## From Sets to Quarks:

```Deriving the Standard Model plus Gravitation
from Simple Operations on Finite Sets
by Tony Smith

Chapter 1 - Introduction.
Chapter 2 - From Sets to Clifford Algebras.
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.```

## Feynman Checkerboards.

(Feynman Checkerboards and conventional Lattice Gauge Theory are two approaches to formulating physics models on lattices.)

```
The 2-dimensional Feynman Checkerboard
is a notably successful and useful representation of
the Dirac equation in 2-dimensional spacetime.

To build a Feynman 2-dimensional Checkerboard,

two future lightcone links and two past lightcone

The future lightcone then looks like
```
` `
```
If the 2-dimensional Feynman Checkerboard is

coordinatized by the complex plane C:

the real axis 1 is identified with the time axis t;

the imaginary axis i is identified with the space axis x; and

the two future lightcone links are
(1 / sqrt2)(1 + i) and (1 / sqrt2)(1 - i).

In cylindrical coordinates t,r with r^2 = x^2,

the Euclidian metric is t^2 + r^2 = t^2 + x^2 and

the Wick-Rotated Minkowski metric with speed of light c is

(ct)^2 - r^2 = (ct)^2 - x^2.

For the future lightcone links on

the 2-dimensional Minkowski lightcone, c = 1.

by complex multiplication by  +/- i.

Now, consider a path in the Feynman Checkerboard.

At a given vertex in the path, denote the future lightcone link in

the same direction as the past path link by 1, and

the (only possible) changed direction by  i.
```
` `
```
The Feynman Checkerboard rule is that

if the future step at a vertex point of a given path
is in a different direction

from the immediately preceding step from the past,

then the path at the point of change gets a weight
of  -i m e ,

where m is the mass
(only massive particles can change directions),and

e is the length of a path segment.

Here I have used the Gersch convention

of weighting each turn by  -im e

rather than the Feynman convention

of weighting by  +im e, because Gersch's
convention gives a better nonrelativistic limit
in the isomorphic 2-dimensional Ising model.

HOW SHOULD THIS BE GENERALIZED TO HIGHER DIMENSIONS?

The 2-dim future light-cone is the 0-sphere S^2-2 = S^0 =
{ i, 1 } ,

with 1 representing a path step to the future in

the same direction as the path step from the past, and

i representing a path step to the future in a

(only 1 in the 2-dimensional Feynman Checkerboard lattice)

different direction from the path step from the past.

The 2-dimensional Feynman Checkerboard lattice spacetime can be

represented by the complex numbers C,

with 1,i representing the two future lightcone directions and

-1,-i representing the two past lightcone directions.

Consider a given path in

the Feynman Checkerboard lattice 2-dimensional spacetime.

At any given vertex on the path in the lattice 2-dimensional
spacetime,

the future lightcone direction representing the
continuation of the path

in the same direction can be represented by 1, and

the future lightcone direction representing the (only 1 possible)

change of direction can be represented by  i since either

of the 2 future lightcone directions can be taken into the other

by multiplication by  +/- i,

+ for a left turn and - for a right turn.

If the path does change direction at the vertex,

then the path at the point of change gets a weight of
-im e,

where i is the complex imaginary,

m is the mass (only massive particles can change directions),
and

e is the timelike length of a path segment,
where the 2-dimensional speed of light is taken to be 1.

Here I have used the Gersch convention

of weighting each turn by  -im e

rather than the Feynman convention

of weighting by  +im e, because Gersch's
convention gives a better nonrelativistic limit
in the isomorphic 2-dimensional Ising model \citeGER.

For a given path, let

C be the total number of direction changes, and

c be the cth change of direction, and

i be the complex imaginary representing
the cth change of direction.

C can be no greater than the timelike Checkerboard distance D

between the initial and final points.

The total weight for the given path is then

the product where c runs from 0 to C of  -ime

or in other words  -ime  to the Cth power.

The product is a vector in the direction  +/- 1 or  +/- i.

Let N(C) be the number of paths with C changes in direction.

The propagator amplitude for the particle to go from
the initial vertex to the final vertex is the sum over all
paths of the weights, that is the path integral sum
over all weighted paths:

the sum over C from 0 to D  of  N(C) times -ime to the Cth power.

The propagator phase is the angle between

the amplitude vector in the complex plane and the complex real axis.

Conventional attempts to generalize the Feynman Checkerboard from

2-dimensional spacetime to k-dimensional spacetime are based on

the fact that the 2-dimensional future light-cone directions are

the 0-sphere S^2-2 = S^0 = { i,1 }.

The k-dimensional continuous spacetime lightcone directions are

the (k-2)-sphere S^k-2.

In 4-dimensional continuous spacetime, the lightcone directions
are S^2.

Instead of looking for a 4-dimensional lattice spacetime, Feynman

and other generalizers went from discrete S^0
to continuous S^2

for lightcone directions, and then tried to construct a weighting

using changes of directions as rotations
in the continuous S^2, and

never (as far as I know) got any generalization that worked.

The HyperDiamond generalization has
discrete lightcone directions.

If the 4-dimensional Feynman Checkerboard is coordinatized by

the quaternions Q:

the real axis 1 is identified with the time axis t;

the imaginary axes i,j,k are identified with the space
axes x,y,z; and

the four future lightcone links are

(1/2)(1+i+j+k),

(1/2)(1+i-j-k),

(1/2)(1-i+j-k), and

(1/2)(1-i-j+k).

In cylindrical coordinates t,r

with r^2 = x^2+y^2+z^2,

the Euclidian metric is t^2 + r^2 = t^2 +
x^2+y^2+z^2 and

the Wick-Rotated Minkowski metric with speed of light c is

(ct)^2 - r^2 = (ct)^2 - x^2 -y^2 -z^2.

For the future lightcone links on

the 4-dimensional Minkowski lightcone, c = sqrt3.

Any future lightcone link is taken into any other future lightcone

link by quaternion multiplication by  +/- i,  +/- j, or  +/- k.

For a given vertex on a given path,

continuation in the same
direction can be represented by the link 1, and

changing direction can be represented by the

imaginary quaternion  +/- i, +/- j, +/- k corresponding to

the link transformation that makes the change of direction.

Therefore, at a vertex where a path changes direction,

a path can be weighted by quaternion imaginaries

just as it
is weighted by the complex imaginary i in the 2-dimensional case.

If the path does change direction at a vertex, then

the path at the point of change gets a weight of
-im e, -jm e, or -km e

where i,j,k is the quaternion imaginary representing
the change of direction,

m is the mass (only massive particles can change directions),
and

sqrt3 e is the timelike length of a path segment,

where the 4-dimensional speed of light is taken to be sqrt3.

For a given path,

let C be the total number of direction changes,

c be the cth change of direction, and

ec be the quaternion imaginary i,j,k representing
the cth change of direction.

C can be no greater than the timelike Checkerboard distance D

between the initial and final points.

The total weight for the given path is then

m sqrt3 ec to the Cth power times the product (c from 0 to C) of -ec

Note that since the quaternions are not commutative,

the product must be taken in the correct order.

The product is a vector in the direction  +/- 1,
+/- i,  +/- j, or  +/- k.

Let N(C) be the number of paths with C changes in direction.

The propagator amplitude for the particle to go

from the initial vertex to the final vertex is

the sum over all paths of the weights,

that is the path integral sum over all weighted paths:```
```
the sum from 0 to D of N(C)
times
the Cth power of m sqrt3 ec
times
the product (c from 0 to C) of -ec```
```

The propagator phase is the angle between

the amplitude vector in quaternionic 4-space and

the quaternionic real axis.

The plane in quaternionic 4-space defined by

the amplitude vector and the quaternionic real axis

can be regarded as the complex plane of the propagator phase.

Since the D4-D5-E6-E7-E8 VoDou Physics model is
fundamentally a Planck Scale HyperDiamond Lattice
Generalized Feynman Checkerboard model,
it violates Lorentz Invariance at the Planck Scale,
affecting Ultra High Energy Cosmic Rays.

From Sets to Quarks:

Chapter 1 - Introduction.
Chapter 2 - From Sets to Clifford Algebras.
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.