfor Q. Since Quaternionic Multiplication is not commutative, you haveto distinguish between Left-Multiplication andRight-Multiplication.

Let { 1, iL, jL, kL } denote Left-Multiplication by 1, i, j, andk, and let { 1, iR, jR, kR } denote Right-Multiplication by 1, i, j,and k. Left- and Right-Multiplication by 1 is the same, but by 1, j,and k it is different.

Let QL denote the Left-Multiplication actions whose basis is { 1,iL, jL, kL } and let QR denote the Right-Multiplication actions whosebasis is { 1, iR, jR, kR }. Note that QL and QR have in common thereal axis spanned by { 1 }.

Since QL is isomorphic to QR and both are isomorphic to Q,describe the tensor product Q x Q by the tensor product QL x QR, sothat the basis elements of Q x Q are formed from

{ 1, iL, jL, kL } x { 1, iR, jR, kR }

or, in other words:

{ 1,

iL, jL, kL,

iR, jR, kR,

iL iR, iL jR, iL kR,

jL iR, jL jR, jL kR,

kL iR, kL jR, kL kR }

These 16 elements are isomorphic to M(4,R), or, in GeoffreyDixon's terminology, to the full adjoint action QA of theQuaternions. M(4,R) has graded structure 1 4 6 4 1 and is generatedby the 4 vector basis elements, which can be written as

1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 -1 0 0 -1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 -1 0 0-1 0 0 0

In this representation, the scalar identity of M(4,R) is

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

and the pseudoscalar of M(4,R) is

0 0 -1 0 0 0 0 1 1 0 0 0 0 -1 0 0

However, Q x Q = M(4,R) = Cl(3,1) is not isomorphic to16-dimensional M(2,Q) = Cl(1,3).

M(2,Q) = Cl(1,3) is generated by its 4 vector generator basiselements

1 0 0 -1 0 i i 0 0 j j 0 0 k k 0

In this representation, the scalar identity of M(2,Q) is

1 0 0 1

and the pseudoscalar of M(2,Q) is

0 1-1 0

Even though the full Minkowski Clifford Algebras Cl(3,1)and Cl(1,3) are not isomorphic, their bivector Lie algebras Spin(3,1)and Spin(1,3) are isomorphic to each other and to the Lie AlgebraSL(2,C) of Lorentz transformations.

If you tensor Cl(3,1) with itself, you get M(4,R) x M(4,R) =M(4x4,RxR) = (since RxR = R) = M(16,R) = Cl(0,8), which is the basisof the D4-D5-E6-E7-E8 Vodou PhysicsModel and is the basis of the 8-foldPeriodicity of Real Clifford Algebras and of Homotopy.

If you tensor Cl(1,3) with itself, you get M(2,Q) x M(2,Q) =M(2x2,QxQ) = (since QxQ = M(4,R)) = M4,M(4,R) = M(16,R) = Cl(0,8),which is also the basis of theD4-D5-E6-E7-E8 VoDou Physics Model and is also the basis ofthe 8-fold Periodicity of Real CliffordAlgebras and of Homotopy.

Since Cl(3,1)and Cl(1,3) provide two Clifford Paths to the D4-D5-E6-E7-E8 VoDouPhysics Model,

there is the question:

Which Signatureis more Physically Realistic ?

John Baez, on theusenetgroup sci.physics.research, said:

"... fermionsare quaternionic and bosons are real ...".

Note that Spin(-+++) = Spin(+---) = SL(2,C), so that you can't usethe Spin group to tell which is theright one, so look at the Clifford algebra from which the Spin groupis derived. The Clifford algebras of candidate physical spacetimesare:

Cl(+---) = Cl(3,1) = M(4,R) the 4x4 real matrix algebra

Cl(-+++) = Cl(1,3) = M(2,Q) the 2x2 quaternionic matrixalgebra.

Both of them are Clifford algebras with a real 4-dim vector space,and graded structure

1 4 6 4 1
• The grade-0 1 is the scalar space (like Higgs boson).
• The grade-1 4 is the vector space (like spacetime).
• The grade-2 6 is the bivector space (like gauge bosons).

Since all these spaces are real, the bosons in both of them arereal.

However, the fermions should correspond to half-spinors,with the fermion particles being +half-spinors and the fermionantiparticles being mirror image -half-spinors.

For Cl(+---) = Cl(3,1), the full spinor space (minimalideal of the Clifford algebra) is 1x4 real column of the 4x4 realmatrix and each half-spinor space is 1x2 real column, so a fermionhalf-spinor would be real.

For Cl(-+++) = Cl(1,3), the full spinor space (minimalideal of the Clifford algebra) is 1x2 quaternionic column of the2x2 quaternionic matrix and each half-spinor space is 1x1 quaternioncolumn, or, in other words, afermion is a quaternion.

Therefore, if "... fermionsare quaternionic and bosons are real ..." then

the Clifford algebra of spacetime is Cl(-+++) = Cl(1,3)and the signature ofphysical spacetime is -+++.

Some interesting structure of Cl(1,3) is shown by

modifying the construction of Minkowski Cl(3,1) = R(4) fromCl(2,0) = R(2) and Cl(1,1) = R(2) of Dennis W. Marks, Larry J.Arbuckle, James Baugh, David Ritz Finkelstein, Jason Looper, andHeinrich Saller as described by DennisMarks in his talk at SESAPS 2002 at Auburn

to a construction of Minkowski Cl(1,3) = Q(2) from Cl(0,2) = Q andCl(1,1) = R(2).

In the notation below:

• t, z, x, and y correspond to 1 time and 3 spatial coordinates;
• Ih corresponds to the quantum i of i hbar for the Dirac equation in Cl(0,2) = Q in which it is a pseudoscalar that is a bivector like the area hbar so that Ih must be combined with a Hodge dual operation in order to act as a scalar multiple;
• Ic not only corresponds to the quantum i of i hbar for the Dirac equation in Cl(1,1) = R(2) but also corresponds to the spacetime i of ict in Cl(1,1) = R(2) in which it is a pseudoscalar bivector that takes vectors like x and t into vectors so that Ic can act directly as a space/time conversion factor;
• Ich corresponds to the gamma-5 pseudoscalar imaginary that lives in Cl(1,3) = Q(2)
• in which it is a pseudoscalar 4-vector that takes area bivectors( like the hbar of the Dirac equation in Cl(1,3)) into area bivectors so that Ich can act directly as quantum Dirac equation factor and
• in which it as a pseudoscalar 4-vector takes vectors like x, t, z, and y into trivectors so that Ich must be combined with a Hodge dual operation in order to act as a scalar multiple;
• and 1, i, j, and k denote a basis for the quaternions Q.
Cl(0,2) = Q

Note that the quaternions are natural variablesfor a Diracequation in 2-dimensional Eucliudean spacewith Planckconstant factor Ih.

basis (1x1 quaternion):  1        i         j         k                        z         y         Ih     2x2 complex Pauli representation basis:            1  0     0  1      1  0      0  i            0  1     1  0      0 -1     -i  0Cl(1,1) = R(2)

Note that (1,1) signature is that of1+1=2-dimspacetime with space/time conversionfactor Ic which also acts as Planckconstant factor Ih for a Diracequation in 2-dimensional Minkowski space.

basis (2x2 real):  1  0     0 -1      1  0      0  1 0  1     1  0      0 -1      1  0           t         x         Ic          Cl(1,1) x Cl(0,2) = Cl(1,3) = Q(2)

Note that the 2-dim Ic and Ih factors map tomiddle-grade bivectors while the factor Ich that is effective in4-dim is the pseudoscalar and that Ich acts as both space/timeconversion factor and Planckconstant factor.

basis 1  0     0 -1      1  0      0  1 0  1     1  0      0 -1      1  0 i  0     0 -i      i  0      0  i 0  i     i  0      0 -i      i  0 j  0     0 -j      j  0      0  j 0  j     j  0      0 -j      j  0 k  0     0 -k      k  0      0  k 0  k     k  0      0 -k      k  0or, taking into account 1 2 1    1ijk1 2 1    R(2) matrices1 2 1  2 4 2    1 2 11 4 6 4 1with graded structure:--------------------------------------- 1  0 0  1---------------------------------------            i  0     j  0 0  i     0  j  z        y          0 -1      1  0          1  0      0 -1           t         x---------------------------------------            k  0 0  k  Ih 0 -i     j  0      0 -i     j  0 i  0     0 -j      i  0     0 -j 0  1 1  0  Ic-------------------------------------- 0  i     0  j i  0     j  0          0 -k      k  0          k  0      0 -k-------------------------------------- 0  k k  0  Ich--------------------------------------In this case of Cl(1,3) = Q(2), the full spinor space would be 8-dimensional: 10i0j0k0and01  0i0j0kEach half-spinor space would be 4-dimensional with quaternionic structure  and would have a triality-like isomorphism with the vector space of Cl(1,3). That triality would correspond to the triality of Cl(1,7) = R(16)with dimensionality 2^8 = 256 = 16x16 and 8-dimensional half-spinorsandgraded structure1   8  28  56  70  56  28   8   1that is used in the D4-D5-E6-E7-E8 VoDou Physics model.

Clifford Structure ofthe D4-D5-E6-E7-E8 VoDou PhysicsModel.

The D4-D5-E6-E7-E8 VoDou physics model is based on representations of Spin(0,8).In 8 dimensions, Cl(0,8) = M16(R) the 16x16 real matrices The 1-dimensional scalar 0-grade subspace is represented as the blue square.  It is a 1-dimensional part of the symmetric part of the upper left 8x8 diagonal subspace.    The 8-dimensional vector 1-grade subspace is represented as the green squares.  It is part of the lower left 8x8 off-diagonal subspace.   The 28-dimensional bivector 2-grade subspace is represented as the red squares.  It is the antisymmetric part of the upper left 8x8 diagonal subspace.     This accounts for the  1   8  28   graded elements of the  1   8  28  56  70  56  28   8   1  = 256-dim Cl(0,8).   The 56-dimensional trivector 3-grade subspace is represented as the white squares in the lower left quadrant.  In the full 8-dimensional theory, the 56 trivectors are related to the structure of 3+1=4-dimensional subspaces of 1+7=8-dimensional spacetime that are connected with the E8 HyperDiamond lattice links that are (normalized) sums of 4 of the basis octonions.  To reduce the dimension of spacetime to 1+3=4 dimensions, an associative 3-form is used.  This effectively fixes a particular trivector, so the 56 trivectors do not play a dynamical role in the 4-dimensional phase of the D4-D5-E6 model.  Here and now, we do not have the technology to do experiments that could test the structure of the full 56-dimensional trivector sector. The 70 4-vectors in the 4-grade subspace are reducible to two sets of 35 4-vectors each. One set of 35 4-vectors is represented  as the grey squares in the upper left quadrant.  They are the symmetric parts of the upper left 8x8 diagonal subspace, remaining after taking out the 1-dimensional scalar 0-vector Higgs scalar.  The 1-dimensional scalar 0-vector representing the Higgs scalar can be thought of as the trace of the full symmetric 1+35=36-dimensional space of symmetric 8x8 real matrices.  The 35  4-vectors are the traceless symmetric 8x8 matrices. They are related to the coassociative 4-form that is fixed in the dimensional reduction process to determine the internal symmetry space.  At our low energy levels, below the Planck-scale at which dimensional reduction occurs in the D4-D5-E6 model, the 35 4-vectors do not play a dynamical role that we can test experimentally here and now.  However, they show that the Higgs mechanism is related fundamentally to BOTH the particles and fields of the internal symmetry space and the spacetime of conformal MacDowell-Mansouri gravity.

According to Dennis W. Marks, in his paper A Binary Index Notationfor Clifford Algebras (revised 27 February 2003):

"... we define the n x n co-identity matrix as

J_n = e_(2^n - 1) = e_(1 ... n ones ...1)

... Duality operations generate isomorphism between grades k and n-k. There are several different duals, including

• the Clifford dual, which we will write as e_m*, defined ... as e_m* = e_m J_n,
• and the Hodge dual, which we will write as m_e_*, defined as m_e_* = (m_e) J_n = (-1)^( k_m ( k_m - 1 ) / 2 ) e_m J_n = (-1)^( k_m ( k_m - 1 ) / 2 ) e_m*.
• ... A dual distinct from the preceding duals is introduced by bit inversion that maps e_m -> e_mbar , where mbar is the bit inverse of m ... In particular, bit inversion transforms vectors (grade 1) ... into covectors (grade n-1) ...

The bit inverse of the bit inverse is the original element: e_mbarbar = e_m.

For the various duals involving J_n, the dual of the dual of e_m is +/- e_m ...

The duals involving J_n also ...[depend on]... the handedness of the base coordinate system, because J_n changes sign with the swapping of two of the bases.

Bit inversion does not ...[ depend on the handedness of the base coordinate system ]...

Complementarity between space-time and momentum-energy is achieved by bit inversion, which interconverts between position representation and momentum representation.

Treating momentum as a Clifford covector has the virtue of automatically enforcing the Heisenberg commutation relation as a consequence of the commutation and anti-commutation propeerties of the Clifford elements. ...".

In the case of 4-dimensional physical space-time, with Cliffordalgebra Cl(1,3) and graded structure

1 4 6 4 1

Marks's bit inversion complementarity acts on the middle-grade6-dimensional bivectors (grade 2) as a map of the 6-dimensionalLorentz Lie Algebra Spin(1,3) = Spin(3) x Spin(3) into itself, actingas an isomorphism between the 3-dimensional Spin(3) rotation Liealgebra (rotations of positions with x, y, and z components) and the3-dimensional Spin(3) boost Lie algebra (transformations of momentawith tx, ty, and tz components).

In the case of 8-dimensional space-time, with Clifford algebraCl(1,7) and graded structure

18 28 56 70 56 28 8 1

Since the 70 decomposes into two 35s, just as the Cl(1,3) 6decomposes into two 3s,

you can write the graded structureas

1 8 28 56(35+35)56 28 8 1

where the blue 1 8 28 56 35 have the8-position-related physical interpretation given above, and DennisMarks's bit inversion gives a complementarity between theposition-related blue 1 8 28 56 35 andthe corresponding momentum-related Clifford co-multivectors ofthe red 35 56 28 8 1. The bitinversion complementarity can be visualized in terms of thefigure as a mapping of the two left-hand quadrants (8-position-related)onto the two right-hand quadrants (8-momentum-related) by reflectionthrough the center (origin) of the figure.

The position-momentum (row vector half-spinor gammas) dualityis also a duality between particlesand antiparticles (column vector half-spinors).

The relevant Duality Structure can be seen in such physicalconcepts as Type IV(2) Domains,Hyperspace, BlackHoles, Wavelets, andConformalSpaceTime.

What about the SPINOR representations?  To see the two 8-dimensional half-spinor spaces of Cl(0,8), look at the even subalgebra Cle(0,8): The minimal-left-ideal column 8-dimensional half-spinors are the blue-green columns, one each in the two 8x8 diagonal subspaces of Cle(0,8).   The minimal-right-ideal row 8-dimensional half-spinors are the yellow-green rows, one each in the two 8x8 diagonal subspaces of Cle(0,8).   The two 8-dimensional half-spinor spaces and the 8-dimensional vector space of Cl(0,8) are all isomorphic to each other.  This is Spin(0,8) TRIALITY, and is evident in the Dynkin diagram of Spin(0,8): The 4 Spin(0,8) representations of the Dynkin diagram are:  the green point is the vector, the two blue points are the two (left-ideal column) half-spinors, and the red center is the 28-dimensional adjoint.
Along with the scalar representation, these representations are used to build the 8-dimensional Lagrangian of the D4-D5-E6 physics model: The vector representation is the 8-dimensional spacetime over which the Lagrangian density is integrated.  The adjoint representation gives the curvature term and (through the covariant derivative) interactions with spinor fermion and scalar particles.  The half-spinor representations give the spinor fermion term. The scalar representation gives the scalar term.  Path-integral quantization can produce gauge-fixing and ghost terms.

Clifford Paths for the D4-D5-E6-E7-E8VoDou Physics Model

To make connection with observable physics in our regime, farbelow the Planck energy of 10^19 GeV, the 8-dimensionalnonassociative octonionic spacetime must be reduced to a4-dimensional associative submanifold.

Since the more realistic Minkowski PhysicalSpaceTime with -+++ Signature has Quaternionic Structure, auseful Clifford path is this Clifford Path that is Quaternionic fromCl(3,5) through Cl(2,4) to Cl(1,3) and from Cl(2,6) through Cl(2,5)and Cl(2,4) to Cl(1,3): In the D4-D5-E6-E7-E8 VoDou Physicsmodel, D5 is taken to be SL(2,O) = Spin(1,9), and spinors are R16+ R16 of Cle(1,9) = Cl(1,8),

where Cle(p,q) = Cl(p,q-1) and Cle(p,0) =Cl(0,p-1)

The D4 can initially be taken to be Spin(0,8), or, with the same Clifford algebra, Spin(1,7) or Spin(4,4). Dimensional Reduction from 8 to 4 dimensions introduces the subspace substructure ofQuaternionic Physical SpaceTime in Octonionic Vector Space.

The (1,7)-dimensional RP1 x S7 = S1 x S7 = U(1) xS7 spacetime of the D4-D5-E6-E7-E8VoDou Physics model prior todimensionalreduction can be represented byQuaternionic Projective 2-space QP2.

Atiyah and Berndtsay in their paper Projective Planes, Serveri Varieties, andSpheres, math.DG/0206135,the S1 x S7 considered as QP2 breaks down into two parts:

• a CP2 acted on by an SU(3), which plays the role of Internal Symmetry Space after dimensional reduction in the D4-D5-E6-E7-E8 VoDou Physics model; and
• an S1 bundle over the S3 that is the complement S7 \ CP2, which plays the role of (1,3) RP1 x S3 Physical Minkowski Spacetime after dimensional reduction in the D4-D5-E6-E7-E8 VoDou Physics model.
Therefore, dimensional reduction changes Real M(16,R) of Cl(0,8), Cl(1,7), and Cl(4,4)   to the Quaternionic M(8,Q) of Cl(2,6) or Cl(3,5).
Cl(2,6) = M(8,Q) Spin*(8) = Spin(2,6) Cl(2,5) = M(4,Q)+M(4,Q) Spin(2,5) Cl(2,4) = M(4,Q) Spin(2,4) = SU(2,2)
Spin(2,4) = SU(2,2) is the connected component with identity of the Conformal(1,3) Group  of Minkowski SpaceTime. Since both 1+1 and 3+1 are even, Conformal(1,3) has 4 components, not 2 components as for Conformal(p,q) where p+1 and/or q+1 are odd. U(2,2) = U(1)xSU(2,2) corresponds to compact U(4) = U(1)xSU(4), the compact isotropy space of the 12-dimensional rank-2 symmetric spaces Spin*(8) / U(4) (non-compact) and Spin(8) / U(4) (compact).
Cl(1,4) = M(2,Q)+M(2,Q) Spin(1,4) = Sp(1,1)
Spin(1,4) is the connected component with identity of the deSitter Group of Minkowski SpaceTime. Although the MacDowell-Mansouri mechanism, as originally formulated, produced gravity from anti-deSitter Spin(2,3) = Spin(0,5) = Sp(2), this Quaternionic chain uses, instead of Sp(2), Sp(1,1) = Spin(1,4).
Cl(1,3) = M(2,Q) Spin(1,3)
Spin(1,3) is the connected component with identity of the Lorentz Group of Minkowski SpaceTime.
Cl(0,3) = Q+Q Spin(0,3) = Sp(1) = SU(2) = S3 Cl(0,2) = Q Spin(0,2) = U(1) Cl(0,1) = C Spin(0,1) = Z2 Cl(0,0) = R Spin(0,0) = I
(Much of the above is taken from Ian Porteous's book   Clifford Algebras and the Classical Groups (Cambridge 1995).)

The path from Cl(1,7) through Cl(2,4) to Cl(1,3)can also be seen from the viewpoint of spinor fermions of theD4-D5-E6-E7-E8 VoDou Physics model:

Half-Spinors for 16x16 real matrix Cl(1,7) are 1x8 real column vectorswhich correspond to first-generation fermion particles:   Octonion       FermionBasis Element    Particle     1           e-neutrino     i         red  up  quark     j       green  up  quark     k        blue  up  quark     I       red  down  quark     J     green  down  quark     K      blue  down  quark     E            electronFull Spinors for 4x4 quaternion matrix Cl(2,4) are 1x4 quaternion column vectors:q1q2q3q4which are reducible to two Half-Spinor mirror image 1x2 quaternion column vectors:q+1q+2q-1q-2Since each of these is a 4-dimensional quaternion,if you let the q+ be fermion particlesand the q- be fermion antiparticles,you have4 fermion particles for q+1, which can be taken tobe electron, red up quark, blue up quark, and green up quark,and4 fermion particles for q+2, which can be taken tobe neutrino, red down quark, blue down quark, and green down quark,and4 fermion antiparticles for q-1, which can be taken tobe positron, red up antiquark, blue up antiquark, and green up antiquark,and4 fermion antiparticles for q-2, which can be taken to be antineutrino,red down antiquark, blue down antiquark, and green down antiquark, so that the quaternionic spinors of Cl(2,4) correspond directly to the fermions of the real spinors of Cl(1,7). For Cl(1,3) the 2x2 quaternionic matrices have Full Spinorsthat are 1x2 quaternion column vectors. Each Half-Spinor space is one quaternion variable, which has a 1-2 correspondence with first generation fermions, and also corresponds 1-1 with the (1,3) vector space of physical Minkowski spacetime, resulting in a quaternionic version of triality (diluted by the 1-2 nature of the fermion correspondence) that is related to the reducibility of the D2 Lie algebra Spin(1,3).

Here are some other useful pathways for theD4-D5-E6-E7-E8 VoDou Physics Model: From Split Cl(4,4):

The Paths can be described as a RealProcess involving M(2,R)

or

as a Quaternionic Process involvingthe Quaternions Q.

Since the tensor product Q x Q = M(4,R) = M(2,R) x M(2,R), theProcesses are Physically Equivalent.

Real Process through Cl(4,3) andCl(4,2) and Cl(3,1)

based on the tensor products M(2,R) x M(2,R) = M(4,R)

and M(2,R) x M(4,R) = M(8,R)

and M(2,R) x M(8,R) = M(16,R)

Real Step 1 of 2:

Start with Cl(4,4) = M(16,R), the 16x16 matrices with entries inthe Real Numbers, and then represent them as M(2,CSG(4,2)) =M(2,M(8,R)), the 2x2 Vahlen Matrices with entries in the CliffordSemiGroup of the Cl(4,2) Clifford Algebra = M(8,R) of 8x8 matriceswith Real entries, which Vahlen Matrices represent the MoebiusTransformations of R^(4,2).

The unit sphere in CSG(4,4) represents Spin(4,4), which isa 2-fold covering group of the Moebius Transformations of R^(4,2).

The Rotations are 15-dimensional Spin(4,2) = SU(2,2), of the form A x A^(-1) , with matrix form
A 0 0 A

In the D4-D5-E6-E7-E8 VoDou PhysicsModel, the 15 dimensions of the Rotations are the Spin(4,2) =SU(2,2) Conformal Group of Physical Minkowski SpaceTime,which,through the MacDowell-Mansouri Mechanism,forms Gravity (with Torsion and ParityViolation) plus the Higgs Mechanism.

The Translations are 2+4 = 6-dimensional, of the form x + B , with matrix form
1 B 0 1

The Transversions are 2+4 = 6-dimensional, of the form
( x + x^2 C ) / (1 + 2 x . C + x^2 C^2 ) , with matrixform
1 0 C 1

In the D4-D5-E6-E7-E8 VoDou PhysicsModel, the 4+4 = 8 dimensions of the Translations andTransversions form the 8-dimensonal SU(3) of the Color Force.

In the D4-D5-E6-E7-E8 VoDou PhysicsModel, the 2+2 = 4 dimensions of the Translations andTransversions form the 4-dimensonal U(2) of the ElectroWeakForce.

The Dilations are 1-dimensional, of the form x D , with matrix form
sqrt(D) 0 0 1/sqrt(D)

In the D4-D5-E6-E7-E8 Vodou PhysicsModel, the 1 dimension of the Dilations represent the U(1)Complex Phase of Particle and Force Propagators.

Real Step 2 of 2:

Now consider the R^(4,2) Clifford Algebra Cl(4,2) = M(8,R), the8x8 matrices with entries in the Real Numbers, and then representthem as M(2,CSG(3,1)) = M(2,M(4,R)), the 2x2 Vahlen Matrices withentries in the Clifford SemiGroup of the Cl(3,1) Clifford Algebra =M(4,R) of 4x4 matrices with Real entries, which Vahlen Matricesrepresent the Moebius Transformations of R^(3,1).

The unit sphere in CSG(4,2) represents Spin(4,2) =SU(2,2), which is a 4-fold covering group of the MoebiusTransformations of Minkowski space R^(3,1).

The Rotations are 6-dimensional Spin(3,1), of the form A x A^(-1) , with matrix form
A 0 0 A

The Translations are 1+3 = 4-dimensional, of the form x + B , with matrix form
1 B 0 1

In the D4-D5-E6-E7-E8 VoDou PhysicsModel, the 6+4 = 10 dimensions of the Rotations and Translationsare the Spin(3,2) anti-deSitter Group of Physical MinkowskiSpaceTime,which, through theMacDowell-Mansouri Mechanism, forms Gravity (with Torsionand Parity Violation).

The Transversions are 1+3 = 4-dimensional, of the form
( x + x^2 C ) / (1 + 2 x . C + x^2 C^2 ) , with matrixform
1 0 C 1

In the D4-D5-E6-E7-E8 Vodou PhysicsModel, the 4 dimensions of the Transversions form the 4 degreesof freedom of the Higgs SU(2) Scalar Particle. They also correspondto the Conformal Moebiustransformations involved in torsion-superluminal solutions ofMaxwell's equations that may be permit longitudinalphotons, thus indicating that such solutions may be related tothe Higgs Mechanism.

Related Conformal Structures mayproduce GraviPhoton Phenomena.

The Dilations are 1-dimensional, of the form x D , with matrix form

sqrt(D) 0 0 1/sqrt(D)

In the D4-D5-E6-E7-E8 VoDou PhysicsModel, the 1 dimension of the Dilations represents theMass/Energy Scale of the Higgs Vacuum Expectation Value, which mayrepresent a measure of the Compressibility of Maxwell's Aether thatcould give rise to torsion-superluminalsolutions of Maxwell's equations that may permit longitudinalphotons.

Quaternionic Process through Cl(3,4)and Cl(2,4) and Cl(1,3),

based on the tensor product Cl(0,8) = Cl(0,4) x Cl(0,4) =M(2,Q) x M(2,Q) = M(16,R).

Start with Cl(4,4) = M(16,R), the 16x16 matrices with entries inthe Real Numbers.

Since the tensor product Q x Q = M(4,R), M(16,R) = Q xM(4,Q).

Quaternionic Step 1 of 2:

The transition from Cl(4,4) = M(16,R) = Q x M(4,Q) to M(4,Q) =Cl(2,4)

is the tensor product by the Quaternions Q,which takes M(4,Q) toM(4,M(4,R)) = M(16,R).

What does the tensor product by Q do to the Spin(2,4) = SU(2,2)part of Cl(2,4) ?

Since the tensor product by Q takes Cl(2,4) to Cl(4,4),

it should take Spin(2,4) = SU(2,2) to Spin(4,4). How does it dothat?

Factor the Quaternions Q into a Real Part Re(Q) and an ImaginaryPart Im(Q),

and then look at how the tensor product by Q acts on the SU(2,2)that is within Cl(2,4).

The image of this action should contain Spin(4,4), but may containother stuff that is within Cl(4,4) (or, to be more nearly technicallyaccurate, CSG(4,4)) but is not in Spin(4,4).

Represent Re(Q) by U(1) = S1 and Im(Q) by SU(2) = S3.

Then, regard the tensor product Q x SU(2,2) as

( U(1) , SU(2) ) x SU(2,2) = ( U(1) x SU(2,2) , SU(2) xSU(2,2) ) =

= ( U(2,2) , SU(2) x SU(2,2) )

The first component is just U(2,2) = U(1) x Spin(2,4), whichincludes the Spin(2,4) of Cl(2,4), and it makes up a 16-dimensionalpart of 28-dimensional Spin(4,4).

What about the other 12 dimensions of Spin(4,4) ?

They should be in the second component, the tensor product SU(2) xSU(2,2),

but they should only be in the part of it that has Lie Groupstructure.

To see what part that is, first observe that it contains the LieGroup SU(2) (the first factor)

Then, look at the fibration SU(2,2) / (U(1) x SU(3)) = CHP3 =Complex Hyperbolic Projective 3-space.

Then, regard SU(2,2) as U(1) x SU(3) x CHP3.

Then, since CHP3 is not a Lie Group, disregard it, leaving the LieGroups U(1) and SU(3).

Therefore,

regard the result of the tensor product by Q of the Spin(2,4) =SU(2,2) part of Cl(2,4) as the 12-dimensional

U(1) x SU(2) x SU(3)

that corresponds to the Standard Model group in theD4-D5-E6-E7-E8 VoDou Physics Model,

plus the U(1) of U(1) x Spin(2,4) = U(2,2) that, in theD4-D5-E6-E7-E8 VoDou Physics Model, corresponds to the ComplexU(1) Propagator Phase.

Quaternionic Step 2 of 2:

Go from Cl(2,4) = M(4,Q) = M(2,M(2,Q)) = M(2,Cl(1,3)) down toM(2,Q) = Cl(1,3).

M(2,Q) M(2,Q)M(2,Q) M(2,Q)

This step starts with the R^(2,4) Clifford Algebra Cl(4,2) =M(4,Q), the 4x4 matrices with entries in the Quaternions, which arerepresented as M(2,CSG(1,3)), the 2x2 Vahlen Matrices with entries inthe Clifford SemiGroup of the Cl(1,3) Clifford Algebra = M(2,Q) of2x2 matrices with Quaternion entries, which Vahlen Matrices representthe Moebius Transformations of R^(1,3).

The unit sphere in CSG(2,4) represents Spin(2,4) =SU(2,2), which is a 4-fold covering group of the MoebiusTransformations of Minkowski space R^(1,3).

The Rotations are 6-dimensional Spin(1,3), of the form A x A^(-1) , with matrix form
A 0 0 A

The Translations are 1+3 = 4-dimensional, of the form x + B , with matrix form
1 B 0 1

In the D4-D5-E6-E7-E8 VoDou PhysicsModel, the 6+4 = 10 dimensions of the Rotations and Translationsare the Spin(2,3) anti-deSitter Group of Physical MinkowskiSpaceTime,which, through theMacDowell-Mansouri Mechanism, forms Gravity (with Torsionand Parity Violation).

The Transversions are 1+3 = 4-dimensional, of the form
( x + x^2 C ) / (1 + 2 x . C + x^2 C^2 ) , with matrixform
1 0 C 1

In the D4-D5-E6-E7-E8 VoDou PhysicsModel, the 4 dimensions of the Transversions form the 4 degreesof freedom of the Higgs SU(2) Scalar Particle. They also correspondto the Conformal Moebiustransformations involved in torsion-superluminal solutions ofMaxwell's equations that may be permit longitudinalphotons, thus indicating that such solutions may be related tothe Higgs Mechanism.

Related Conformal Structures mayproduce GraviPhoton Phenomena.

The Dilations are 1-dimensional, of the form x D , with matrix form
sqrt(D) 0 0 1/sqrt(D)

In the D4-D5-E6-E7-E8 VoDou PhysicsModel, the 1 dimension of the Dilations represents theMass/Energy Scale of the Higgs Vacuum Expectation Value, which mayrepresent a measure of the Compressibility of Maxwell's Aether thatcould give rise to torsion-superluminalsolutions of Maxwell's equations that may permit longitudinalphotons.

From Euclidean Cl(0,8) through Cl(0,6) andCl(0,4):

The dimensional reduction process  starts with the Clifford algebra Cl(0,8) = M16(R),  then goes to its even subalgebra Cle(0,8) which is the Clifford algebra Cl(0,7) = M8(R)+M8(R) + then goes to its even subalgebra Cle(0,7) which is the Clifford algebra Cl(0,6) = M8(R) then goes to its even subalgebra Cle(0,6) which is the Clifford algebra Cl(0,5) = M4(C). + The Clifford algebra chain of dimensional reduction is shown below in green: The chain starts with a real Clifford algebra, Cl(0,8), and proceeds through other real ones until it reaches its end at COMPLEX Cl(0,5) = M4(C).  Its even subalgebra is Cle(0,5) = Cl(0,4) = M2(Q) the 2x2 Quaternionic matrix algebra. Since M2(Q) is also the Clifford Algebra Cl(1,3), you get both Euclidean Cl(0,4) and Minkowski Cl(1,3) by taking the even subalgebra of the Cl(0,5) Clifford Algebra of Spin(0,5).  In this way, the D4-D5-E6-E7-E8 VoDou physics model allows you to go back and forth by Wick rotation between Euclidean (0,4) and Minkowski (1,3).   Further, Cl(0,5) = M4(C) is the complexification of each and every one of the five 4-dimensional real Clifford algebras, shown in blue in the figure above, some of which have Dirac algebra real structure M4(R), and some of which have quaternionic structure M2(Q) (sometimes, as in the figure, denoted M2(H))  In the D4-D5-E6-E7-E8 VoDou physics model the real part of Cl(0,5) = M4(C) describes gravity, and the imaginary part of Cl(0,5) = M4(C) describes the color, weak, and electromagnetic forces, with the Higgs mechanism affecting both sectors.   For details, see the references in this home page to the D4-D5-E6-E7-E8 VoDou physics model, as well as the papers linked or cited therein.

References and Background Material for theClifford Paths for the D4-D5-E6-E7-E8 VoDou PhysicsModel:

| Ahlfors | Porteous| Lounesto | Gilbertand Murray | Krausshar and Ryan | Dixon|

Lars Ahlfors, in his paper Clifford Numbersand Moebius Transformations in R^n, published on pages 167-175 ofthe book Clifford Algebras and Their Applications in MathematicalPhysics, Proceedings of NATO and SERC Workshop, Canterbury, Kent,1985, edited by Chisholm and Common, NATO ASI Series (Reidel 1986),says:

"... Moebius transformations in any dimension can beexpressed through 2x2 matrices with Clifford numbers as entries. Thistechnique is relatively unknown in spite of having been introduced asearly as 1902. ...".

What are Clifford Numbers? Clifford Numbers are theClifford Semigroup CSG(n) of theClifford Algebra Cl(n).

Ian Porteous, in his book CliffordAlgebras and the Classical Groups (Cambridge 1995), describes thetensor product

Cl(p,q+4) = Cl(p,q) x Cl(0,4) = Cl(p,q) xM(2,Q)

and the tensor product

Q x Q = M(4,R)

PerttiLounesto, in his book CliffordAlgebras and Spinors (Cambridge 1997), describes theConformal Compactification of R^(p,q) by Moebius Transformations interms of the p-sphere Sp and the q-sphere Sq as:

Sp x SqZ2

Pertti Lounesto'sChapter 16 describes the tensor product relation

Cl(p+1,q+1) = Cl(p,q) x Cl(1,1) = M(2,Cl(p,q)

The cases of Cl(2,4) = M(2,Cl(1,3)) and Cl(4,2) = M(2,Cl(3,1)) areuseful in physics, since Spin(2,4) = Spin(4,2) = SU(2,2) is theConformal Group of Conformal Transformations of MinkowskiSpacetime.

That leads into his discussion of Moebius Transformations andVahlen Matrices in his Chapter 19, in which he describes them as 2x2matrices with Clifford entries. RelatedConformal Structures may produce GraviPhoton Phenomena.

Pertti Lounesto'sChapter 21 deals with Binary Index Setsand Walsh Functions.

John E. Gilbert and Margaret A. M.Murray, in their book Clifford Algebras and Dirac Operators inHarmonic Analysis (Cambridge 1991), describe the MoebiusTransformations, in both Spherical and Hyperbolic Space, bySL(2,CSGn) and by SU(2,CSGn), respectively, where CSGn denotes theClifford Semigroup of Cl(n). RelatedConformal Structures may produce GraviPhoton Phenomena.

Gilbert and Murray describe theClifford Semigroup of the Clifford Algebra Cl(n) as being madeup of all x in Cl(n) such that for each y in R^(n+1) there exists zin R^(n+1) xy = zx', and they they say:

"... Denote by CSG(n) the Clifford semigroup ...
CSG(n) = { x y ... z : x, y, z, ... in R^(n+1) }

generated by the non-zero elements in R^(n+1), and then denote by CG(n) the Clifford group

CG(n) = { x y ... z : x, y, z, ...in R^(n+1) \ {0} }

Of course, CSG(n) = CG(n) u {0}. ...

... [The Clifford norm is denoted by | | and the Clifford operator norm, defined by regarding the Clifford algebra Cl(n) as an algebra of operators on itself, is denoted by || || . They coincide on CSG(n).] ...

For each n = 0, 1, ...

Spin(n+1) = { a in CSG(n) : ||a|| = 1 }

... Now O(n) can be identified with the subgroup of SO(n+1) fixing the subspace R(1,0,...,0) of R^(n+1) ...

... [where Cl(n) has the direct sum graded structure Cl(n) = Cl(0)(n) + Cl(1)(n) + ... Cl(n)(n) and R^(n+1) is regarded as Cl(0)(n) + Cl(1)(n). Let n denote intersection and Cle(n) be the even subalgebra of Cl(n)] ...

For each n = 1, 2, ...,

Pin(n) = { a in CSG(n) : a a* = 1, ||a|| = 1 }

Spin(n) = { a in CSG(n) n Cle(n) : a a* = 1, ||a|| = 1 }

... It is well-known that the unit sphere in R^(n+1) ha a multiplicative structure if n = 0, 1, or 3, when it coincides with Pin(1), Spin(2) and Spin(3) respectively. The results above show what to do for larger n:

one takes the unit sphere in the Clifford semi-group CSG(n), not just the unit sphere in R^(n+1). ...".

Gilbertand Murray describe the PeriodicityTheorem, saying:

"... For each n > 0 there is a realization of Cl(n+8) as the algebra
Cl(n+8) = M(16, Cl(n))

of all 16x16 matrices having entries from Cl(n) ...

... For each n > 0 there is a realization of Cl(n+4) as the tensor product

Cl(n+4) = Cl(n) x M(2,Q) = Cl(n) x Cl(4)

over R of Cl(n) and M(2,Q) ... [the 2x2 matrices with entries in Q, the quaternions] ...

... two-fold application of [the realization] gives

Cl(n+8) = Cl(n) x M(2,Q) x M(2,Q)

with respect to the usual tensor product of algebras. Since M(2,Q) x M(2,Q) = M(16,R), ... [ ...

Cl(n+8) = Cl(n) x M(16,R)

... ] ...".

Gilbert andMurray describe the Dirac Operator, which is the naturaldifferential operator acting on spinors ( and therefore the naturaldifferential operator for the fermion terms in the Lagrangianof the D4-D5-E6-E7-E8 VoDou Physicsmodel ), saying:

"... A Clifford algebra bundle can be constructed over any Riemannian manifold, but topological obstructions may prevent the construction of Clifford modules. ... the unit sphere in C has two inequivalent spin structures. But the unit sphere in Rn, n > 3, has exactly one spin structure; its principal Spin(n) bundle is diffeomorphic to Spin(n). ... [ for k > 1 ] real projective space RP(4k-1) has two inequivalent spin structures, whereas RP(4k+1) does not have any [ although it is orientable ] ... RP(2k) is not orientable ... [ and, according to Spin Geometry, by H. Blaine Lawson and Marie-Louise Michelsohn (Princeton 1989), CPn is spin iff n is odd and QPn is spin for all n ] ... An oriented riemannian manifold X admits a spin structure if and only if its second Steifel-Whitney class is zero. Furthermore, if w2(X) = 0, then the spin structures on X are in one-to-one correspondence with elements of H1(X;Z2). ]...
... h ... will denote ...[ a Cl(n)-module ]... on which there is a representation t of Spin(n) such that
t(a) e_i t(a)^(-1) = s(a) e_i ( 1 < i < n )

for all a in Spin(n), where s : Spin(n) -> SO(n) is the two-fold covering homomorphism. ...[ and where e_i (x) = SUM(1 to n) gamma^ij (x) e_i so that

e_j(x) e_k(x) + e_k(x) e_j(x) = - 2 g^jk(x)

where gamma is the square root of g ]...

... the passage from R and C to more general Clifford algebras allows the notion of fractional linear transformations on R and C to be defined naturally on higher-dimensional Euclidean space. This in turn is used ... to give ... representation theory for Spino(n,1) in parallel with the well-known description of the unitary representations of SL(2,R), thinking of SL(2,R) as a transformation group on C. We derive in the process explicit realizations of the Dirac operator on hyperbolic and spherical space. ...

... The Dirac operator on ... real n-dimensional hyperbolic space Hn ... is defined ... as a differential operator on [ C(infinity) ( Rn+ with coordinates , h ) ]... by

D = ( SUM(1 to n-1) e_i ( y (d/dxi) + (1/2) dt( e_i e_n ) ) + e_n y (d/dy )

...[where d denotes partial derivative]...

the multiplicative structure on ...[ Cl(n) describes ]... the isometry group of Hn by the action of Spin(n,1) as 'fractional linear tranformations' on Rn+ ...

... The Dirac operator on ... real n-dimensional spherical space Sn ... is defined ... as a differential operator on ...[ C(infinity) ( Rn u {infinity} , h ) ]... by

D = ( 1 + |x|^2 ) ( SUM(1 to n) e_i ( d/dxi) + (1/(1+|x|^2)) dt( e_i x ) ) )

...[where d denotes partial derivative]...

... the fundamental Bochner-Weizenboeck theorem expresses (-D^2) in general as a second-order Laplacian together with a zero-order curvature operator. ...".

R. S. Krausshar and John Ryan, in math.AP/0212086,Some Conformally Flat Spin Manifolds, Dirac Operators andAutomorphic Forms, say:

"... In this paper we study Clifford and harmonic analysis on some conformal flat spin manifolds.... manifolds treated here include RPn and S1 x S(n-1). Special kinds of Clifford-analytic automorphic forms associated to the different choices of are used to construct Cauchy kernels, Cauchy Integral formulas, Green's kernels and formulas together with Hardy spaces and Plemelj projection operators for Lp spaces of hypersurfaces lying in these manifolds. ...

... Solutions to the Dirac equation are called Clifford holomorphic functions or monogenic functions. Such functions are covariant under Mobius transformations acting over Rn u {oo} . In fact this covariance is an automorphic invariance best described by a method using Clifford algebras and due to Ahlfors and Vahlen. In fact all solutions to the equation D^k f = 0 where k is in N exhibit a similar automorphic invariance under Mobius transformations. Given this natural method for describing conformal or Mobiustransformations and the automorphic invariance of solutions to D^k f = 0 under actions of the Mobius group a natural choice of generalization of Riemann surfaces from one complex variable to the present context would be conformally flat manifolds. ... Conformally flat manifolds are manifolds with atlases whose transition functions are Mobius transformations. ... one fruitful way of constructing conformally flat manifolds is to factor out a simply connected subdomain U of either the sphere Sn or Rn by a Kleinian subgroup G of the Mobius group where G acts discontinuously on U. This gives rise to the conformally flat manifold U / G. Examples of such manifolds include for example n-tori, cylinders, real projective space and S1 x S(n-1). ...".

Goeffrey Dixon has written DivisionAlgebras: Octonions, Quaternions, Complex Numbers, and the AlgebraicDesign of Physics (Kluwer 1994), in which he describes DivisionAlgebras and Clifford Algebras. He discusses in some detail how thetensor product RxR = R and CxC = C+C, but QxQ is M(4,R), the matrixalgebra of 4x4 matrices of Real Numbers. He shows howLeft-Multiplication of Quaternions is distinct fromRight-Multiplication, so that the action QL of Left-Multiplication isnot identical with the action QR of Right-Multiplication. He alsoshows:

• Real Multiplication generates 1-dimensional M(1,.R);
• Complex Multiplication does not fully generate 4-dimensional M(2,R), but requires also conjugation and multiplication by the Complex Imaginary i to do so;
• Quaternion Multiplication, being non-commutative, has distinct Left-Multiplication that generates QL and Right-Multiplication that generates QR, and neither QL nor QR alone fully generates M(4,R), but both together produce the Quaternion full Adjoint action QA = M(4,R); and
• Octonion Multiplication, both non-commutative and non-associative, but alternative, has distinct Left-Multiplication generating OL and Right-Multiplication generating OR, and OL = OR = M(8,R) = Cl(0,6) = Cl(4,2) = Cl(3,3).

For more about Clifford algebras, particularly with respect to Cl(0,8), see Why not SEDENIONS?and From Sets to Quarks, particularly Chapter 2.    All Lie algebras can be built from Clifford Algebras.   Also, it is conjectured that MetaClifford Algebras might be useful.
REFERENCES:  Three books that are good places to start reading about the details of Clifford algebras and Spinors are: Clifford Algebras and Spinors, by Pertti Lounesto (London Mathematical Society Lecture Note Series, No 239) Spinors and Calibrations, by F. Reese Harvey, Academic Press (1990).  Clifford Algebras and the Classical Groups, by Ian Porteous, Cambridge University Press (1995).  Errors can creep into published books and papers in any field. So that errors don't propagate and become widely-held misconceptions, it is important to find them and point them out. The best person I know at doing that is Pertti Lounesto, who has a web page of counterexamples that not only points out errors, but shows why they were probably made, so that the error-correction process not only corrects, but also gives deeper insight into the fundamental structures that led the original authors astray. It is important that authors not be condemned for making errors, so long as they are willing to acknowledge them and correct them.   OY! Barry Simon has written YABOGR!The official title is: Representations of Finite and Compact Groups (AMS 1996)What is YABOGR? Read the Book! What does YABOGR do? It, together with ideas of Onar Aam and Ben Goertzel about XOR and set theory, inspires me to write THIS PAGE ABOUT SETS, CLIFFORD GROUPS AND ALGEBRAS, AND THE McKAY CORRESPONDENCE. (any errors you see are due to me, not to Barry Simon, Onar Aam, or Ben Goertzel)