Introduction to my May 2002 Cookeville Clifford Algebra talk:


Complex Clifford Periodicity

Cl(2N;C) = Cl(2;C) x ...(N times tensor product)... x Cl(2;C)

Cl(2;C) = M2(C) = 2x2 complex matrices

spinor representation = 1x2 complex column spinors

Hyperfinite II1 von Neumann Algebra factor is the completion of the union of all the tensor products

Cl(2;C) x ...(N times tensor product)... x Cl(2;C)

By looking at the spinor representation, you see that "the hyperfinite II1 factor is the smallest von Neumann algebra containing the creation and annihilation operators on a fermionic Fock space of countably infinite dimension."

In other words, Complex Clifford Periodicity leads to the complex hyperfinite II1 factor which represents Dirac's electron-positron fermionic Fock space.

Now, generalize this to get a representation of ALL the particles and fields of physics.


Use Real Clifford Periodicity to construct a Real Hyperfinite II1 factor as the completion of the union of all the tensor products

Cl(1,7;R) x ...(N times tensor product)... x Cl(1,7;R)

where the Real Clifford Periodicity is

Cl(N,7N;R) = Cl(1,7;R) x ...(N times tensor product)... x Cl(1,7;R)

The components of the Real Hyperfinite II1 factor are each


[ my convention is (1,7) = (-+++++++) ]

Cl(1,7) is 2^8 = 16x16 = 256-dimensional, and has graded structure

1 8 28 56 70 56 28 8 1
What are the physical interpretations of its representations?

There are two mirror image half-spinors, each of the form of a real (1,7) column vector with octonionic structure.

The 1 represents:

The 7 represent:
One half-spinor represents first-geneneration fermion particles, and its mirror image represents first-generation fermion antiparticles.

Second and third generation fermions come from dimensional reduction of spacetime, so that

There is a (1,7)-dimensional vector representation that corresponds to an 8-dimensional high-energy spacetime with octonionic structure

that reduces at lower energies to quaternionic structures that are

There is a 28-dimensonal bivector representation that corresponds to the gauge symmetry Lie algebra Spin(1,7)

that reduces at lower energies to:

There is a 1-dimensional scalar representation for the Higgs mechanism.

The above structures fit together to form a Lagrangian

that reduces to a Lagrangian for Gravity plus the Standard Model.

Representations have geometric structure related to E6

E6 is an exceptional simple graded Lie algebra of the second kind:

E6 = g = g-2 + g-1 + g0 + g1 + g2

g0 = so(1,7) + R + iR

dim g-1 = 16

dim g-2 = 8

This gives real Shilov boundary geometry of S1xS7 for (1,7)-dimensional high-energy spacetime representation and for the first generation half-spinor fermion representations.

The geometry of the representation spaces, along with combinatorial structure of second and third generation fermions, allows calculation of relative force strengths and particle masses:





Fermilab says that the T-quark mass is about 170 GeV.

?? Which is the True T-quark mass: 130 or 170 ??






The quote is from John Baez's web page week 175 at

E6 GLA structure is from Soji Kaneyuki's writing in Analysis and Geometry on Complex Homogeneous Domains, by Jacques Faraut, Soji Kaneyuki, Adam Koranyi, Qi-keng Lu, and Guy Roos (Birkhauser 2000).

Frank D. (Tony) Smith, Jr., Cartersville, GA, March 2002