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Michael Gibbs's unpublished model of fermions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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