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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.
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.
Octonions and E8 lattices.
The E8 lattice is made up of one
hypercubic Checkerboard D8 lattice plus
another D8 shifted by a glue vector
(1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2),
so that E8 = D8 u ([1] + D8)
is an 8-dimensional HyperDiamond 8HD lattice.
If octonionic coordinate are chosen so that
a given minimal vector in E8 is +1,
the vectors in E8 that are perpendicular to +1 make up
a spacelike E7 lattice.
The E8 lattice nearest neighbor vertices have
only 4 non-zero coordinates,
like 4-dimensional spacetime with speed of light
c = sqrt3,
rather than 8 non-zero coordinates,
like 8-dimensional spacetime with speed of light c = sqrt7,
so the E8 lattice light-cone structure appears to be
4-dimensional rather than 8-dimensional.
To build the E8 Lattice:
Begin with an 8-dimensional octonionic spacetime R^8,
where a basis for the octonions is { 1,i,j,k,E,I,J,K } .
The vertices of the E8 lattice are of the form
(a01 + a1E + a2i + a3j + a4I + a5K + a6k + a7J)/2 ,
where the ai may be either all even integers, all odd integers,
or four of each (even and odd),
with residues mod 2 in the four-integer cases being
(1;0,0,0,1,1,0,1)
or (0;1,1,1,0,0,1,0)
or the same with the last seven cyclically permuted.
E8 forms an integral domain of integral octonions.
The E8 lattice integral domain has 240 units:
+/- 1, +/- i, +/- j, +/- k +/- E, +/- I, +/- J +/- K,
( +/- 1 +/- I +/- J +/- K)/2, ( +/- E +/- i +/- j +/- k)/2,
and the last two with cyclical permutations of
{ i,j,k,E,I,J,K } in the order (E, i, j, I, K, k, J).
The cyclical permutation (E, i, j, I, K, k, J)
preserves the integral domain E8,
but is not an automorphism of the octonions
since it takes the associative triad { i,j,k }
into the anti-associative triad { j,ie,je }.
The cyclical permutation (E, I, J, i, k, K, j)
is an automorphism of the octonions but
takes the E8 integral domain defined above
into another of seven integral domains.
Denote the integral domain described above as 7E8,
and the other six by iE8 , i = 1, ... , 6.
The 240 units of the 7E8 lattice corresponding to
the integral domain 7E8 represent the 240 lattice points
in the shell at unit distance (also commonly normalized as 2):
+/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,
(+/- 1 +/- I +/- J +/- K)/2
(+/- e +/- i +/- j +/- k)/2
(+/- 1 +/- K +/- E +/- k)/2
(+/- i +/- j +/- I +/- J)/2
(+/- 1 +/- k +/- i +/- J)/2
(+/- j +/- I +/- K +/- E)/2
(+/- 1 +/- J +/- j +/- E)/2
(+/- I +/- K +/- k +/- i)/2
(+/- 1 +/- E +/- I +/- i)/2
(+/- K +/- k +/- J +/- j)/2
(+/- 1 +/- i +/- K +/- j)/2
(+/- k +/- J +/- E +/- I)/2
(+/- 1 +/- j +/- k +/- I)/2
(+/- J +/- E +/- i +/- K)/2
The other six integral domains iE8 are:
1E8:
+/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,
(+/- 1 +/- J +/- i +/- j)/2
(+/- k +/- E +/- I +/- K)/2
(+/- 1 +/- j +/- I +/- K)/2
(+/- i +/- k +/- E +/- J)/2
(+/- 1 +/- K +/- k +/- i)/2
(+/- j +/- E +/- I +/- J)/2
(+/- 1 +/- i +/- E +/- I)/2
(+/- j +/- k +/- J +/- K)/2
(+/- 1 +/- I +/- J +/- k)/2
(+/- i +/- j +/- E +/- K)/2
(+/- 1 +/- k +/- j +/- E)/2
(+/- i +/- I +/- J +/- K)/2
(+/- 1 +/- E +/- K +/- J)/2
(+/- i +/- j +/- k +/- I)/2
2E8:
+/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,
(+/- 1 +/- i +/- k +/- E)/2
(+/- j +/- I +/- J +/- K)/2
(+/- 1 +/- E +/- J +/- j)/2
(+/- i +/- k +/- I +/- K)/2
(+/- 1 +/- j +/- K +/- k)/2
(+/- i +/- E +/- I +/- J)/2
(+/- 1 +/- k +/- I +/- J)/2
(+/- i +/- j +/- E +/- I)/2
(+/- 1 +/- J +/- i +/- K)/2
(+/- j +/- k +/- E +/- I)/2
(+/- 1 +/- K +/- E +/- I)/2
(+/- i +/- j +/- k +/- J)/2
(+/- 1 +/- I +/- j +/- i)/2
(+/- k +/- E +/- J +/- K)/2
3E8:
+/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,
(+/- 1 +/- k +/- K +/- I)/2
(+/- i +/- j +/- E +/- J)/2
(+/- 1 +/- I +/- i +/- E)/2
(+/- j +/- k +/- J +/- K)/2
(+/- 1 +/- E +/- j +/- K)/2
(+/- i +/- k +/- I +/- J)/2
(+/- 1 +/- K +/- J +/- i)/2
(+/- j +/- k +/- E +/- I)/2
(+/- 1 +/- i +/- k +/- j)/2
(+/- e +/- I +/- J +/- K)/2
(+/- 1 +/- j +/- I +/- J)/2
(+/- i +/- k +/- E +/- K)/2
(+/- 1 +/- J +/- E +/- k)/2
(+/- i +/- j +/- I +/- K)/2
4E8:
+/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,
(+/- 1 +/- K +/- j +/- J)/2
(+/- i +/- k +/- E +/- I)/2
(+/- 1 +/- J +/- k +/- I)/2
(+/- i +/- j +/- E +/- K)/2
(+/- 1 +/- I +/- E +/- j)/2
(+/- i +/- k +/- J +/- K)/2
(+/- 1 +/- j +/- i +/- k)/2
(+/- e +/- I +/- J +/- K)/2
(+/- 1 +/- k +/- K +/- E)/2
(+/- i +/- j +/- I +/- J)/2
(+/- 1 +/- E +/- J +/- i)/2
(+/- j +/- k +/- I +/- K)/2
(+/- 1 +/- i +/- I +/- K)/2
(+/- j +/- k +/- E +/- J)/2
5E8:
+/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,
(+/- 1 +/- j +/- E +/- i)/2
(+/- k +/- I +/- J +/- K)/2
(+/- 1 +/- i +/- K +/- J)/2
(+/- j +/- k +/- E +/- I)/2
(+/- 1 +/- J +/- I +/- E)/2
(+/- i +/- j +/- k +/- K)/2
(+/- 1 +/- E +/- k +/- K)/2
(+/- i +/- j +/- I +/- J)/2
(+/- 1 +/- K +/- j +/- I)/2
(+/- i +/- k +/- E +/- J)/2
(+/- 1 +/- I +/- i +/- k)/2
(+/- j +/- E +/- J +/- K)/2
(+/- 1 +/- k +/- J +/- j)/2
(+/- i +/- E +/- I +/- K)/2
6E8:
+/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K,
(+/- 1 +/- E +/- I +/- k)/2
(+/- i +/- j +/- J +/- K)/2
(+/- 1 +/- k +/- j +/- i)/2
(+/- e +/- I +/- J +/- K)/2
(+/- 1 +/- i +/- J +/- I)/2
(+/- j +/- k +/- E +/- K)/2
(+/- 1 +/- I +/- K +/- j)/2
(+/- i +/- k +/- E +/- J)/2
(+/- 1 +/- j +/- E +/- J)/2
(+/- i +/- k +/- I +/- K)/2
(+/- 1 +/- J +/- k +/- K)/2
(+/- i +/- j +/- E +/- I)/2
(+/- 1 +/- K +/- i +/- E)/2
(+/- j +/- k +/- I +/- J)/2
The vertices that appear in more than one lattice are:
+/- 1, +/- i, +/- j, +/- k, +/- E, +/- I, +/- J, +/- K in all of them;
(+/- 1 +/- i +/- j +/- k)/2 and (+/- e +/- I +/- J +/- K)/2 in 3E8, 4E8, and 6E8;
(+/- 1 +/- i +/- E +/- I)/2 and (+/- j +/- k +/- J +/- K)/2 in 7E8, 1E8, and 3E8;
(+/- 1 +/- j +/- E +/- J)/2 and (+/- i +/- k+/- I +/- K)/2 in 7E8, 2E8, and 6E8;
(+/- 1 +/- k +/- E +/- K)/2 and (+/- i +/- j +/- I +/- J)/2 in 7E8, 4E8, and 5E8;
(+/- 1 +/- i +/- J +/- K)/2 and (+/- j +/- k +/- E +/- I)/2 in 2E8, 3E8, and 5E8;
(+/- 1 +/- j +/- I +/- K)/2 and (+/- i +/- k+/- e +/- J)/2 in 1E8, 5E8, and 6E8;
(+/- 1 +/- k +/- I +/- J)/2 and (+/- i +/- j +/- E +/- K)/2 in 1E8, 2E8, and 4E8;
The 240 unit vertices in the E8 lattices do not include
any of the 256 E8 light cone vertices,
of the form (+/- 1 +/- i +/- j +/- k +/- E +/- I +/- J +/- K)/2.
They appear in the next layer out from the origin,
at radius sqrt 2, which layer contains in all 2160 vertices.
The E8 lattice is, in a sense,
fundamentally 4-dimensional.
For instance:
the E8 lattice nearest neighbor vertices have
only 4 non-zero coordinates,
like
4-dimensional spacetime with speed of light c = sqrt(3),
rather than 8 non-zero coordinates,
like
8-dimensional spacetime with speed of light c = sqrt(7),
so
the E8 lattice light-cone structure appears to be
4-dimensional rather than 8-dimensional;
the representation of the E8 lattice by quaternionic
icosians, as described by Conway and Sloane;
the Golden ratio construction of the E8 lattice from
the D4 lattice, which has a 24-cell nearest neighbor
polytope
The construction starts with the 24 vertices of a 24-cell,
then adds Golden ratio points on each of the 96 edges of
the 24-cell, then extends the space to 8 dimensions
by considering the algebraicaly independent sqrt(5)
part of the coordinates to be geometrically independent,
and
finally doubling the resulting 120 vertices in 8-dimensional
space by considering both the D4 lattice and
its dual D4*
to get the 240 vertices of the E8 lattice nearest neighbor
polytope (the Witting polytope); and
the fact that the 240-vertex Witting polytope,
the E8 lattice nearest neighbor polytope,
most naturally lives in 4 complex dimensions,
where it is self-dual, rather than in 8 real dimensions.
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.
Tony Smith's Home Page
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