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From Sets toQuarks:

Deriving the Standard Model plus Gravitation from Simple Operations on Finite Setsby Tony Smith Table of Contents: 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.


FeynmanCheckerboards.

(Feynman Checkerboards and conventionalLattice Gauge Theory are two approaches to formulating physicsmodels on lattices.)

 The 2-dimensional Feynman Checkerboardis a notably successful and useful representation ofthe Dirac equation in 2-dimensional spacetime. To build a Feynman 2-dimensional Checkerboard, start with a 2-dimensional Diamond Checkerboard with two future lightcone links and two past lightconelinks at each vertex. 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. Either link is taken into the other linkby 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 future lightcone link inthe (only possible) changed direction by  i.   
 The Feynman Checkerboard rule is that if the future step at a vertex point of a given pathis in a different direction from the immediately preceding step from the past, then the path at the point of change gets a weightof  -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'sconvention gives a better nonrelativistic limitin 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-dimensionalspacetime, the future lightcone direction representing thecontinuation 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'sconvention gives a better nonrelativistic limitin 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 representingthe 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 fromthe initial vertex to the final vertex is the sum over allpaths of the weights, that is the path integral sumover 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 directionsare S^2. Instead of looking for a 4-dimensional lattice spacetime, Feynman and other generalizers went from discrete S^0to continuous S^2 for lightcone directions, and then tried to construct a weighting using changes of directions as rotationsin the continuous S^2, and never (as far as I know) got any generalization that worked.   The HyperDiamond generalization hasdiscrete 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 spaceaxes 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 samedirection 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 itis 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 representingthe 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 representingthe 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: Table of Contents: 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.
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