### Clifford Tensor Product Universe

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

... x Cl x Cl x Cl x Cl x Cl x Cl x Cl x ...

where Cl = Mat2(C) to a similar structure with Cl = Mat16(R).

• Begin by considering the Clifford tensor product as a linear chain of Cl's.
• Consider each Cl in the linear chain as a node in a linear pregeometry.
• Let the linear pregeometry, like a long line of yarn, "fold" or "weave" it into a higher-dimensional "array" or "tapestry" of Cl's.
• Prior to the folding/weaving, each Cl node in the linear pregeometry would have 2 nearest neighbors in the chain
... Cl--Cl--Cl--Cl--Cl--Cl--Cl ...
that corresponds to the 1-dim lattice of Natural Numbers.
• After the folding/weaving, each Cl pregeometry node in the tapestry could have more nearest neighbors. For an oversimpified visualization, consider each Cl pregeometry node as having 4 "arms" or "hooks" corresponding to { -x, +x, -t, +t } (i.e., this oversimplification is sort of like Cl(2,R) for which the 4 arms/hooks of each Cl would correspond to + and - in the Cl(2,R) 2-dimensional vector space), so, for the purpose of this visualization, denote each Cl pregeometry node by the 4-armed symbol + to get the linear chain
`... +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+-- ...`

which might be folded/woven roughly as follows:

```... +--+--+--+--+--+--+--+--+
|
+--+--+--+--+--+--+--+--+--+-- ...

... +--+--+--+--+--+--+--+--+--+
|
... +--+--+--+--+--+--+--+--+  +
|  |
+--+

+--+--+--+
|        |
+  +--+  +
|  |  |  |
+  +  +--+
|  |
+  +--+--+--+--+--+--+--+--+--+--+-- ...
|
+--+--+--+--+--+--+--+--+--+--+--+-- ...
```
• After formation of natural nearest-neighbor-connections among the folded/woven pregeometry nodes, then you might get:
```... +--+--+--+--+--+--+--+--+
|
+--+--+--+--+--+--+--+--+--+-- ...

... +--+--+--+--+--+--+--+--+--+
|  |  |  |  |  |  |  |  |  |
... +--+--+--+--+--+--+--+--+--+
|  |
+--+

```

+--+--+--+

```                      |  |  |  |
+--+--+--+
|  |  |  |
+--+--+--+
|  |  |  |
+--+--+--+--+--+--+--+--+--+--+--+-- ...
|  |  |  |  |  |  |  |  |  |  |  |
+--+--+--+--+--+--+--+--+--+--+--+-- ...
```

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.

The Ising model ( shown by Hal Gersch (Int. J. Theor. Phys. 20 (1981) 491) to be equivalent to a Feynman checkerboard for the 2-dim Dirac equation ) is related to von Neumann algebras:
• Adrian Ocneanu, in his article Quantized Groups, String Algebras and Galois Theory for Algebras, at pages 119-172 in Operator Algebras and Applications, Volume 2, edited by David E. Evans and Masamichi Takesaki (Cambridge 1988), said:
• "... We introduce a Galois type invariant for the position of s subalgebra inside an algebra, called a paragroup, which has a group-like structure. Paragroups are the natural quantization of (finite) groups. ... harmonic analysis for the paragroup corresponding to the group Z2 is done in the Ising model ...".

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:

• "... The planar Ising model has become one of the most important statistical mechanical systems for the study of phase transitions and critical phenomena. ...It is ... the purpose of this article ... to discuss two ... mathematical aspects ... The first ... is the Yang-Baxter system of equations for the planar Ising model in zero field with transfer in the (1,1) direction. This work shows that the Clifford-algebraic structure of the exact solution is a natural consequence of the star-triangle equations. The second item is a Fredholm system which turns out to be of crucial importance in unerstanding surface and interface problems, as well as the pair correlation function. ...".
• There are many possible different ways of folding/weaving, and they might be related to each other in ways that can be described mathematically by braids. As to braids:

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:

• "... The "algebraic quantum field theory" of Haag, Kastler and others ... is an attempt to approach quantum field theory by seeing what constraints are imposed on the underlying operator algebras by general physical principles such as relativistic invariance and positivity of the energy. A von Neumann algebra of "localised observables" is postulated for each bounded region of space-time. Causality implies that these von Neumann algebras commute with each other if no physical signal can travel between the regions in which they are localised. The algebras act simultaneously on some Hilbert space which carries a unitary representation of the Poincare (=Lorentz plus 4-d translations) group. The amount of structure that can be deduced from this data is quite remarkable. ... Just as remarkably, more than one type II1 factor (up to isomorphism) was constructed ... and ... uncountably many were shown to exist and the classification of factors is not at all straightforward. That is the bad news.

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:

• "... In quantum mechanics, symmetry groups can be realized by projective, as well as by ordinary unitary, representations. For the permutation symmetry relevant to quantum statistics of N indistinguishable particles, the simplest properly projective representation is highly non-trivial, of dimension 2^{(N-1)/2)\$, and is most easily realized starting with spinor geometry. Quasiparticles in the Pfaffian quantum Hall state realize this representation. Projective statistics is a consistent theoretical possibility in any dimension. ...[A]... very basic quantum mechanical symmetry concerns the interchange, or permutation, of indistinguishable particles. It is natural to ask whether the permutation symmetry is realized projectively in Nature. The mathematical theory of projective representations of the group SN of permutations of N elementary particles was developed in classic papers by I. Schur ...[ in 1907 and 1911]... , prior to the discovery of either modern quantum mechanics or spinors. The simplest (irreducible) non-trivial projective representations of SN are already surprisingly intricate and have dimensions which grow exponentially with N. They are intimately related to spinor representations of SO(N) ... For even N = 2p one can construct an irreducible representation of the G [my substitution for capital gamma] matrices of dimension 2^p iteratively ... This is not irreducible for SO(2p) ... By projecting onto the eigenvalues of k = G1 G2 ... G2p we get irreducible spinor representations. k, of course, does not commute with the representatives of the permutation group. But k' = k( G1 - G2 + G3 - G4 ... ) does. By projecting onto its eigenvalues, we obtain irreducible (projective) representations of S2p. ... Schur demonstrated that all the non-trivial, irreducible projective representations of SN realize the modified algebra ...[and]... may be classified using Young diagrams, but with the additional restriction that row lengths must be strictly decreasing. In this construction, the spinorial representation constructed above corresponds to a single row, analogous to bosons. ... In recent work on the Pfaffian nu = 1/2 quantum Hall state, it was shown that 2n quasiparticles at fixed positions span a 2^(n-1) dimensional Hilbert space, and that braiding such quasiparticles around one another generated operations closely analogous to spinor representations ... (In addition, there are exp( 2 pi i / 8 ) "anyonic" phase factors.) The concepts explained above allow one to formulate the results in a different way: the exchange of these quasiparticles realizes the simplest projective representation of the symmetric group.

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

Cl(8n,R) = Cl(8,R) x ...(n times tensor)... x Cl(8,R)

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).

• The special dimensions 1,2,4,8 in which there exist real division algebras may contain special folding/weaving configurations that are likely to emerge from pregeometry to physical spacetime geometry, perhaps based on lattices such as: