The D4-D5-E5-E7-E8 VoDou Physics HyperDiamond Feynman Checkerboard model can be constructed from the 8-fold Periodicity of Real Clifford Algebras
Cl(8N) = Cl(8) x ..N.. x Cl(8)
where x ..N.. x denotes N-fold tensor product, by using a very large Clifford Algebra Cl(8N) as a starting point, and using 8-fold Periodicity to factor Cl(8N) into N copies of Cl(8).
To see what to do with the N copies of Cl(8), look at how Feynman's 2-dimensional Checkerboard might be constructed from N copies of Cl(2).
Given N copies of Cl(2), how can they be put in a useful order?
Look at Cl(2) = C(2), with graded structure
1 2 1
The Cl(2) vector space is 2-dimensional, so the N copies of Cl(2) should be put in a 2-dimensional array. The most natural such array would be the Complex Gaussian integers, which make a 2-dim Feynman checkerboard with all the Cl(2) at the vertices of the checkerboard connected to other Cl(2) in such a way as to make a 2-dim lattice that is consistent with complex number multiplication.
Cl(2) has 1-dim U(1) as its bivector Lie algebra, which is consistent with the electromagntism gauge group, and with complex number multiplication, and Cl(2) has full spinor space of dimension sqrt(4) = 2, and so half-spinor spaces that are 1-dimensional, one for the electron and one for the positron to move around on the checkerboard, so a Complex Gaussian lattice with a Cl(2) at each vertex defines a 2-dimensional Feynman Checkerboard with half-spinor particles moving around on it and a U(1) gauge group providing the i that Feynman used to weight changes of direction.
If you look at the N copies of Cl(8) the same way, you see that Cl(8) has graded structure
1 8 28 56 70 56 28 8 1
so that Cl(8) has an 8-dim vector space, so that the N copies of Cl(8) wants to be connected as an 8-dim checkerboard, so the N copies of Cl(8) should be ordered as an array of integral octonions, that is, they should live on an E8 lattice, which is the Octonionic 8-dim correspondent of the 2-dim Complex Gaussian lattice.
You can look at the natural 4-dim physical spacetime sublattice, and see that it is the 4-dim HyperDiamond lattice.
Since Cl(8) half-spinors are 8-dim, you get (where Cl(2) gives you the electron and positron) for fermions to move on the lattice the 8 first generation fermion particles
neutrino
red up quark
blue up quark
green up quark
red down quark
blue down quark
green down quark
electron
and their 8 antiparticles. The projection of an 8-dim E8 lattice onto the 4-dim HyperDiamond lattice gives the second and third generation fermions.
At each vertex, instead of the Cl(2) gauge group U(1), you get the Cl(8) gauge group Spin(8), which gives Gravity and the Standard Model from the point of view of the 4-dim HyperDiamond lattice, and which is consistent with octonion multiplication.
The weighting of changes of direction in the 4-dm HyperDiamond lattice is given by Quaternion Imaginaries of the change of direction, in this way:
There are 7 different charged first-generation fermion particles {electron, rgb up quarks, rgb down quarks}. There are 7 different E8 lattices in the full 8-dimensional SpaceTime prior to Dimensional Reduction. All of the 7 E8 lattices produce, by Dimensional Reduction, the same 4-dimensional Physical SpaceTime and 4-dimensional Internal Symmetry Space.
The HyperDiamond generalization has discrete lightcone directions. If the 4-dimensional Feynman Checkerboard is coordinatized by the quaternions Q:
the real axis 1 is identified with the time axis t;
the imaginary axes i,j,k are identified with the space axes x,y,z; and
the four future lightcone links are
(1/2)(1+i+j+k),
(1/2)(1+i-j-k),
(1/2)(1-i+j-k), and
(1/2)(1-i-j+k).
In cylindrical coordinates t,r with r^2 = x^2+y^2+z^2, the Euclidian metric is
t^2 + r^2 = t^2 + x^2+y^2+z^2
and the Wick-Rotated Minkowski metric with speed of light c is
(ct)^2 - r^2 = (ct)^2 - x^2 -y^2 -z^2.
For the future lightcone links on the 4-dimensional Minkowski lightcone, c = sqrt3.
Any future lightcone link is taken into any other future lightcone link by quaternion multiplication by +/- i, +/- j, or +/- k.
For a given vertex on a given path, continuation in the same direction can be represented by the link 1, and changing direction can be represented by the imaginary quaternion +/- i, +/- j, +/- k corresponding to the link transformation that makes the change of direction.
Therefore, at a vertex where a path changes direction, a path can be weighted by quaternion imaginaries just as it is weighted by the complex imaginary i in the 2-dimensional case.
If the path does change direction at a vertex, then the path at the point of change gets a weight of -im e, -jm e, or -km e where i,j,k is the quaternion imaginary representing the change of direction, m is the mass (only massive particles can change directions), and sqrt3 e is the timelike length of a path segment, where the 4-dimensional speed of light is taken to be sqrt3.
For a given path, let C be the total number of direction changes, c be the cth change of direction, and ec be the quaternion imaginary i,j,k representing the cth change of direction.
C can be no greater than the timelike Checkerboard distance D between the initial and final points.
The total weight for the given path is then m sqrt3 ec to the Cth power times the product (c from 0 to C) of -ec
Note that since the quaternions are not commutative, the product must be taken in the correct order.
The product is a vector in the direction +/- 1, +/- i, +/- j, or +/- k.
Let N(C) be the number of paths with C changes in direction. The propagator amplitude for the particle to go from the initial vertex to the final vertex is the sum over all paths of the weights, that is the path integral sum over all weighted paths:
the sum from 0 to D of N(C)
times
the Cth power of m sqrt3 ec
times
the product (c from 0 to C) of -ec
The propagator phase is the angle between the amplitude vector in quaternionic 4-space and the quaternionic real axis. The plane in quaternionic 4-space defined by the amplitude vector and the quaternionic real axis can be regarded as the complex plane of the propagator phase.
ManyWorlds of the Feynman Checkerboard can be represented by Surreal Numbers.
