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From Sets toQuarks:

Deriving the Standard Model plus Gravitation from Simple Operations on Finite Sets by Tony Smith Table of Contents: Chapter 1 - Introduction. Chapter 2 - From Sets to Clifford Algebras.                  Sets, Reflections, Subsets, and XOR.                 Discrete Clifford Algebras.                  Real Clifford and Division Algebras.                 Discrete Division Algebra Lattices.                 Spinors.                 Signature.                 Periodicity 8MANY-WORLDS QUANTUM THEORY.  Chapter 3 - Octonions and E8 lattices. Chapter 4 - E8 spacetime and particles. Chapter 5 - HyperDiamond Lattices. Chapter 6 - Spacetime and Internal Symmetry Space. Chapter 7 - Feynman Checkerboards. Chapter 8 - Charge = Amplitude to Emit Gauge Boson. Chapter 9 - Mass = Amplitude to Change Direction. Chapter 10 - Protons, Pions, and Physical Gravitons. Appendix A - Errata for Earlier Papers. References.


From Sets to CliffordAlgebras.

Sets, Reflections, Subsets, and XOR.

  Start with von Neumann's Set Theoretical definition of the Natural Numbers N: 0 = 0 , 1= { 0 } , 2 = { 0 , { 0 } } , ... , n + 1 = n u { n }  .  in which each Natural Number n is a set of n elements. By reflection through zero, extend the Natural Numbers Nto include the negative numbers, thus getting an Integral Domain,the Ring Z. Now, following the approach of Barry Simon,consider a set Sn = { e1, e2, ... , en }  of n elements. Consider the set 2^Sn of all of its 2^n subsets,with a product on 2^Sn definedas the symmetric set difference XOR. Denote the elements of 2^Sn by mA where A is in 2^Sn.  

Discrete CliffordAlgebras.

The Discrete HyperDiamond GeneralizedFeynman Checkerboard and ContinuousManifolds are related by QuantumSuperposition:
 To go beyond set theory to Discrete Clifford Algebras,enlarge 2^Sn to DClG(n) by: order the basis elements of Sn, and then give each element of 2^Sn a sign,either +1 or -1, so that DClG(n) has 2^(n+1) elements. This amounts to orientation of the signed unit basis of Sn. Then define a product on DClG(n) by (x1 eA) (x2 eB) = x3 eA  XOR  B) where A and B are in 2^Sn with ordered elements,and x1, x2, and x3 determine the signs. For given x1 and x2, x3 = x1 x2 x(A,B)where x(A,B) is a function that determines sign by using the rules ei ei = +1 for i in Sn and ei ej = - ej ei for i \neq j in Sn , then writing (A,B) as an ordered set of elements of Sn, then using ei ej = - ej ei to moveeach of the B-elements to the left until it: either meets a similar element and then cancelling itwith the similar element by using ei ei = +1 or it is in between two A-elements in the proper order. DClG(n) is a finite group of order 2^n + 1. It is the Discrete Clifford Group of n signedordered basis elements of Sn. Now construct a discrete Group Algebra of DClG(n)by extending DClG(n) by the integral domain ring Z and using the relations ei ej + ej ei = 2 delta(i,j) 1 where delta(i,j) is the Kronecker delta. Since DClG(n) is of order 2^(n+1),and since two elements differing only by sign correspond to the same Group Algebra basis element, the discrete Group Algebra of DClG(n)is 2^n - dimensional. The vector space on which itacts is the n-dimensional hypercubic lattice Z^n. The discrete Group Algebra of discrete Clifford Group DClG(n)is the discrete Clifford Algebra DCl(n).   Here is an explicit example showing how to assign the elements of the Clifford Group to the basis elements of the Clifford Algebra Cl(3): First, order the 2^(3+1) = 16 group strings into rows lexicographically:  0000000100100011010001010110011110001001101010111100110111101111 Then discard the first bit of each string, because it corresponds to sign which is redundant in defining the Algebra basis.  This reduces the number of different strings to 2^3 = 8:  000001010011100101110111 Then separate them into columns by how many 1's they have:  000      001      010            011      100            101            110                  111 Now they are broken down into the 1 3 3 1 graded pattern of the Clifford Algebra Cl(3). The associative Cl(3) product can be deformedinto the 1 x 1 Octonion nonassociative product  by changing EE from 1 to -1,and IJ from K to -k, JK from I to -i, KI from J to -j,and changing the cross-terms accordingly.  If you want to make an Octonion basis without the graded structure, and with the 7 imaginary octonions all on equal footing,  all you have to do is to assign them, one-to-one in the order starting from the left column and from the top of each row, to the 8 Octonion Algebra strings:   1  000     0000000I  001     1000000J  010     0100000K  100     0010000i  110     0001000j  101     0000100k  011     0000010E  111     0000001 To give an example of how to write an octonion product in terms of XOR operations, look at the 7 associative triangles:                        j                    / \                   i---k      J     j     J     I     J     K    / \   / \   / \   / \   / \   / \   i---K I---K I---k E---i E---j E---k   which, in string terms, are each represented by 3 element strings  and 1 Asssociative Triangle string:   Octonion    3 Elements        Associative Triangle   Coassociative Square                  i 0001000   I         J 0100000               0111000                1000111              K 0010000              I 1000000   J         j 0000100               1010100                0101011              K 0010000              I 1000000   K         J 0100000               1100010                0011101              k 0000010              E 0000001   i         I 1000000               1001001                0110110              i 0001000              E 0000001   j         J 0100000               0100101                1011010              j 0000100              E 0000001   k         K 0010000               0010011                1101100              k 0000010              i 0001000   E         j 0000100               0001110                1110001                 k 0000010   Here is Onar Aam's method for calculating the octonion product ab of octonion basis elements a and b in terms of these strings:  To multiply using triangles, note that there are 7 octonion imaginary elements andthat XOR of two triangles give a square and  that the Hodge dual (within imaginary octonions) of that square is the triangle that represents the product of the two triangles, so that:    ab = *(a XOR b) For instance, here is an example of multiplying by triangles (up to +/- sign determined by ordering):   EI = *( 0001110 XOR 0111000 ) = *(0110110) = 1001001 = i On the other hand, the XOR of two squares is a square, so that multiplying by squares simply becomes an XOR. and we have (up to +/- sign determined by ordering): ij = 0110110 XOR 1011010 = 1101100 = k  

For a related method of calculating the imaginaryoctonion cross-product, see thispart of my updated Corvallis 97 talk.

    The Clifford Algebra product . combines the vector space exterior /\ product and the vector space interior product |_ so that, if a is a 1-vector and B is a k-vector, a.B  =  a/\B  -  a|_B The Clifford Algebra has the same graded structure as the exterior /\ algebra of the vector space, and the underlying exterior product antisymmetry rule that A/\B  =  (-1)^pq  B/\A    for p-vector A and q-vector B.     The associative Cl(3) product can be deformedinto the 1 x 1 Octonion nonassociative product  by changing EE from 1 to -1,and IJ from K to -k, JK from I to -i, KI from J to -j,and changing the cross-terms accordingly.  A fundamental reason for the deformation is that the graded structure of the Clifford algebra gives it the underlying exterior product antisymmetry rule that A/\B  =  (-1)^pq  B/\A    for p-vector A and q-vector B, while for Octonions, you want for all unequal imaginary octonions to have the antisymmetry rule  AB  =  - BA  and for equal ones to have  AA = -1.     The discrete Clifford Algebra DCl(n) acts onthe n-dimensional hypercubic lattice Z^n.   

Real Cliffordand Division Algebras.

  Compare the Discrete Clifford Algebra, withthe Real Group Algebra of DClG(n) made byextending it by the real numbers R, which is the usual realEuclidean Clifford Algebra Cl(n) of dimension 2^n,acting on the real n-dimensional vector space R^n.  The empty set 0 corresponds to a vector space of dimension -1,so that Cl(-1) corresponds to the VOID.   Cl(0) has dimension 2^0 = 1 andcorresponds to the real numbers and to time.Its even subalgebra is EMPTY.Cl(0) is the 1 x 1 real matrix algebra. Cl(1) has dimension 2^1 = 2 = 1 + 1 = 1 + i  andcorresponds to the complex numbers and to 2-dim space-time.Its even subalgebra is Cl(0).Cl(1) is the 1 x 1 complex matrix algebra. Cl(2) has dimension 2^2 = 4 = 1 + 2 + 1 = 1 + { j,k } + i  andcorresponds to the quaternions and to 4-dim space-time.Its even subalgebra is Cl(1).Cl(2) is the 1 x 1 quaternion matrix algebra. Cl(3) has dimension 2^3 = 8 = 1 + 3 + 3 + 1 =1 + { I,J,K } + { i,j,k } + E  andcorresponds to the octonions and to 8-dim space-time.Its even subalgebra is Cl(2).Cl(3) is the direct sum of two 1 x 1 quaternion matrix algebras.  The associative Cl(3) product can be deformedinto the 1 x 1 octonion nonassociative product  by changing EE from 1 to -1,and IJ from K to -k, JK from I to -i, KI from J to -j,and changing the cross-terms accordingly.  A fundamental reason for the deformation is that the graded structure of the Clifford algebra gives it the underlying exterior product antisymmetry rule that A/\B  =  (-1)^pq  B/\A    for p-vector A and q-vector B, while for octonions, you want for all unequal imaginary octonions to have the antisymmetry rule  AB  =  - BA  and for equal ones to have  AA = -1.    

Compare the Cl(N) associative gradedClifford Algebra with the not-necessarily-associative ungraded2^N-ion Algebra. In particular, compare Cl(8) with Voudon2^8-ions.

  There are a number of choices you can make in writing the octonion multiplication table:  Given 1, i, and j,there are 2 inequivalent quaternion multiplication tables,one with ij = k and the reverse with ji = k, or ij = -k. To get an octonion multiplication table,start with an orthonormal basis of 8 octonions{ 1,i,j,k,E,I,J,K }, and pick a scalar real axis 1 and pick (2 sign choices) a pseudoscalar axis E or -E. Then you have 6 basis elements to designate as i or -i,which is 6 element choices and 2 sign choices. Then you have 5 basis elements to designate as j or -j,which is 5 element choices and 2 sign choices. Then the underlying quaternionic product fixes ij as k or -k,which is 2 sign choices.   How many inequivalent octonion multiplication tables are there? You had 6 i-element choices, 5 j-element choices, and 4 sign choices, for a total of 6 x 5 x 2^4 = 30 x 16 = 480 octonion products.    Cl(4) has dimension 2^4 = 16 = 1 + 4 + 6 + 4 + 1 = = 1 + { S,T,U,V } + { i,j,k,I,J,K } + { W,X,Y,Z } + E  Cl(4) corresponds to the sedenions.Its even subalgebra is Cl(3).Cl(4) is the 2 x 2 quaternion matrix algebra. Cl(4) is the first Clifford algebra that is NOTmade of 1 x 1 matrices or the direct sum of 1 x 1 matrices,and the Cl(4) sedenions do NOT form a division algebra.  
For more details see these WWW URLs, web pages, and references therein:  Clifford Algebras:  http://www.innerx.net/personal/tsmith/clfpq.html  McKay Correspondence:  http://www.innerx.net/personal/tsmith/DCLG-McKay.html  Octonions:  http://www.innerx.net/personal/tsmith/3x3OctCnf.html  Sedenions: http://www.innerx.net/personal/tsmith/sedenion.html  Cross-Products:  http://www.innerx.net/personal/tsmith/clcroct.html  ZeroDivisor Algebras:   http://www.innerx.net/personal/tsmith/NDalg.html  
 

Discrete Division AlgebraLattices.

 The discrete Clifford Algebras DCl(n) can beextended from hypercubic lattices to latticesbased on the Discrete Division Algebras. For n = 2,the 2-dimensional hypercubic lattice Z^2 can bethought of as the Gaussian lattice of the complex numbers. For n greater than 2,the action of the discrete Clifford Algebra DCl(n) onthe n-dimensional hypercubic lattice Z^n can beextended to action on the Dn latticewith 2n(n - 1) nearest neighbors to the origin,corresponding to the second-layer, or norm-square 2,vertices of the hypercubic lautice Zn.The Dn lattice is called the Checkerboard lattice,because it can be represented as one half of the vertices of Z^n. For n = 4,the extension of the 4-dimensional hypercubic lattice Z^4to the D4 lattice produces the lattice of integral quaternions,with 24 vertices nearest the origin, forming a 24-cell. For n = 8,the 8-dimensional lattice D8 can beextended to the E8 lattice of integral octonions,with 240 vertices nearest the origin, forming a Witting polytope,by fitting together two copies of the D8 lattice,each of whose vertices are at the center of the holes of the other. For n = 16,the 16-dimensional lattice D16 can beextended to the \Lambda16 Barnes-Wall lattice.with 4,320 vertices nearest the origin. For n = 24,the 24-dimensional lattice D24 can beextended to the \Lambda24 Leech lattice.with 196,560 vertices nearest the origin.  

Spinors.

 The discrete Clifford Group DClG(n) is a subgroupof the discrete Clifford Algebra DCl(n). There is a 1-1 correspondence betweenthe representations of DCl(n) andthose representations of DClG(n) such that U(-1) = -1. DClG(n) has 2^n 1-dimensional representations,each with U(-1) = +1. The irreducible representations of DClG(n)with dimension greater than 1 have U(-1) = -1,and are representations of DCl(n). If n is even,there is one such representation, of degree 2^n/2,the full spinor representation of DCl(n).It is reducible into two half-spinor representations,each of degree 2^(n - 1)/2. If n is odd,there are two such representations, each of degree 2^(n - 1)/2,One of them is the spinor representation of DCl(n).  

Signature.

 So far, we have been discussing only Euclidean space withpositive definite signature, DCl(n) = DCl(0,n). Symmetriesfor the general signature cases include: DCl(p-1,q) = DCl(q-1,p); The even subalgebras of DCl(p,q) and DCl(q,p) are isomorphic; DCl(p,q) is isomoprphic to both the even subalgebra of DCl(p+1,q)and the even subalgebra of DCl(p,q+1). Signature is not meaningful for complex vector spaces.The complex Clifford algebra DCl(2p)C is thecomplexification DCl(p,p) xR C  

Periodicity8.

 DCl(p,q) has the periodicity properties:  DCl(n,n) = DCl(n-4,n+4)  DCl(n,n+8) = DCl(n,n) x M(R,16) = DCl(n,n) x DCl(0,8) = DCl(n+8,n)  Therefore any discrete Clifford algebra DCl(p,q) of any sizecan first be embedded in a larger one with p and q multiples of 8,and then converted to one of the form DCl(0,p+q)which then can be "factored" into DCl(0,p+q) = DCl(0,8) x ... x DCl(0,8) so that the fundamental building block of the real discreteClifford algebras is DCl(0,8). Each of the vector, +half-spinor, and -half-spinorrepresentations of DCl(0,8) is 8-dimensionaland can be represented by an octonionic E8 lattice.  

MANY-WORLDSQUANTUM THEORY.

 

To see how Many-Worlds Quantum Theory arises naturally in the D4-D5-E6 HyperDiamond Feynman Checkerboard physics model, note that the model is basically built by using the discrete Clifford Algebra DCL(0,8) as its basic building block, due to the Periodicity 8 property, so that the model looks like a tensor product of Many Copies of DCl(0,8):      DCl(0,8) x DCl(0,8) x DCl(0,8) x ... x DCl(0,8)  How do the Many Copies of DCl(0,8) fit together?   Consider the structure of each Copy of DCl(0,8).   Its graded structure is:      1   8  28  56  70  56  28   8   1    
The vector 8 space corresponds to an 8-dimensional spacetime that is a discrete E8 lattice.
 More about the Clifford structure, including position-momentum duality, is HERE. Take any two Copies of DCl(0,8) and consider the origin of the E8 lattice of each Copy.   From each origin, there are 240 links to nearest-neighbor vertices.  The two Copies naturally fit together if theorigin of the E8 lattice of the vector 8 space of one Copy and the origin of the E8 lattice of the vector 8 space of the other Copy are nearest neighbors, one at each end of a single link in the E8 lattice.   If you start with a Seed Copy of DCl(0,8), and repeat the fitting-together process with other copies, the result is one large E8 lattice spacetime, with one Copy of DCl(0,8) at each vertex.    Since there are 7 TYPES OF E8 LATTICE,  7 different types of E8 lattice spacetime neighborhoods can be constructed.  WHAT HAPPENS at boundaries of different E8 neighborhoods?  ALL the E8 lattices have in common links of the form                      +/- V(where V = 1,i,j,k,E,I,J,K)but they DO NOT AGREE for all links of the form      ( +/- W  +/- X  +/- Y  +/- Z ) / 2 (where W = 1,E;  X = i,I; Y = j,J; Z = k,K)    ALL the E8 lattices BECOME CONSISTENT if they are DECOMPOSED into two 4-dimensional HyperDiamond latticesso that  E8 = 8HD = 4HDa + 4HDca where 4HDa is the 4-dimensional associative Physical Spacetime and 4HDca is the 4-dimensional coassociative Internal Symmetry space.    Therefore, the D4-D5-E6 HyperDiamond Feynman Checkerboard model is physically represented on a 4HD lattice Physical Spacetime, with an Internal Symmetry space that is also another 4HD.    
BIVECTOR GAUGE BOSON STATES ON LINKS:
 The bivector 28 space corresponds to the 28-dimensional D4 Lie algebra Spin(0,8), which,  after Dimensional Reduction of Physical Spacetime,  corresponds to 28 gauge bosons:       12 for the Standard Model,     15 for Conformal Gravity and the Higgs Mechanism, and      1 for propagator phase.  More about the Clifford structure, including position-momentum duality, is HERE. Define a Bivector State of a given Copy of DCl(0,8) to be a configuration of the 28 gauge bosons at its vertex.  Now look at any link in the E8 lattice, and at the two Copies of DCl(0,8) at each end.  The gauge boson state on that link is given by the Lie algebra bracket product of the Bivector States of the two Copies of DCL(0,8) at each end.   Now define the Total Superposition Bivector State of a given Copy of DCl(0,8) to be the superposition of all configurations of the 28 gauge bosons at its vertex and   the Total Superposition Gauge Boson State on a link to be the superposition of all gauge boson states on that link.   
SPINOR FERMION STATES AT VERTICES:
 The 8+8 = 16 fermions corresponding to spinors do not correspond to any single grade of DCl(0,8)      1   8  28  56  70  56  28   8   1 but correspond to the entire Clifford algebra as a whole.  Its total dimension is  2^8 = 256=16x16 and there are, in the first generation, 8 half-spinor fermion particles and 8 half-spinor fermion antiparticles, for a total of 16 fermions.   More about the Clifford structure, including position-momentum duality, is HERE.Dimensional Reduction of Physical Spacetime produces 3 Generations of spinor fermion particles and antiparticles.   Define a Spinor Fermion State at a vertex occupied by a given Copy of DCl(0,8) to be a configuration of the spinor fermion particles and antiparticles of all 3 generations at its vertex.  Now define the Total Superposition Spinor Fermion State at a vertex to be the superposition of all Spinor fermion states at that vertex.    Now we have:  4HD HyperDiamond lattice Physical Spacetime and for each link, a Total Superposition Gauge Boson State and for each vertex, a Total Superposition Spinor Fermion State.    Now:  

Define Many-Worlds Quantum Theory by specifying each ofits many Worlds, as follows:

Each World of the Many-Worlds is determinedby:

for each link, picking one Gauge Boson State from theTotal Superposition

and

for each vertex, picking one Spinor Fermion State fromthe Total Superposition.

 

To get an idea of how to think about the D4-D5-E6 HyperDiamond Feynman Checkerboard lattice model, here is a rough outline of how the Uncertainty Principle works:   Do NOT (as is conventional) say that a particle is sort of "spread out" around a given location in a given space-time                        |                      x                     xxx                   xxxxxxx              xxxxxxxxxxxxxxxxxdue to "quantum uncertainty".   Instead, say that the particle is really at a point in space-time                      |                      xBUT that the "uncertainty spread" is not a property of the particle, but is due to dynamics of the space-time, in which particle-antiparticle pairs x-o are being createdsort of at random.  For example, in one of the Many-Worlds,the spacetime might not be just                       | but would have created a particle-antiparticle pair                      |                x - o  If the original particle is where we put it to start with, then in this World we would have                       |                x - o x  Now, if the new o annihilates the original x, we would have                       |                x                             and, since the particles x are indistinguishable from each other, it would APPEAR that the original particle x was at a different location, and the probabilities of such appearances would look like the conventional uncertainty in position.    In the D4-D5-E6-E7-E8 Vodou Physics model, correlated states, such as a particle-antiparticle pair coming from the non-trivial vacuum, or an amplitude for two entangled particles, extend over a part of the lattice that includes both particles. The stay in the same World of the Many-Worlds until they become uncorrelated.    

From Sets to Quarks: Table of Contents: Chapter 1 - Introduction. Chapter 2 - From Sets to Clifford Algebras.   MANY-WORLDS QUANTUM THEORY.  Chapter 3 - Octonions and E8 lattices. Chapter 4 - E8 spacetime and particles. Chapter 5 - HyperDiamond Lattices. Chapter 6 - Spacetime and Internal Symmetry Space. Chapter 7 - Feynman Checkerboards. Chapter 8 - Charge = Amplitude to Emit Gauge Boson. Chapter 9 - Mass = Amplitude to Change Direction. Chapter 10 - Protons, Pions, and Physical Gravitons. Appendix A - Errata for Earlier Papers. References.


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