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. 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. What Factors Determine Charge? Volume of Target Internal Symmetry Space. Volume of Link to Target. Effective Mass Factors. Gravity. Color Force. Renormalization. Weak Force. Electromagnetism. Chapter 9 - Mass = Amplitude to Change Direction. Chapter 10 - Protons, Pions, and Physical Gravitons. Appendix A - Errata for Earlier Papers. References.

- Elements of a Discrete Clifford algebra correspond to Basis Elements of a Real Clifford algebra.
- General Elements of a Real Clifford
algebra correspond to Superpositions of Basis Elements =
Elements of the underlying Discrete
Clifford algebra.
- Volumes of Spaces of Superpositions of some given Sets of Basis Elements correspond to Mass or Charge of Particles or Forces represented by those Basis Elements.
- Volumes of Spaces of Superpositions of other given Sets of Basis Elements correspond to Volume of Physical SpaceTime and Volume of Internal Symmetry Space represented by those Basis Elements.

In the HyperDiamond Feynman Checkerboard model the charge of a particle is the amplitude for a particle to emit a gauge boson of the relevant force. Neutral particles do not emit gauge bosons. Force strengths are probalilities, or squares of amplitudes for emission of gauge bosons, or squares of charges, so that calculation of charges is equivalent to calculation of force strengths. Three factors determine the probability for emission of a gauge boson from an origin spacetime vertex to a target vertex: the part of the Internal Symmetry Space of the target spacetime vertex that is available for the gauge boson to go to from the origin vertex; the volume of the spacetime link that is available for the gauge boson to go through from the origin vertex to the target vertex; and an effective mass factor for forces (such as the Weak force and Gravity) that, in the low-energy ranges of our experiments, are carried effectively by gauge bosons that are not massless high-energy. In the D4-D5-E6 Lagrangian continuum version of this physics model, force strength probabilities are calculated in terms of relative volumes of bounded complex homogeneous domains and their Shilov boundaries. The relationship between the D4-D5-E6 Lagrangian continuum approach and the HyperDiamond Feynman Checkerboard discrete approach is that: the bounded complex homogeneous domains correspond to harmonic functions of generalized Laplacians that determine heat equations, or diffusion equations; while the amplitude to emit gauge bosons in the HyperDiamond Feynman Checkerboard is a process that is similar to diffusion, and therefore also corresponds to a generalized Laplacian. Details of the D4-D5-E6 Lagrangian continuum approach can be found on the World Wide Web at URLs http://xxx.lanl.gov/abs/hep-ph/9501252 which is also available as a pdf file on this web site. For the discrete HyperDiamond Feynman Checkerboard approach of this paper, the only free charge parameter is the charge of the Spin(5) gravitons in the MacDowell-Mansouri formalism of Gravity. Note that these Spin(5) gravitons are NOT the ordinary spin-2 gravitons of the low-energy region in which we live. The charge of the Spin(5) gravitons is taken to be unity, 1, so that its force strength is also unity, 1. All other force strengths are determined as ratios with respect to the Spin(5) gravitons and each other. The four forces of the HyperDiamond Feynman Checkerboard model are Gravity, the Color force, the Weak force, and Electromagnetism. The charge of each force is the amplitude for one of its gauge bosons to be emitted from a given origin vertex of the spacetime HyperDiamond lattice and go to a neighboring target vertex. The force strength of each force is the square of the charge amplitude, or the probablility for one of its gauge bosons to be emitted from a given origin vertex of the spacetime HyperDiamond lattice and go to a neighboring target vertex. The HyperDiamond Feynman Checkerboard model calculations are actually done for force strengths, or probablilities, because it is easier to calculate probablilities. The force strength probability for a gauge boson to be emitted from an origin spacetime HyperDiamond vertex and go to a target vertex is the product of three things: the volume Vol(MISforce) of the target Internal Symmetry Space, that is, the part of the Internal Symmetry Space of the target spacetime vertex that is available for the gauge boson to go to from the origin vertex; the volume Vol(Qforce) / Vol(Dforce)^( 1 / mforce ) of the spacetime link to the target spacetime vertex from the origin vertex; and an effective mass factor 1 / Mforce^2 for forces (such as the Weak force and Gravity) that, in the low-energy ranges of our experiments, are carried effectively by gauge bosons that are not massless high-energy SU(2) or Spin(5) gauge bosons, but are either massive Weak bosons due to the Higgs mechanism or effective spin-2 gravitons. For other forces, the effective mass factor is taken to be unity, 1. Therefore, the force strength of a given force is alphaforce = (1 / Mforce^2 \) ( Vol(MISforce)) ( Vol(Qforce) / Vol(Dforce)^( 1 / mforce )) The symbols have the following meanings: alphaforce represents the force strength; Mforce represents the effective mass; MISforce represents the part of the target Internal Symmetry Space that is available for the gauge boson to go to; Vol(MISforce) stands for volume of MISforce; Qforce represents the link from the origin to the target that is available for the gauge boson to go through; Vol(Qforce) stands for volume of Qforce; Dforce represents the complex bounded homogeneous domain of which Qforce is the Shilov boundary; mforce is the dimensionality of Qforce, which is 4 for Gravity and the Color force, 2 for the Weak force (which therefore is considered to have two copies of QW for each spacetime HyperDiamond link), and 1 for Electromagnetism (which therefore is considered to have four copies of QE for each spacetime HyperDiamond link) Vol(Dforce)^( 1 / mforce ) stands for a dimensional normalization factor (to reconcile the dimensionality of the Internal Symmetry Space of the target vertex with the dimensionality of the link from the origin to the target vertex). The force strength formula is stated in terms of continuum structures, such as volumes of manifolds, Shilov Boundaries, etc., rather than in discrete terms. We recognize that a discrete version of the calculations would be the fundamentally correct way to calculate in the discrete HyperDiamond Feynman Checkerboard model, but it is easier for us to look up relevant manifold volumes than to write and execute computer code to do the discrete calculations. However, since the HyperDiamond Feynman Checkerboard lattice spacing is Planck length and therefore much smaller than the relevant distances for any experiments that we want to describe, we think that the continuum calculations are good approximations of the fundamental discrete HyperDiamond Feynman Checkerboard calculations. The geometric volumes needed for the calculations, mostly taken from Hua, are Force M Vol(M) Q Vol(Q) D Vol(D) gravity S^4 8pi^2/3 RP^1xS^4 8pi^3/3 IV5 pi^5/2^4 5! color CP^2 8pi^2/3 S^5 4pi^3 B^6(ball) pi^3/6 weak S^2xS^2 2x4pi RP^1xS^2 4pi^2 IV3 pi^3/24 e-mag T^4 4x2pi - - - -

Note that I normalize the size of CP2 so that its volume equals the volume of a unit 4-sphere S4. My reasoning is that:

- the high-energy physical 8-dim spacetime breaks into two 4-dimensional structures, one a 4-dim Physical Spacetime and the other a 4-dim Internal Symmetry Space;
- if the 4-dim Physical Spacetime were Eulidean with S4 structure, it would inherit unit radius normalization from the parent 8-dim spacetime;
- the 4-dim Physical Spacetime regarded as Euclidean S4 should have volume equal to the CP2 4-dim Internal Symmetry Space, which is the other half of 8-dim spacetime; and
- I am effectively regarding CP2 as an S4 everywhere except for structure at infinity, which for CP2 = C2 u CP1 = C2 u C1 u {point} is CP1 = C1 u {point} and for S4 = R4 u {point} is {point}, and I am identifying C2 with R4 in the obvious way and saying that the structure at infinity only affects topological structure, not 4-dim volume.
- Atiyah and Berndt say in their
paper Projective Planes, Serveri Varieties, and
Spheres, math.DG/0206135,
"... There are natural diffeomorphisms
- CP2 / O(1) = S4 ... S0 = O(1) ...
- HP2 / U(1) = S7 ... S1 = U(1) ...
- OP2 / Sp(1) = S13 ... S3 = Sp(1) ...".

Since O(1) = S0 = { -1 , +1 } I identify CP2 with S4 by that 2-to-1 mapping and consider that the volume of CP2 is the same as that of S4 with the difference being that the CP2 "boundary" of the volume is "twice as thick" as the S4 "boundary" of that volume.

My volume for CP2 differs from some others stated in the literature, such as the paper Volumes of Compact Manifolds, math-ph/0210033, by Boya, Sudarshan, and Tilma, where they say:

"... We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or normalization of the manifold, that is, the generalization of the unit radius condition for spheres. ... even ... when we have a manifold described as a product of spheres, the volume will not always be the product of the natural sphere volumes! ... Compact group manifolds are symmetric spaces and therefore the curvature is covariantly constant. There is a natural metric (induced from the Killing form in the Lie algebra) which generates an invariant measure (Haar, or more appropriately Hurwitz), but also there is a structure close to the product of odd spheres: the two criteria clash. ... We have not attempted to calculate the canonical volume for the E6,7,8 groups ...... The sphere S(d-1) is a maximally symmetric space ... Explicit useful formulas for the even/odd cases are Vol(S(2n+1)) = 2 pi^n+1) / n! and Vol(S(2n)) = 2 (2 pi )^n / (2n-1)!! ...

... Notice the volume is maximal for S6, and lim( n -> infinity ) Vol(Sn) -> 0 ...

... Our convention for the radius R = 1 is extrinsic, that is, depends on the embedding S(d-1) in Rd; an equivalent intrinsic criterion is the length of the geodesic: geodesic length = 2 pi <-> radius R = 1 applicable because all geodesics in the sphere are equivalent ...

... the volume of the unit ball is ... Vol(S(n-1)) / n ...

... We shall define projective spaces as quotient of spheres, and we shall see that the natural scale for the geodesics is to have length pi, not 2 pi as in the spheres.

... We ... define the volume of CPn as a quotient

Vol(CPn) = Vol(S(2n+1)) / Vol(S1) = ( 2 pi^(n+1) / n! ) / ( 2 pi ) = pi^n / n! ... for n = 1 we have Vol(CP1) = pi, whereas CP1 = S3 / S1 = S2, and Vol(S2) = 4 pi: obviously the geodesic length of our CP1 is only pi, whereas the volume of the equivalent S2 space is 4 = 2x2 times as big ...

... We compute Vol(Spin(6)) = 2 x Vol(S1 x S2 x ... x S5) = 256 pi^9 / 3

whereas Vol(SU(4)) = sqrt(2) pi^9 / 3 even though SU(4) = Spin(6)! ...".

Note that N!! = N (N-2) (N-4) ... .

I got confused trying to compare their sphere volume formulas with those of Conway and Sloane, but Todd Tilma was nice enough to explain to me by e-mail their equivalence by using formulas such as

I prefer a normalization that makes Vol(Spin(6)) = Vol(SU(4)). Also, using my approach gives Vol(CP1) = Vol(S2), which I think is natural because CP1 = S2. Further, I am more partial than Boya, Sudarshan, and Tilma to the idea that a manifold that is topologically like a product of spheres should be normalized so that its volume is the product of the volumes of those spheres.

Using these numbers, the results of the calculations are the relative force strengths at the characteristic energy level of the generalized Bohr radius of each force: Gauge Force Characteristic Geometric Total Group Energy Force Force Strength Strength Spin(5) gravity approx 10^19 GeV 1 GGmproton^2 approx 5 x 10^-39 SU(3) color approx 245 MeV 0.6286 0.6286 SU(2) weak approx 100 GeV 0.2535 GWmproton^2 approx 1.05 x 10^-5 U(1) e-mag approx 4 KeV 1/137.03608 1/137.03608 The force strengths are given at the characteristic energy levels of their forces, because the force strengths run with changing energy levels. The effect is particularly pronounced with the color force. The color force strength was calculated at various energies according to renormalization group equations, with the following results: Energy Level Color Force Strength 245 MeV 0.6286 5.3 GeV 0.166 34 GeV 0.121 91 GeV 0.106 Shifman in a paper at http://xxx.lanl.gov/abs/hep-ph/9501222 has noted that Standard Model global fits at the Z peak, about 91 GeV, give a color force strength of about 0.125 with LambdaQCD approx 500 MeV, whereas low energy results and lattice calculations give a color force strength at the Z peak of about 0.11 with LambdaQCD approx 200 MeV. In the remainder of this Chapter 8, we discuss further the concepts of Target Internal Symmetry Space, Link to Target, and Effective Mass Factors, and then we discuss in more detail each of the four forces. In this HyperDiamond Feynman Checkerboard model, all force strengths are represented as ratios with respect to the geometric force strength of Gravity (that is, the force strength of Gravity without using the Effective Mass factor).

- Elements of a Discrete Clifford algebra correspond to Basis Elements of a Real Clifford algebra.
- General Elements of a Real Clifford
algebra correspond to Superpositions of Basis Elements =
Elements of the underlying Discrete
Clifford algebra.
- Volumes of Spaces of Superpositions of some given Sets of Basis Elements correspond to Mass or Charge of Particles or Forces represented by those Basis Elements.
- Volumes of Spaces of Superpositions of other given Sets of Basis Elements correspond to Volume of Physical SpaceTime and Volume of Internal Symmetry Space represented by those Basis Elements.

MISforce represents the part of the target Internal Symmetry Space that is available for the gauge boson to go to; and Vol(MISforce) stands for volume of MISforce. What part of the target Internal Symmetry Space is available for the gauge boson to go to? Each vertex of the spacetime HyperDiamond lattice is not just a point, but also contains its own Internal Symmetry Space (also a HyperDiamond lattice), so the amplitude for a gauge boson to go from one vertex to another depends not only on the spacetime link between the vertices but also on the degree of connection the gauge boson has with the Internal Symmetry Spaces of the vertices. If the gauge boson can connect any vertex in the origin Internal Symmetry Space with any vertex in the destination Internal Symmetry Space, then the gauge boson has full connectivity between the Internal Symmetry Spaces. However, if the gauge boson can connect a vertex in the origin Internal Symmetry Space only with some, but not any, of the vertices in the destination Internal Symmetry Space, then the gauge boson has only partial connectivity between the Internal Symmetry Spaces, and has a lower amplitude to be emitted from the origin spacetime vertex to the destination spacetime vertex. The amount of connectivity between the Internal Symmetry Spaces is the geometric measure of the charge of a force, and therefore of its force strength. To represent the gauge boson connectivity between Internal Symmetry Spaces, it is useful to label the basis of the Internal Symmetry Space by the degrees of freedom of the forces: electric charges { +1, -1 } ; and color (red, green, blue) charges { +r, -r; +g, -g; +b, -b } . so that the total basis of the Internal Symmetry Space is { +1, -1, +r, -r; +g, -g; +b, -b } This connectivity can be measured by comparing the full target Internal Symmetry Space HyperDiamond lattice with the subspace of the target Internal Symmetry Space HyperDiamond lattice that is the image of a given point of the origin Internal Symmetry Space under all the transformations of the Internal Symmetry Space of the gauge group of the force. The MISforce target manifolds are Spacetime for Spin(5) Gravity and Internal Symmetry Space for gauge groups SU(3), SU(2), and U(1). Each irreducible component has dimension mforce. The MISforce target manifolds for the four forces are: Gauge Group Symmetric Space mforce MISforce Spin(5) Spin(5) / Spin(4) 4 S^4 SU(3) SU(3) / SU(2)xU(1) 4 CP^2 SU(2) SU(2) / U(1) 2 S^2xS^2 U(1) U(1) 1 T^4

- Elements of a Discrete Clifford algebra correspond to Basis Elements of a Real Clifford algebra.
- General Elements of a Real Clifford
algebra correspond to Superpositions of Basis Elements =
Elements of the underlying Discrete
Clifford algebra.
- Volumes of Spaces of Superpositions of some given Sets of Basis Elements correspond to Mass or Charge of Particles or Forces represented by those Basis Elements.
- Volumes of Spaces of Superpositions of other given Sets of Basis Elements correspond to Volume of Physical SpaceTime and Volume of Internal Symmetry Space represented by those Basis Elements.

If, as in the case of the Electromagnetic U(1) photon, there is only one gauge boson that can go through the link from an origin spacetime HyperDiamond vertex to a target vertex, then the link to the target vertex is like a one-lane road. For the Weak SU(2) force, the Color SU(3) force, and Spin(5) Gravity, there are, respectively, 3, 8, and 10 gauge bosons that can go through the link from an origin spacetime HyperDiamond vertex to a target vertex, so that the link to the target vertex is like a multi-lane highway. The volume of the link to the target vertex is not measured just by the number of gauge bosons, but by the volume of the minimal manifold Qforce that can carry two things: the gauge bosons, with gauge group Gforce; and the U(1) phase of the propagator from origin to target. The first step in constructing Qforce is to find a manifold whose local isotropy symmetry group is Gforce x U(1) so that it can carry both Gforce and U(1). To do that, look for the smallest Hermitian symmetric space of the form K / (Gforce x U(1)). Having found that Hermitian symmetric space, then go to its corrresponding complex bounded homogeneous domain, Dforce. Then take Qforce to be the Shilov boundary of Dforce. Qforce is then the minimal manifold that can carry both Gforce and U(1), and so is the manifold that should represent the link from origin to target vertex. The Qforce, Hermitian symmetric space, and Dforce manifolds for the four forces are: Gauge Hermitian Type mforce Qforce Group Symmetric of Space Dforce Spin(5) Spin(7) / Spin(5)xU(1) IV5 4 RP^1xS^4 SU(3) SU(4) / SU(3)xU(1) B^6(ball) 4 S^5 SU(2) Spin(5) / SU(2)xU(1) IV3 2 RP^1xS^2 U(1) - - 1 - The geometric volumes of the target Internal Symmetry Space MISforce, the link volume Qforce, and the bounded complex domains Dforce of which the link volume is the Shilov boundary, mostly taken from Hua, are:

Force M Vol(M) Qgravity S4 8 pi^2 / 3 RP1xS4 color CP2 8 pi^2 / 3 S5 weak S2xS2 2x4 pi RP1xS2 e-mag T4 4x2 pi -

Force Vol(Q) D Vol(D)gravity 8 pi^3 / 3 IV5 pi^5 / 2^4 5! color 4 pi^3 B^6(ball) pi^3 / 6 weak 4 pi^2 IV3 & pi^3 / 24 e-mag - - -

The geometric part of the force strength is formed from the product of the volumes of the target Internal Symmetry Space MISforce and the link volume Qforce. To take the product properly, we must take into account how the target Internal Symmetry Space MISforce, which lives at the target vertex, and the link volume Qforce, which lives on the link connecting the origin vertex to the target vertex and so can be regarded as containing both the origin vertex and the target vertex, fit together. The dimension mforce of each irreducible component of the target Internal Symmetry Space MISforce is less than the dimension of the link volume Qforce, which in turn is less than the dimension of the bounded complex domain Dforce of which the link volume Qforce is the Shilov boundary. Since the link volume Qforce is a Shilov boundary, it can be regarded as a shell whose Shilov interior is the bounded complex domain Dforce. If we were to merely take the product of the volume of the target Internal Symmetry Space MISforce with the volume of the link volume Qforce, we would be overcounting the contribution of the link volume Qforce because of its dimension is higher than the dimension of each irreducible component of the target Internal Symmetry Space MISforce. To get rid of the overcounting, we should make the link volume Qforce compatible with each irreducible component of the target Internal Symmetry Space MISforce. Since the target Internal Symmetry Space MISforce has fundamentally a 4-dimensional HyperDiamond structure, each irreducible component is fundamentally made up of mforce-dimensional hypercubic cells, where the irreducible component hypercubic cells are 4-dimensional for Gravity and the Color force, 2-dimensional for the Weak force, and 1-dimensional for Electromagnetism. Since Dforce is the Shilov interior of Qforce, Qforce would be compatible with MISforce if Dforce were mapped 1-1 onto an mforce-dimensional hypercubic cell in an irreducible component of the target Internal Symmetry Space MISforce. To do this, first construct a hypercube of the same dimension mforce as an irreducible component of the target Internal Symmetry Space MISforce and the same volume as Dforce. The edge length of such a hypercube is the mforce-th root of the volume of the bounded complex domain Dforce. Since the hypercubes in the fundamental HyperDiamond structures of the target Internal Symmetry Space MISforce and its irreducible components are unit hypercubes with edge length 1, we must, to make the link volume Qforce compatible with each irreducible component of the target Internal Symmetry Space MISforce, divide the volume of the link volume Qforce by the mforce-th root of the volume of the bounded complex domain Dforce to make Dforce the right size to fit the hypercubes in the fundamental HyperDiamond structures of the target Internal Symmetry Space MISforce and its irreducible components. In other words, to reconcile the dimensionality of the link volume Qforce to the dimensionality of the target Internal Symmetry Space, divide Qforce by Vol(Dforce)^( 1 / mforce ) . The resulting volume Vol(Qforce) / Vol(Dforce)^( 1 / mforce ) is then the correctly normalized volume of the spacetime link that is available for the gauge boson to go through to get to the target spacetime vertex from the origin vertex, which correctly normalized volume should be used in multiplying by the volume of he target Internal Symmetry Space MISforce to get the geometric part of the force strength.

In the low-energy ranges of our experiments, the Weak force and Gravity are carried effectively by gauge bosons that are not massless high-energy SU(2) or Spin(5) gauge bosons, but are either massive Weak bosons due to the Higgs mechanism or effective spin-2 gravitons. Mforce represents the effective mass for the Weak force and Gravity in the low-energy range. In the low-energy range, the forece strengths of the Weak force and Gravity have an effective mass factor 1 / Mforce^2. For other forces, the Color force and Electromagnetism, the effective mass factor is taken to be unity, 1.

Effective mass factors for the weak force and gravity have a visualization (arising from e-mail discussion with Dick Andersen): Gauge bosons are visualized as going from a source through a medium to a target:

- The weak force mass factor is related to the Higgs mechanism. The Higgs scalar field absorbs some of the weak bosons as they go through the medium, thus weakening the weak force and producing the weaker effective weak force that is observed by experiments;
- The graviton by itself is long-range and massless, but virtual Planck-mass black holes in spacetime absorb some of the gravitons as they go through the spacetime medium, thus weakening the gravitational force and producing the weaker effective gravitational force that is observed by experiments;

Gravity comes from the conformal group Spin(6) and its de Sitter subgroup Spin(5). For Gravity, Spin(5) gravitons can carry all of the Internal Symmetry Space charges { +1, -1; +r, -r; +g, -g; +b, -b } Spin(5) gravitons act transitively on the 4-dimensional manifold S^4 = Spin(5) / Spin(4), so that they can take any given point in the origin Internal Symmetry Space into any point in the target Internal Symmetry Space. The full connectivity of the Gravity of Spin(5) gravitons is geometrically represented by MISG as the 4-dimensional sphere S^4 = Spin(5) / Spin(4) whose volume is 8pi^2/3. The link manifold to the target vertex is QG = RP^1 x S^4 = ShilovBdy(DG) with volume 8pi^3/3 The bounded complex homogeneous domain DG is of type IV5 with volume pi^5/2^4 5!. It corresponds to the Hermitian Symmetric Space Spin(7)/(Spin(5) x U(1). For Spin(5) Gravity, mG is 4. For Spin(5) Gravity, MG = mPlanck, so that 1 / MG^2 = 1 / mPlanck^2. Therefore, the force strength of Gravity is: alphaG = (1 / MG^2 ) ( Vol(MISG) ) ( Vol(QG) / Vol(DG)^( 1 / mG )) which is GGmproton^2 approx 5 x 10^-39 at the characteristic energy level of approx 10^19 GeV, the Planck energy. The only factor different from 1 in the force strength of Gravity is the Effective Mass factor, because force strengths in the HyperDiamond Feynman Checkerboard model are represented as ratios with respect to the strongest geometric force, Gravity, so that the geometric force strength factors for Gravity are cancelled to unity by taking the ratio with themselves.

A physical interpretation for the value of 1 for the geometric strength of gravity is that gravity is what holds spacetime together, and the amplitude for a graviton to be emitted/absorbed from one point of spacetime to a neighboring (with respect to Planck-length lattice structure) point of spacetime is 1, so that, with probability 1^2 = 1, each spacetime point is connected to its neighboring points, and spacetime holds together.

The mass factor for gravitation has a visualization (arising from e-mail discussion with Dick Andersen). Gauge bosons are visualized as going from a source through a medium to a target. The graviton by itself is long-range and massless, but virtual Planck-mass black holes in spacetime absorb some of the gravitons as they go through the spacetime medium, thus weakening the gravitational force and producing the weaker effective gravitational force that is observed by experiments.

For an estimate of the Planck mass calculated in the spirit of the HyperDiamond Feynman Checkerboard model, see http://www.innerx.net/personal/tsmith/Planck.html The action of Gravity on Spinors is given by the Papapetrou Equations.

In the Near Field Induction/Static Region, the gravitons effectively bypass the virtual Planck-mass black holes in spacetime that absorb some Far Field gravitons as they go through the Far Field Region of the spacetime medium.

For the Color force, SU(3) gluons carry the Internal Symmetry Space color charges { +r, -r; +g, -g; +b, -b } SU(3) gluons act transitively on the 4-dimensional manifold CP^2 = SU(3) / (SU(2) x U(1)), so that they can take any given point in the origin Internal Symmetry Space into any point in the target Internal Symmetry Space. The full connectivity of the Color force of SU(3) gluons is geometrically represented by MISC as CP^2 = SU(3) / (SU(2) x U(1)) whose volume is 8pi^2/3. The link manifold to the target vertex is QC = S^5 = ShilovBdy(B^6) with volume 4pi^3 The bounded complex homogeneous domain DC is of type B^6 (ball) with volume pi^3/6 For the SU(3) Color force, mC is 4. For the SU(3) Color force, MC = 1 , so that for the SU(3) Color force, 1 / MC^2 = 1 Therefore, the force strength of the Color force, which is sometimes conventionally denoted by alphaS (for strong) as well as by alphaC, is: alphaC = alphaS = ( 1 / MC^2 ) ( Vol(MISC) ) ( Vol(QC) / Vol(DC)^( 1 / mC )) which, when divided by the geometric force strength of Gravity, is 0.6286 at the characteristic energy level of about 245 MeV.

Force strength constants and particle masses are not really "constant" when you measure them, as the result of your measurement will depend on the energy at which you measure them. Measurements at one energy level can be related to measurements at another by renormalization equations. In the D4-D5-E6-E7 model, Dilatation Scale Transformations of the Conformal Group provide a natural setting for the Renormalization Group Process. The lightest experimentally observable particle based on the color force is the pion, which is a quark-antiquark pair made up of the lightest quarks, the up and down quarks. A quark-antiquark pair is the carrier of the strong force, and mathematically resembles a bivector gluon, which is the carrier of the color force. The charactereistic energy level of pions is the square root of the sum of the squares of the masses of the two charged and one neutral pion. It is about 245 MeV (to more accuracy 241.4 MeV). The gluon-carried color force strength is renormalized to higher energies from about 245 MeV in the conventional way.

The Color force strength was calculated at various energies according to lowest order renormalization group equations, as also shown at http://www.innerx.net/personal/tsmith/cweRen.html with the following results: Energy Level Color Force Strength 245 MeV 0.6286 5.3 GeV 0.166 34 GeV 0.121 91 GeV 0.106 Shifman, in a paper at http://xxx.lanl.gov/abs/hep-ph/9501222 has noted that Standard Model global fits at the Z0 peak, about 91 GeV, give a color force strength of about 0.125 with LambdaQCD approx 500 MeV, whereas low energy results and lattice calculations give a color force strength at the Z0 peak of about 0.11 with LambdaQCD approx 200 MeV. The low energy results and lattice calculations are closer to the tree level HyperDiamond Feynman Checkerboard model value at 91 GeV of 0.106. Also, the D4-D5-E6 HyperDiamond Feynman Checkerboard model has LambdaQCD equal to about 245 MeV. For the pion mass, upon which the LambdaQCD calculation depends, see http://www.innerx.net/personal/tsmith/SnGdnPion.html

For the Weak force, SU(2) Weak bosons carry the Internal Symmetry Space electric charges { +1, -1 } SU(2) Weak bosons act transitively on the 2-dimensional manifold S^2 = SU(2) / U(1), so that two copies of S^2, in the form of S^2 x S^2, are required so that they can take any given point in the origin Internal Symmetry Space into any point in the target Internal Symmetry Space. Each of the two connectivity components of the Weak force of SU(2) Weak bosons is geometrically represented by MISW as S^2 = SU(2) / U(1) whose volume is 4pi. The total manifold MISW is S^2 x S^2, with volume 2 x 4pi The link manifold to the target vertex is QW = RP^1 x S^2 = ShilovBdy(DW) with volume 4pi^3. The bounded complex homogeneous domain DW is of type IV3 with volume 4pi^2. It corresponds to the Hermitian Symmetric Space Spin(5)/(SU(2) x U(1). For the SU(2) Weak force, mW is 2. For the SU(2) Weak force, due to the Higgs mechanism, MW = sqrt(mW+^2 + mW-^2 + mW0^2), so that 1 / MW^2 = 1 / (mW+^2 +mW-^2 + mW0^2).

The mass factor for the weak force has a visualization (arising from e-mail discussion with Dick Andersen). Gauge bosons are visualized as going from a source through a medium to a target. The weak force mass factor is related to the Higgs mechanism. The Higgs scalar field absorbs some of the weak bosons as they go through the medium, thus weakening the weak force and producing the weaker effective weak force that is observed by experiments;

Therefore, the force strength of the Weak force is: alphaW = ( 1 / MW^2 ) ( Vol(MISW)) ( Vol(QW) / Vol(DW)^( 1 / mW )) which, when divided by the geometric force strength of Gravity, is GWmproton^2 = approx 1.05 x 10^-5 at the characteristic energy level of approx 100 GeV. The geometric component of the Weak force strength, that is, everything but the effective mass factor, has the value 0.2535 when divided by the geometric force strength of Gravity,

For Electromagnetism, U(1) photons carry no Internal Symmetry Space charges. U(1) Weak bosons act transitively on the 1-dimensional manifold S^1 = U(1), so that four copies of S^1, in the form of the 4-torus T^4, are required so that they can take any given point in the origin Internal Symmetry Space into any point in the target Internal Symmetry Space. Each of the four connectivity components of the Electromagnetism of U(1) photons is geometrically represented by MISE as S^1 = U(1) whose volume is 2pi. The total manifold MISE is T^4 = S^1 x S^1 x S^1 x S^1, with volume 4 x 2pi The link manifold to the target vertex is trivial for the abelian neutral U(1) photons of Electromagnetism, so we take QE and DE to be equal to unity. For U(1) Electromagnmetism, mE is 1. For U(1) Electromagnmetism, ME = 1 , so that 1 / ME^2 = 1. Therefore, the force strength of Electromagnetism is: alphaE = ( 1 / ME^2 ) ( Vol(MISE) ) ( Vol(QE) / Vol(DE)^( 1 / mE )) = 8 pi which, when divided by the geometric force strength of Gravity, is 8 pi Vol(IV5)^( 1 / 4 )) / ( Vol(S^4) Vol(ShilovBdy(IV5)) ) and gives the electromagnetic fine structure constant alphaE = 1/137.03608 at the characteristic energy level of about 4 KeV.

The electromagnetic fine structure constant was first (to my knowledge) calculated in a similar way by Armand Wyler in the 1960s - early1970s.

In March 2003, **Walter Smilga** wrote a
paper entitled **Higher order terms in the contraction of
SO(3,2)**, in which he said:

"... The contraction of a spin-1/2 representation of the de Sitter group SO(3,2) yields a translation operator that consists of the usual momentum operator plus a second order term, the "momentum spin" as described by F. Gursey. The contribution of momentum spin to the kinematics of a multiparticle system in a tangential space of anti de Sitter space is analyzed. It is shown that it can be described by a perturbation term with the structure of the interaction term of quantum electrodynamics. An evaluation of the corresponding coupling constant reproduces Wyler's heuristic formula for the electromagnetic coupling constant. ...... We start from a system of spin-1/2 ... massive, structureless, lepton-like ... particles in a tangential space-time with a Minkowski structure. ...

... we ... have to incorporate a sujitable p;orjection mechanism onto our two-particle basic. ... only such terms with q = q' will be involved, as a consequence of momentum entanglement within two-particle states. ... we can drop the restriction to a fixed ... momentum P and collect all contributions that belong to the same p and q. Hence, we can write [using y for gamma]

... bbar( p + q ) y_u b(p) atilde^u(q) ... (26) ... with

atilde^u(q) = INTEGRAL dV(p') bbar( p' - q ) p^u b(p') , (27)

where dV(p') indicates a summation over all contribution that belong to the same p and q. ... Let us replace the operator atilde^u(q) of (27) by a new operator a^u(q) ... atilde^u(q) -> e a^u(q) ... We have to insert a ... normaliztion factor e ... Then ... a^u(k) ... are ... absorption operators for quanta with momentum k. ...

... We ... have explicitly constructed an interaction term from known elements of the electrons Fock space and also have defined the bookkeeping field in terms of these elements. ... our approach does not leave room for any free parameter. This means that the coupling constant e is determined by the theory and, therefore, should be calculable. ...

... This coupling constant is defined by the normalization of atilde^u(q) in (27) relative to the emission operator a^u(q) ... This ratio can ... be determined by correctly "counting" all contributions to (27) - in other words: by a careful analysis of the volume element of the integral in (27). ...

... The basis for the evaluation of the integration volume in (27) are the particle momenta adn the homgeneous Lorentz group acting on the particle momenta. The SO(3,1) acts transitively on a particle mass shell p0^2 - p1^2 - p2^2 - p3^2 = m^2. (34) The independent parameters p1,p2,p3 span a 3-dimensional parameter space. For a two-aprticle state ... we have ...

p0^2 + p'0^2 - p1^2 - p'1^2 - p2^2 - p'2^2 - p3^2 - p'3^2 = M^2 - k^2 . (38)

The symmetry group of this quadratic form is SO(6,2). ...

[ Note that the corresponding Clifford Algebra is Cl(2,6) = M8(Q) = 8x8 Quaternionic matrix algebra. ]

...[ Invariance of k^2]... reduces the number of independent parameters from 6 to 5 and thereby SO(6,2) to SO(5,2). SO(5,2) acts transitively on this 5-dimensional parameter spce. Each point in this parameter space corresponds to a state in the two-particle state space H_M. ... Given a point Q in this parameter space, then other points can be reached by applying a linear transformation of SO(5,2) to Q. There are certain transformations that do not change the point Q. These transformations form the subgroup S(O(5)xO(2)). This is the isotropy subgroup or stabilizer of Q. Therefore, to obtain the multiplicity of states, we have to start from the coset space D5 = SO(5,2) / S(O(5)xO(2)) rather than from SO(5,2).D5 is a symmetric space. By construction D5 is isomorphic to H_M. ... D5 ... is isomorphic to the real hyperball R_R(5,2) = { X in R^(5x2) | I - XX' > 0 }. (39) ...

... all states with given P0 are confined to a volume Dbar5 inside of ... the border sphere C5 of D5 ... and including C5. The subspace Dbar5 of D5 is finite and can be mapped onto R_R(5,2) by an isometric mapping. If Q = (q1, ... , q5) is a point of Dbar5 that is mapped into a point S = (s1, ... , s5) of R_R(5,2), then ... qi = r si , (40) where r is a ... scaling factor. ... We will use the notation V(D5) for the volume that corresponds to Dbar5 but is calculated in R_R(5,2) and will remember that we have to apply the correct number of scaling factors r.

[ Note that in the D4-D5-E6-E7-E8 VoDou Physics model, C5 is a Shilov Boundary of a Type IV5 Complex Domain related to Gravity. ]

C5 has another important property: If particle 1 is initially at rest and a second particle with a given momentum p' is added to form a two-particle state, then this state corresponds to a point on C5 ... Other states can be generated from this "initial" state by the exchange of momentum. Therefore, to determine all states that are eventually involved we have first to collater all initial states. This means, we have to perform an integration over C5 with a volume element of d4s / V(C5). This delivers a first volume factor of 1 / V(C5).To collect all possible momentum changes of particle 1 we have to integrate over Dbar5. ... only momentum exchanges in the subspace perpendicular to P have to be considered. Since the direction of the total momentum is undetermined .. we have to keep the integration over D5. We compensate for this by a volume factor of 1 / V(S4) where S4 = SO(5,2) / SO(4,2) = SO(5) / SO(4) is the unit sphere in 4 dimensions ...

[ Note that in the D4-D5-E6-E7-E8 VoDou Physics model, S4 is a Target Manifold related to Gravity. ]

... This reduces the number of independent parameters to 4. Let (s1, ..., s4) be a new set of independent parameters corresponding to a new set (q1, ... , q4). If we integrate over Dbar5 using this new parameter set, each si will be responsible for a contribution of V(D5)^(1/4) to the volume of Dbar5. Three of these parameters can be mapped onto the transferred momentum q. The fourth parameter s4 ... corresponds to a momentum transfer within each of the particle momenta, without any momentum transfer between the particles. such transitions contribute to the volume of C5. We can perform the integration over s4 and obtain a correcting factor to the aready calculated volume V(C5) of V(D5)^(1/4).There are two more factors that contribute to the multiplicity of momentum states. One is related to the spin components of the particle states, which give each momentum state a multiplicity of 4 pi because of the periodicity of spin states.

The other factor is related to the (relative) phases of the momentum states within multiparticle states. By adding another factor of 2 pi we take into account this degree-of-freedom.

Collecting all these factors we end up with

8 pi^2 V(D5)^(1/4) / ( V(S4) V(C5) ).

This is essentially Wyler's formula. ... we obtain a value for alpha

alpha = 1 / 137.03608245

... We can easily convince ourselves that the scaling factors r either cancel or are absorbed in the volume element d3q. ...

.. It is ... very tempting to identify Gursey's momentum spin with the current of quantum electrodynamics. ... There may exist essential contributions of other higher order terms. ... Such terms still deserve more attention. ...".

With respect to the mass of the photon, according to hep-ph/0306245 by Eric Adelberger, Gia Dvali, and Andrei Gruzinov:

"...The possibility of a non-zero photon mass remains one of the most important issues in physics, as it would shed light on fundamental questions such as charge conservation, charge quantization, the possibility of the charged black holes and magnetic monopoles, etc. The most stringent upperbounds on the photon mass listed by the Particle Data Group ... m < 3 X 10^(-27) eV and m < 2 x 10^(-16), are based on the assumption that a massive photon would cause large-scale magnetic fields to be accompanied by an energy density ... associated with the Proca field ... that describes the massive photon ... This manifests itself in two different ways. The first limit comes from the potential astrophysical effects ... and the second from an experiment that used a toroidally magnetized pendulum ... to measure the magnetic field gradient in a magnetically shielded vacuum. A recent experiment ... using an improved technique obtained a similar result. ... Both experiments actually measured the product m^2 Abar where the ambient Proca vector potential [ Abar] is presumably dominated by the field of the galaxy. The value assumed ... Abar = R B = 1 uG x kpc, is astronomicallyreasonable as the large-scale, R = 1 kpc, galactic field has a strength B = 1 uG. ...If m arises from a Higgs effect, both limits are invalid because the Proca vector-potential of the galactic magnetic field may be neutralized by vortices giving a large-scale magnetic field that is effectively Maxwellian. ...Crudely speaking, if m is due to a Higgs effect, then the Universe is effectively a type II superconductor where magnetic fields create Abrikosov vortices. ... We claim that photon mass bounds cannot be established without specifying the microscopic origin of the mass. In particular, if m arises from the commonly accepted Higgs mechanism, the above bounds do not apply over a large portion of the parameter space. ... It is quite possible for large-scale magnetic fields to be effectively Maxwellian, even if photons are massive. In this case observations of large-scale fields, say from the galaxy or from Jupiter, are not sensitive to m. This leaves us withthe upper bound from laboratory tests of Coulomb's law, m... It is also possible that the large-scale fields do remain in the Proca regime. But then, the available information about the large-scale magnetic field of the galaxy, and the gas pressure in the galaxy, actually gives a much more stringent bound, m^(-1)<10^(-14) eV>R = 1 kpc, orm...".<10^(-26) eV

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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