The beginning of our physical universe may be described by a D4-D5-E6-E7-E8 VoDou Physics model generalization ( related to loopoids ) of the von Neumann hyperfinite II1 Clifford tensor product
where Cl = Mat2(C) to a similar structure with Cl = Mat16(R).
that corresponds to the 1-dim lattice of Natural Numbers.
... +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+-- ...
which might be folded/woven roughly as follows:
... +--+--+--+--+--+--+--+--+ | +--+--+--+--+--+--+--+--+--+-- ... ... +--+--+--+--+--+--+--+--+--+ | ... +--+--+--+--+--+--+--+--+ + | | +--+ +--+--+--+ | | + +--+ + | | | | + + +--+ | | + +--+--+--+--+--+--+--+--+--+--+-- ... | +--+--+--+--+--+--+--+--+--+--+--+-- ...
... +--+--+--+--+--+--+--+--+ | +--+--+--+--+--+--+--+--+--+-- ... ... +--+--+--+--+--+--+--+--+--+ | | | | | | | | | | ... +--+--+--+--+--+--+--+--+--+ | | +--+
+--+--+--+
| | | | +--+--+--+ | | | | +--+--+--+ | | | | +--+--+--+--+--+--+--+--+--+--+--+-- ... | | | | | | | | | | | | +--+--+--+--+--+--+--+--+--+--+--+-- ...
If you continue that pattern of folding/weaving indefinitely in a natural way, you might end up with a 2-dim square lattice that could be taken to be an Ising model, or, equivalently, a Feynman checkerboard for the 2-dim Dirac equation. (That equivalence has been shown by Hal Gersch (Int. J. Theor. Phys. 20 (1981) 491).)
Note that in the case of the example show above where each vertex neighborhood + looks in the continuum limit like a unit disk of the complex plane, all the vertex neighborhoods are the same, and the total 2-dim lattice space looks in the continuum limit like a big complex plane with its unique differential structure,
while in the case of each vertex neighborhood + looking like an E8 lattice (in the continuum limit like the 8-dimensional vector space of the Cl(8,R) = Mat16(R)) there can be 7 different kinds of vertex neighborhoods, corresponding to the 7 different E8 latttices. Further, if in the continuum limit the boundary of each vertex neighborhood looks like a 7-sphere S7, then, since each S7 can have 28 different differential structures, the total 8-dim space can have a very complicated structure, whether viewed as a lattice (with varying types of E8 neighborhoods) or in the continuum limit (as a manifold with complicated Riemannian structure).
It is interesting that the 2-dimensional weave structure looks a lot like a Ulam spiral.
According to an Abarim web page:
"... Stanislaw Ulam was attending some boring meeting, and to divert himself somewhat he began to scribble on a piece of paper. ... He put down the number 1 as the bright shining center of a universe of numbers that Big Banged outwardly in a spiral:Much to his amazement the prime numbers appeared to gravitate towards diagonal lines emanating from the central 1. ... Most of them sat on or in the vicinity of a diagonal, but some obviously didn't. ...".
According to a Prime Number Spiral web page:
"... Consider a rectangular grid. We start with the central point and arrange the positive integers in a spiral fashion (anticlockwise) as at right. The prime numbers are then marked ... There is a tendency for the prime numbers to form diagonal lines. This can be seen more clearly in the image below,which shows a window onto a square array of 640 x 640 numbers, with the primes marked by white pixels. ...".
According to a M. Watkins web page:
"... There is currently no explanation for the distinct diagonal lines which appear when the primes are marked out along a particular 'square spiral' path. ...".
Physically, if the 2-dim weave corresponds to 2-dim spacetime, the diagonal lines would correspond to light-cone correlations of points of spacetime.
In dimensions greater than 2, the weaving should produce something like a Moore space-filling curve. Acccording to a web page of V. B. Balayoghan:
"... The Hilbert and Moore curves use square cells -- the level n curve has 4^n cells (and hence 4^n - 1 lines). The Moore curve has the same recursive structure as the Hilbert curve, but ends one cell away from where it started. The Hilbert curve starts and ends at opposite ends of a side of the unit square. ...".
According to a web page by William Gilbert:
"... We exhibit a direct generalization of Hilbert's curve that fills a cube. The first three iterates of this curve are shown.In constructing one iterate from the previous one, note that the direction of the curve determines the orientation of the smaller cubes inside the larger one.
The initial stage of this three dimensional curve can be considered as coming from the 3-bit reflected Gray code which traverses the 3-digit binary strings in such a way that each string differs from its predecessor in a single position by the addition or subtraction of 1. The kth iterate could be considered a a generalized Gray code on the Cartesian product set {0,1,2,...,2^k-1}^3.
The n-bit reflected binary Gray code will describe a path on the edges of an n-dimensional cube that can be used as the initial stage of a Hilbert curve that will fill an n-dimensional cube. ...".
According to Numerical Recipes in C, by Press, Teukolsky, Vettering, and Flannery (2nd ed, Cambridge 1992):
"... A Gray code is a function G(i) of the integers i, that for each integer N > 0 is one-to-one for 0 < i < 2^N -1, and that has the following remarkable property: The binary representation of G(i)and G(i+1) differ in exactly one bit. an example of a Gray code ... is the sequence ...[0000 ( 0=0000), 0001 ( 1=0001), 0011 ( 2=0010), 0010 ( 3=0011), 0110 ( 4=0100), 0111 ( 5=0101), 0101 ( 6=0110), 0100 ( 7=0111), 1100 ( 8=1000), 1101 ( 9=1001), 1111 (10=1010), 1110 (11=1011), 1010 (12=1100), 1011 (13=1101), 1001 (14=1110), 1000 (15=1111)]... for i = 0, ... 15. The algorithm for generating this code is simply to form ... XOR of i with 1/2 (integer part). ... G(i) and G(1+1) differ in the bit position of the rightmosst zero bit of i ... Gray codes can be useful when you need to do some task that depends intimatelyu on the bits of i, looping over many values of i. Then, if there are economies in repeating the task for values differing only by one bit, it makes sense to do things in Gray code order rather than consecutive order. ...".
According to some MathWorld web pages:
"... The binary reflected Gray code is closely related to the solutions of the towers of Hanoi and baguenaudier, as well as to Hamiltonian circuits of hypercube graphs ...[ A Hamiltonian Circuit is]... A graph cycle (i.e., closed loop) through a graph that visits each node exactly once ... The number of Hamiltonian circuits on an n-hypercube is 2, 8, 96, 43008, ...".
If you look at a 2-dimensional slice of the n-dimensional Moore curve including the time axis and one spatial axis, you see something like a Ulam Spiral and also like a 2-dimensional Feynman checkerboard.
D. B. Abraham, in his article Some Recent Results for the Planar Ising Model, at pages 1-22 in Operator Algebras and Applications, Volume 2, edited by David E. Evans and Masamichi Takesaki (Cambridge 1988), said:
Vaughan F. R. Jones, in his review of the book Quantum symmetries on operator algebras, by D. Evans and Y. Kawahigashi, Oxford Univ. Press, New York, 1998, Bull. (N.S.) Am. Math. Soc., Volume 38, Number 3, Pages 369-377, said:
Now the good news. A von Neumannn algebra is called hyperfinite if it contains an increasing dense sequence of finite dimensional *-subalgebras ... it was shown that there is a unique hyperfinite II1 factor. (It can be realised as U(G) where G is the group of all finite permutations of [the natural numbers] N .) ...
...The ideal result would be that to each standard invariant there is a unique subfactor of the hyperfinite II1 factor. This is partly true. There is an amenability condition for a subfactor defined in terms of the random walk on the principal graph. For amenable subfactors (in particular finite depth ones) and standard invariants Popa has shown that the ideal result holds true. This is a deep theorem and implies among other things the Connes-Ocneanu classification of actions of discrete amenable groups on the hyperfinite II1 factor ... Outside the amenable world things go wrong in both directions. Using actions of free groups it is easy to construct families of subfactors with the same standard invariant, and an unpublished result of Popa implies that even the simplest case (the "Temperley-Lieb" algebra in planar algebra terminology) is not always obtainable from a hyperfinite subfactor. ...
... Ocneanu has shown that subfactors (of finite index and depth) are equivalent to Topological Quantum Field Theories and so give a wealth of unitary representations of mapping class groups and braid groups. ...".
The group G of all finite permutations of the natural numbers N can interchange any pregeometry node Cl = Mat16(R) = Cl(8,R) Clifford algebra with any other node on the pregeometry line parameterized by the natural numbers.
Frank Wilczek, in his paper Projective Statistics and Spinors in Hilbert Space, hep-th/9806228, said:
Another perspective on the projective statistics arises from realizing the Clifford algebra in terms of fermion creation and annihilation operators ... we find for the interchange of an odd [2j-1] index particle with the following even [2j] index particle ... is ... simply the operation of changing the occupation of the j th mode. This makes contact with an alternative description of the nu = 1/2 quasiparticles using antisymmetric polynomial wave-functions, which can be considered to label occupation numbers of fermionic states ... Thus projecting to eigenvalues of k amounts to restricting attention to either even or odd mode occupations. This is adequate to get irreducible representations of the rotation group or of the even permutations. If we want to get an irreducible representation of all permutations we must allow both even and odd occupations, with a peculiar global relation between them. Since the definition of projective statistics refers to interchanges of particles, as opposed to braiding, this concept is not in principle tied to 2+1 dimensional theories. Also, no violation of the discrete symmetries P, T is implied. ...".
By taking the limit as n goes to infinity of the real-Clifford-periodicity tensor factorization of order 8
the full hyperfinite II1 von Neumann algebra R can be denoted as the real Clifford algebra Cl(infinity,R) whose half-spinors are sqrt(2^(infinity))-dimensional. In other words, since the halfspinors of Cl(2n,R) are 2^(n-1)-dimensional, the dimension of the full spinors grows exponentially with the dimension of the vector space of the Clifford algebra.
Note that, unlike vectors of a Clifford algebra (which define a vector space on which actions take place) and bivectors of a Clifford algebra (which define a Lie algebra of rotations on that vector space),
spinors of a Clifford algebra encode information from all parts of the Clifford algebra (such as the orientation/entanglement relations of spin 1/2 fermions with respect to physical vector spacetime), so that Projective Permutation symmetry of the entire Clifford algebra Cl(infinity,R) of pregeometry nodes Cl(8,R) can be represented by the half-spinors of Cl(infinity, R) which in turn can be represented by the infinite tensor product of 8-dimensional half-spinors of Cl(8,R) = Mat16(R).
......