Introduction to my May 2002 Cookeville Clifford Algebra talk:
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
There are two mirror image half-spinors, each of the form of a real (1,7) column vector with octonionic structure.
The 1 represents:
Second and third generation fermions come from dimensional reduction of spacetime, so that
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:
?? Which is the True T-quark mass: 130 or 170 ??
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).