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 8.
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.
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 N
to 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 defined
as the symmetric set difference XOR.
Denote the elements of 2^Sn by mA where A is in 2^Sn.
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 move
each of the B-elements to the left until it:
either meets a similar element and then cancelling it
with 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 signed
ordered 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 it
acts 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:
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
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:
000
001
010
011
100
101
110
111
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 deformed
into 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 0000000
I 001 1000000
J 010 0100000
K 100 0010000
i 110 0001000
j 101 0000100
k 011 0000010
E 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
and
that 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
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 deformed into 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 on the n-dimensional hypercubic lattice Z^n.
Compare the Discrete Clifford Algebra, with
the Real Group Algebra of DClG(n) made by
extending it by the real numbers R, which is the usual real
Euclidean 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 and
corresponds 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 and
corresponds 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 and
corresponds 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 and
corresponds 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 deformed
into 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.
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 NOT
made 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
The discrete Clifford Algebras DCl(n) can be extended from hypercubic lattices to lattices based on the Discrete Division Algebras. For n = 2, the 2-dimensional hypercubic lattice Z^2 can be thought of as the Gaussian lattice of the complex numbers. For n greater than 2, the action of the discrete Clifford Algebra DCl(n) on the n-dimensional hypercubic lattice Z^n can be extended to action on the Dn lattice with 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^4 to 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 be extended 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 be extended to the \Lambda16 Barnes-Wall lattice. with 4,320 vertices nearest the origin. For n = 24, the 24-dimensional lattice D24 can be extended to the \Lambda24 Leech lattice. with 196,560 vertices nearest the origin.
The discrete Clifford Group DClG(n) is a subgroup of the discrete Clifford Algebra DCl(n). There is a 1-1 correspondence between the representations of DCl(n) and those 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).
So far, we have been discussing only Euclidean space with positive definite signature, DCl(n) = DCl(0,n). Symmetries for 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 the complexification DCl(p,p) xR C
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 size can 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 discrete Clifford algebras is DCl(0,8). Each of the vector, +half-spinor, and -half-spinor representations of DCl(0,8) is 8-dimensional and can be represented by an octonionic E8 lattice.
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 the origin 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 lattices so 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:
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
xxxxxxxxxxxxxxxxx
due to "quantum uncertainty".
Instead, say that the particle is really at a point in space-time
|
x
BUT 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 created
sort 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.
......