Here is my interpretation of my Quantum Set Theory. It is NOT to be confused with other Quantum Set Theories, such as the Quantum Set Theory of David Finkelstein or anyone else. However, the identification of quantum arrows with fermions in my Quantum Set Theory is based on Michael Gibbs's unpublished model of fermions being identified with the 2^n vertices of an n-dimensional hypercube. In my Quantum Set Theory, the 2^n hypercube vertices correspond to a the spinors of a 2^(2n) dimensional Clifford algebra, and I use the identification in my D4-D5-E6 physics model. In my interpretation, quantum arrows are used to: 1. describe the types of fermion particles and antiparticles that can possibly exist at any one given point in spacetime; and 2. to assign to each fermion particle or antiparticle a gamma-spinor orientation. Then, the spaces of fermion particles and of the fermion antiparticles are taken to be column spinors (particles and antiparticles each being column half-spinors), and the spaces of particle and antiparticle gamma-matrices are taken to be row spinors (particle gammas and antiparticle gammas each being row half-spinors). Then, the column spinors and row spinors are used to build a Clifford algebra. The columns are left ideals and the rows are right ideals, so the particle column x row plus the antiparticle column x row give the even subalgebra of the Clifford algebra. Then, the underlying graded Grassmann algebra of the Clifford algebra is used to construct spacetime and gauge bosons and a Lagrangian for the physical theory. Spacetime is constructed by extending the single point from which we started by the vectors of the Clifford algebra 1-vectors. The 2-vectors of the Clifford algebra are the gauge bosons. The 0-vector of the Clifford algebra is the Higgs scalar field. For more details of the physical interpretations of the Clifford algebra elements, see my Hodge star page, and my D4-D5-E6 model page.

To build a fundamental theory, start with a set denoted S1 containing only one element, denoted 1. S1 = {1}. Define the set Ac1 of classical arrows of S1 as the set of maps from S1 to S1. Ac1 has only one member, the identity map from 1 to 1. To define the set Aq1 of quantum arrows of S1, use the set of subsets of S1. (Note that the empty set 0 is a subset of S1, although it is NOT an element, or member, of S1.) S1 has 1+1 = 2 subsets {0} {1} Let the element 1 represent fermion antiparticles. Then, the subsets containing 1 correspond to antiparticles, and the subsets not containing 1 correspond to particles. Now, define the set Aq1 of quantum arrows as the set of maps from particle subsets of S1 (interpreted as column fermion types) to particle subsets of S1 (interpreted as row spinor gamma orientations) plus the set of maps from antiparticle subsets of S1 (interpreted as column fermion types) to antiparticle subsets of S1 (interpreted as row spinor gamma orientations) Aq1 has 2^0 x 2^0 + 2^0 x 2^0 = 1+1 = 2 members. Aq1 generates the even Clifford algebra of the 4-dimensional Clifford algebra Cl(0,2). Cl(0,2) is the quaternion algebra Q, with graded structure 1 2 1 The 1-vectors of Cl(0,2) are 2-dimensional, so the S1 Quantum Set Theory has 2-dim spacetime. The 2-vectors of Cl(0,2) are 1-dimensional, and the S1 Quantum Set Theory has a Spin(0,2) = U(1) gauge group. Since there is only one type of fermion particle, and one antiparticle, the Aq1 column particles can be taken to be the electron plus the positron, the S1 Quantum Set Theory gives 2-dim Quantum Electodynamics. Although it is an interesting theory, representable by a Feynman checkerboard, it is not big enough to describe our physical universe.

Go to the 2-element set S2 = {1,i}. Ac2 has 2 x 2 = 4 elements. S2 has 2+2 = 4 subsets: {0} {1} {i} {1,i} Aq2 has 2^1 x 2^1 + 2^1 x 2^1 = 4+4 = 8 members. Aq2 generates the even Clifford algebra of the 16-dimensional Clifford algebra Cl(0,4). Cl(0,4) is the 2x2 matrix algebra of quaternions, with graded structure 1 4 6 4 1 The 1-vectors of Cl(0,4) are 4-dimensional, so the S2 Quantum Set Theory has 4-dim spacetime. The 2-vectors of Cl(0,4) are 6-dimensional, and the S2 Quantum Set Theory has a Spin(0,4) = SU(2)xSU(2) gauge group, which is a compact version of the Lorentz group. Since there are two types of fermion particles, and two antiparticles, the Aq2 column particles can be taken to be the neutrino and electron plus the antineutrino and positron, the S2 Quantum Set Theory gives a 4-dim spacetime with Lorentz rotations and boosts, plus neutrinos, electrons, and their antiparticles. If the U(1) gauge group of electromagnetism is added by considering the winding numbers of topological holes in the 4-dim spacetime, the S2 Quantum Set Theory gives Wheeler's geometrodynamic model of gravity plus electromagnetism. Although it is an interesting theory, it is not big enough to describe our physical universe.

Go to the 3-element set S3 = {1,i,j}. Ac3 has 3 x 3 = 9 elements. S3 has 4+4 = 8 subsets: {0} {1} {i} {j} {1,i} {1,j} {i,j} {1,i,j} Aq3 has 2^2 x 2^2 + 2^2 x 2^2 = 16+16 = 32 members. Aq3 generates the even Clifford algebra of the 64-dimensional Clifford algebra Cl(0,6). Cl(0,6) is the 8x8 real matrix algebra, with graded structure 1 6 15 20 15 6 1 The 1-vectors of Cl(0,6) are 6-dimensional, so the S3 Quantum Set Theory has 6-dim spacetime. The 2-vectors of Cl(0,6) are 15-dimensional, and the S3 Quantum Set Theory has a Spin(0,6) = SU(4) gauge group, which is the compact version of the conformal group of 4-dim spacetime. Since there are four types of fermion particles, and four antiparticles, the Aq3 column particles can be taken to be the neutrino and red, blue, and green quarks plus the antineutrino and red, blue, and green antiquarks, the S3 Quantum Set Theory gives a 6-dim spacetime with a 4-dim physical spacetime submanifold, with a Spin(0,6) conformal gauge group whose Spin(0,5) de Sitter subgroup can produce Einstein gravity by the MacDowell-Mansouri mechanism, and whose 5 Spin(0,6)/Spin(0,5) degrees of freedom can be gauge-fixed to set the scale of the Higgs mechanism, plus quarks and antiquarks with SU(3) color symmetry. Although it is an interesting theory, illustrating dimensional reduction to 4-dim physical spacetime, MacDowell-Mansouri gravity, the Higgs mechanism, and quarks with SU(3) color symmetry, it is not big enough to describe our physical universe.

Go to the 4-element set S4 = {1,i,j,E}. Ac4 has 4 x 4 = 16 elements. S4 has 8+8 = 16 subsets: {0} {1} {i} {j} {E} {1,i} {1,j} {1,E} {i,j} {i,E} {j,E} {1,i,j} {1,i,E} {1,j,E} {i,j,E} {1,i,j,E} Aq4 has 2^3 x 2^3 + 2^3 x 2^3 = 64+64 = 128 members. Aq4 generates the even Clifford algebra of the 256-dimensional Clifford algebra Cl(0,8). Cl(0,8) is the 16x16 real matrix algebra, with graded structure 1 8 28 56 70 56 28 8 1 The 1-vectors of Cl(0,8) are 8-dimensional, so the S4 Quantum Set Theory has 8-dim spacetime, which is the spacetime of the D4-D5-E6 model prior to dimensional reduction. The 2-vectors of Cl(0,8) are 28-dimensional, and the S4 Quantum Set Theory has a Spin(0,8) gauge group, which is the gauge group of the D4-D5-E6 model prior to dimensional reduction. Since there are eight types of fermion particles, and eight antiparticles, the Aq4 column particles can be taken to be the neutrino, electron, red, blue, and green up quarks, and red, blue and green down quarks, plus their eight antiparticles, the S4 Quantum Set Theory gives the eight first generation fermion particles and antiparticles of the D4-D5-E6 model prior to dimensional reduction. The Aq4 row spinor gammas can be taken to be Particle gammas:

Antiparticle gammas:

Therefore, S4 Quantum Set Theory gives the D4-D5-E6 model.

The D4-D5-E6 model gives values of force strength constants based on relative volumes of bounded complex homogeneous domains and their Silov boundaries. Michael Gibbs uses an approach to calculating force strength constants based on diffusion equations. The two approaches seem to be equivalent ways of doing the same thing, in that: Diffusion equations, or heat equations, are based on generalized Laplacians that correspond to harmonic functions on the bounded complex homogeneous domains that represent the spaces through which diffusion occurs; Values of functions on the Shilov boundaries of bounded complex homogeneous domains determine, through the Poisson kernel, the values of solutions to the diffusion equations for the bounded complex homogeneous domains, so that the Shilov boundary can be considered to be the physically important part of the bounded complex homogeneous domain diffusion space; There are 4 types of bounded complex homogeneous domain, one for each symmetry group of each of the 4 forces, in the D4-D5-E6 model, and there are 4 different types of gauge bosons that can diffuse, at different rates, according to their gauge symmetry, based on different measures that are related to the relative volumes of the different bounded complex homogeneous domains and their Shilov boundaries.

Compare these quantum sets to my Simplex Physics and to MetaClifford Algebras, as well as the construction of Clifford Algebras from Set Theory.

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