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.

From finite sets and some simple operations on sets, construct a Feynman Checkerboard physics model that allows computation of the ratios of force strength constants and masses of elementary particles. Since particle masses can only be observed experimentally for particles that can exist in a free state ("free" means "not strongly bound to other particles, except for virtual particles of the active vacuum of spacetime"), and since quarks do not exist in free states, interpret the calculated quark masses as constituent masses (not current masses). In hep-ph/9802425, Di Qing, Xiang-Song Chen, and Fan Wang, of Nanjing University, present a qualitative QCD analysis and a quantitative model calculation to show that the constituent quark model [after mixing a small amount (15%) of sea quark components] remains a good approximation even taking into account the nucleon spin structure revealed in polarized deep inelastic scattering. The effectiveness of the NonRelativistic model of light-quark hadrons is explained by, and affords experimental Support for, the Quantum Theory of David Bohm. Consitituent particles are Pre-Quantum particles in the sense that their properties are calculated without using sum-over-histories Many-Worlds quantum theory. ("Classical" is a commonly-used synonym for "Pre-Quantum".) However, the natural Clifford algebra structure of the D4-D5-E6 model produces a Many-Worlds Quantum Theory. Since experiments are quantum sum-over-histories processes, experimentally observed particles are Quantum particles. Consider the experimentally observed proton. A proton is a Quantum particle containing 3 constituent quarks: two up quarks and one down quark; one Red, one Green, and one Blue. The 3 Pre-Quantum constituent quarks are called "valence" quarks. They are bound to each other by SU(3) QCD. The constituent quarks "feel" the effects of QCD by "sharing" virtual gluons and virtual quark-antiquark pairs that come from the vacuum in sum-over-histories quantum theory. Since the 3 valence constituent quarks within the proton are constantly surrounded by the shared virtual gluons and virtual quark-antiquark pairs, the 3 valence constituent quarks can be said to "swim" in a "sea" of virtual gluons and quark-antiquark pairs, which are called "sea" gluons, quarks, and antiquarks.

This internal structure of the Proton can be described in terms of a Compton Radius Vortex.

In the model, the proton is the most stable bound state of 3 quarks, so that the virtual sea within the proton is at the lowest energy level that is experimentally observable. The virtual sea gluons are massless SU(3) gauge bosons. Since the lightest quarks are up and down quarks, the virtual sea quark-antiquark pairs that most often appear from the vacuum are up or down pairs, each of which have the same constituent mass, 312.75 MeV. If you stay below the threshold energy of the strange quark, whose constituent mass is about 625 MeV, the low energy sea within the proton contains only the lightest (up and down) sea quarks and antiquarks, so that the Quantum proton lowest-energy background sea has a density of 312.75 MeV. (In the model, "density" is mass/energy per unit volume, where the unit volume is Planck-length in size.) Experiments that observe the proton as a whole do not "see" the proton's internal virtual sea, because the paths of the virtual sea gluon, quarks, and antiquarks begin and end within the proton itself. Therefore, the experimentally observed mass of the proton is the sum of the 3 valence quarks, 3 x 32.75 MeV, or 938.25 MeV which is very close to the experimental value of about 938.27 MeV. To study the internal structure of hadrons, mesons, etc., you should use sum-over-histories quantum theory of the SU(3) color force SU(3). Since that is computationally very difficult (For instance, the internal structure of a proton looks like a nonperturbative QCD soliton. See WWW URL http://www.innerx.net/personal/tsmith/SolProton.html ) you can use approximate theories that correspond to your experimental energy range. For high energy experiments, such as Deep Inelastic Scattering, you can use Perturbative QCD. For low energies, you can use Chiral Perturbation Theory. To do calculations in theories such as Perturbative QCD and Chiral Perturbation Theory, you need to use effective quark masses that are called current masses. Current quark masses are different from the Pre-Quantum constituent quark masses of the model. The current mass of a quark is defined in the model as the difference between the constituent mass of the quark and the density of the lowest-energy sea of virtual gluons, quarks, and antiquarks, or 312.75 MeV. Since the virtual sea is a quantum phenomenon, the current quarks of Perturbative QCD and Chiral Perturbation Theory are, in our view, Quantum particles. The relation between current masses and constituent masses may be explained, at least in part, by the Quantum Theory of David Bohm. Therefore, the model is unconventional in that: the input current quarks of Perturbative QCD and Chiral Perturbation Theory are Quantum, and not Pre-Quantum, so that Perturbative QCD and Chiral Perturbation Theory are effectively "second-order" Quantum theories (rather than fundamental theories) that are most useful in describing phenomena at high and low energy levels, respectively; and a current quark is a composite combination of a fundamental constituent quark and Quantum virtual sea gluon, quarks, and antiquarks (compare the conventional picture of, for example, hep-ph/9708262, in which current quarks are Pre-Quantum and constituent quarks are Quantum composites).

Assuming the accepted values of the gravitational force strength constant (Newton's constant) and the electron mass (0.511 MeV), then the calculationed ratios give values of all the other force strength constants and particle masses. These calculated force strengths and particle masses agree with conventionally accepted experimental results within at most about 10 percent in all but one case: the mass of the Truth quark (sometimes called the Top quark). The tree level constituent mass of the Truth quark is computed to be roughly 130 GeV, as opposed to the roughly 175 GeV figure advocated by FermiLab. In my opinion, the FermiLab figure is incorrect. The Fermilab figure is based on analysis of semileptonic events. I think that the Fermilab semileptonic analysis does not handle background correctly, and ignores signals in the data that are in rough agreement with the tree level constituent mass of about 130 GeV. Further, I think that dileptonic events are more reliable for Truth quark mass determination, even though there are fewer of them than semileptonic events. I think that analysis of dileptonic events gives a Truth quark mass that is in rough agreement with the tree level constituent mass of about 130 GeV. More details about these issues, including gif images of Fermilab data histograms and other relevant experimental results, can be found on the World Wide Web at URL http://www.innerx.net/personal/tsmith/TCZ.html I consider the mass of the Truth quark to be a good test of the model, as the model can be falsified if I turn out to be wrong in my interpretation of experimental results. A strong point of the theory is that calculations give ratios of ALL the masses and force strenghs in a way that is MUTUALLY CONSISTENT. The mutually consistent set of values may not be the only mutually consistent set of values, but as far as I know, they are the only mutually consistent set of values that come from one unified model. The purpose of this paper is to describe the model in some detail. After this introductory overview,Chapter 2describes the construction of discrete Clifford algebras from set theory and some simple natural operations. Begin with set theory to get the Natural Numbers N. Then use reflection to get the integers Z. Then use the set of subsets and the XOR operator to get the Discrete Clifford Group DClG(n). DClG(n) is extended to its Z-Group Algebra, thus producing a discrete real Clifford Algebra DCl(0,n) over the Zn hypercubic lattice. For some n, DCl(0,n) is naturally extended from the Zn hypercubic lattice to larger lattices, such as the D4 quaternionic integer lattices for n=4 and an E8 octonionic integer lattices for n=8. The real Clifford Algebras have periodicity 8, so the fundamental real Clifford Algebra produced by this process is DCl(0,8). The vector, +half-spinor and -half-spinor representations DCl(0,8) are all isomorphic by triality to the discrete integral octonions.Chapter 3describes the octonionic structure of E8 lattices.Chapter 4describes the scalar representation of DCl(0,8) as physically representing the Higgs scalar particle; the vector representation of DCl(0,8) as an E8 lattice physically representing an 8-dimensional spacetime; the bivector representation of DCl(0,8) as having 28 basis bivectors that represent the 28 gauge boson infinitesimal generators of a Spin(0,8) gauge group; and the two half-spinor representations of DCl(0,8) as two E8 lattices, in which the 8 octonion basis vectors of each physically represent 8 fundamental first-generation fermion particles (neutrino; red, blue, green up quarks; red, blue green down quarks; electron) and 8 fermion antiparticles.Chapter 5describes a 4-dimensional HyperDiamond lattice spacetime that comes from a 4-dimensional sub-lattice of the E8 lattice spacetime. 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.Chapter 6describes a 4-dimensional internal symmetry space that comes from the rest of the original E8 lattice spacetime. A separate copy of the internal symmetry space lives on each vertex of the spacetime 4-dim HyperDiamond lattice. Each copy of the internal symmetry space looks like a 4-dim HyperDiamond lattice.Chapter 7describes how a Feynman Checkerboard construction on the HyperDiamond structures gives the physics of the Standard Model plus Gravity.Chapter 8describes the numerical calculation of charge as the amplitude for a particle to emit a gauge boson, with the force strength constant being the square of the charge.Chapter 9describes the numerical calculation of particle mass as the amplitude for a particle to change direction.Chapter 10describes HyperDiamond Feynman Checkerboard configurations that represent protons as triples of confined quarks, pions as confined quark-antiquark pairs, and physical gravitons as quadruples of confined Spin(0,5) gravitons.Appendix Adescribes some earlier papers, including some errata for them.Referencesare some references. The D4-D5-E6 model that is described in some detail on the World Wide Web at URL http://xxx.lanl.gov/abs/hep-ph/9501252 is a continuum version of the HyperDiamond Feynman Checkerboard model at URL http://xxx.lanl.gov/abs/hep-ph/9708379. Both of those papers, and all my papers written prior to June 1998, contain an incorrect value (about 260 GeV) of the Higgs scalar mass, which should be about 146 GeV. The D4-D5-E6-E7-E8 VoDou Physics model is also described on the World Wide Web at URL http://www.innerx.net/personal/tsmith/d4d5e6hist.html Briefly, roughly, and non-rigorously, the D4-D5-E6 model is constructed from E6, D5 = Spin(10, and D4 = Spin(8): The first generation of fermions are constructed from E6 / Spin(10) x U(1), whose dimension is 78-45-1=32, the real dimensionality of a bounded complex domain whose Shilov boundary has real dimension 16=8+8 for 8 fermion particles and 8 fermion antiparticles. An 8-dimensional spacetime is constructed from Spin(10) / Spin(8) x U(1), whose dimension is 45-28-1=16, the real dimensionality of a bounded complex domain whose Shilov boundary has real dimension 8 for 4-dimensional physical spacetime plus 4-dimensional internal symmetry space. 28 gauge bosons are constructed directly from 28-dimensional Spin(8). Then reduce spacetime from 8 dimensions to 4 dimensions, by choosing a quaternionic subspace of octonionic 8-dimensional spacetime. The result of that spacetime symmetry breaking is: The fermions get 3 generations, corresponding to E6, E7, and E8; The 28 = 16+12 gauge bosons split into two parts: 16 of them form U(4), which is U(1) x SU(4), the U(1) to give a complex phase to propagator amplitudes, the SU(4) = Spin(6) to give the conformal group, the Spin(6) (compact version of Spin(4,2) conformal group has 5 dimensions of conformal and scale transformations that give a mass scale and a Higgs scalar and has 10 dimensions that give the Spin(5) deSitter group, which is gauged to give Gravity; 12 = 8+3+1 of them form SU(3) x SU(2) x U(1) of the Standard Model.

**The ****D4-D5-E6-E7
model**** coset spaces ****E6****
/ (****D5**** x U(1)) and
****D5**** / (****D4****
x U(1)) are Conformal Spaces. ** You can continue the chain to
D4 / (D3
x U(1)) where D3 is the 15-dimensional Conformal Group whose compact
version is Spin(6), and to D3 /
(D2 x U(1)) where D2 is the
6-dimensional Lorentz Group whose compact version is Spin(4).
Electromagnetism, Gravity, and the
ZPF all have in common the symmetry of the 15-dimensional D3
Conformal Group whose compact version is Spin(6), as can be seen by
the following structures with D3 Conformal Group symmetry:

- Maxwell's equations of Electromagnetism
- Gravity derived from the Conformal Group using the MacDowell-Mansouri mechanism
- the Quantum Theoretical Hydrogen atom
- the Lie Sphere geometry of SpaceTime Correlations in the Many-Worlds picture

Further, the 12-dimensional Standard Model Lie Algebra U(1)xSU(2)xSU(3) may be related to the D3 Conformal Group Lie Algebra in the same way that the 12-dimensional Schrodinger Lie Algebra is related to the D3 Conformal Group Lie Algebra.

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.

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