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From Sets to Quarks:

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.                  Column Spinors and Fermions.                  Row Spinors and Vector Spacetime.                  Dimensional Reduction                         Spinors and Papapetrou Equations.                      Bivector Gauge Bosons and Higgs Scalar.                          Conformal Moebius Clifford Transformations                         Clifford Algebras                         Division Algebras                         Weyl Groups and Root Vectors                              gluon confinement                 Casimir operators and questions of signature.                 Complex Domains and Shilov Boundaries.                 Other Multivectors.                  Subtle Triality Supersymmetry.                     Ultraviolet Finiteness and Renormalization.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.

 


After DimensionalReduction to 4-dimensional spacetime, Bivector GaugeBosons and the Higgs Scalar take the form of the Standard Modelplus Gravity.

( Global Group Structure of the Standard Model is discussedhere.)

Here is how it works from various viewpoints:

The Gauge Boson dimensionalreduction process of the D4-D5-E6-E7physics model can be described from a WeylGroup - Root Vector Space point of view:


Here is a brief summary (usingEuclidean signature for expository clarity):

We have 8 gammas (1,2,3,4,5,6,7,8) in our Cl(8) Cliffordalgebra.

The 15 with index 6 or less give the conformalgroup and gravity.

The 13 left over do not form any subgroup of the 28-dim Spin(8)rotations of 8-dim spacetime, so break the symmetry of 8-dimspacetime into

4-dim physical spacetime

plus

a 4-dim internal symmetry space that is thespace of colors (red, green, blue, and neutral=black).

Then notice that 8 of the 13 involve one spacetime index and onecolor internal symmetry index. Let them be the 8 color force gluons.Since they are rotations in one spacetime dimension and one colorinternal symmetry dimension, you might ask:

Why don't we see gluons flying around changingcolors?

The answer is: Gluons are confined to theinterior of protons, pions, etc. You have to look inside them to seegluons flying around changing colors.

The remaining 5 of the 13 do not involve any spacetime index, sothat they can act as conventional gauge bosons whose rotations areconfined to the internal symmetry space.

Let 4 of them be the generators of the U(2) = SU(2)xU(1) electroweakforce.

As for the last one, the 13th, let it represent the U(1) phase ofparticle propagators.

 


Here are moredetails:

Let 4 of the 8 basis elements of the Cl(8) Cliffordalgebra, say (0,2,3), correspond to 4-dim physical spacetime, ina straightforward way:

Now, let the other 4, that is, (4,5,6,7) correspond to color spacewhich will be an internal symmetry space. Take as colors red, green,and blue(r,g,b).Let the 4th color be black (k, todistinguish its letter from b=blue). Make this correspondence alittle more complicated than (0,1,2,3,) = (x,y,z,t) as follows:

This way, we have 2 colorless basis elements(k,w) and 2colored ones(r,b). Ofcourse, you can make green from w and r and b, so you have allcolors. The basis is

(0,1,2,3,4,5,6,7) =(x,y,z,t,k,w,r,b)

Now look at the structure of the 28 bivectors, the generators ofSpin(8).

Just as the 8+8 = 16first-generation fermion particles and anti-particles, and8-dimensional spacetime (reduced to 4-dimensionalphysical spacetime plus 4-dimensional internal Symmetry Space)can be represented by a 24-cell,

24 of the 28 gauge bosonSpin(8) generators of the D4-D5-E6-E7-E8VoDou Physics model can be represented by the vertices of adual 24-cell:

The 24 vertices of the 24-cell are the24-vertex root vectors of the D4 Spin(8) Liealgebra in 4-dimensional root vector space with 4 origin rootvectors. The 28 infinitesimal generators correspond to the bivectorindices

01 02 03 04 05 06 07 12 13 14 15 16 17 23 24 25 26 27 34 35 36 37 45 46 47 56 57 67

which can be regarded as the upper triangular part of an 8x8antisymmetric real matrix.

The 4 origin root vectors can correspond to 
01 23 45 67
and the 24-cell vertices can correspond to 
02 03 04 05 06 07 12 13 14 15 16 17 24 25 26 27 34 35 36 37 46 47 56 57

In terms of the basis(x,y,z,t,k,w,r,b),they are

xy xz xt xk xw xr xb yz yt yk yw yr yb zt zk zw zr zb tk tw tr tb kw kr kb wr wb rb

The 6 that involve only (0,1,2,3) = (x,y,z,t)form the Lorentz boosts and rotations of physical spacetime:

xy xz xt yz yt zt

Note that (01,02,12) = (xy,xz,yz) are rotations and (03,13,23 ) =(zt,yt,zt) are Lorentz boosts.

Now let them interact with 4 = k.Since it is colorless, moving in 5-space (k-space) does not changethe color of anything, so it can be regarded as just an "extra"dimension added to 4-dim spacetime in which "rotations" look liketranslations, and we then get the 10 with indices 0,1,2,3,4 which aretaken to be generators of deSitter/Poincare gravity:

01 02 03 04 12 13 14 23 24 34

or

xy xz xt xk yz yt yk zt zk tk

Note that (04,14,24,34) = (xk,yk, zk,tk) are translations. Next let the 10interact with 5 =(r+g+b) =w. Since it is also colorless, moving in6-space (w-space) does not change thecolor of anything, so it can also be regarded as just an "extra"dimension. In it "rotations" look like conformal transformations (1dilation and 4 special conformal transformationsi), and we then getthe 15 with indices 0,1,2,3,4,5 which are taken to be generators ofSegal's conformal Spin(2,4) = SU(2,2)which can also be gauged to produce gravity:

01 02 03 04 05 12 13 14 15 23 24 25 34 35 45

or

xy xz xt xk xw yz yt yk yw zt zk zw tk tw kw

Note that the 45, which is only in(k,w) space, corresponds to the dilationand (05,15, 25, 35) = (xw,yw, zw,tw) are the 4 special conformaltransformations.

Also note that the Weyl root vectorpolytope of D3=A3 Spin(2,4) = SU(2,2) is the 3-dimensionalcuboctahedron, and that it can be taken to be the central(green) cuboctahedron of the 4-dim24-cell root vector polytope of Spin(8).

That leaves 13 generators left over, thosewith at least one index 6 or 7

06 0716 1726 2736 3746 4756 57 67

or

xr xbyr ybzr zbtr tbkr kbwr wb rb

and it would be nice if 12 of them could represent the generatorsof the SU(3) x SU(2) x U(1) Standard Model.

( Global Group Structure of the Standard Model is discussedhere.)


Why don't we just proceed to go on from 15-dim Spin(6) to21-dim Spin(7), in the same way that we went from 10-dim Spin(5) to15-dim Spin(6) ?

(In these remarks I am using Euclideansignature.)

For a first reason, physically at low energies you don't want to see color charges, which should be confined to the interior of hadrons. Therefore, since 15-dim Spin(6)
xy xz xt xk xw yz yt yk yw zt zk zw tk tw kw
does NOT involve the indices r and b that are not color-neutral, you can use it, but you cannot go to 21-dim Spin(7) becaus, if you do, you must involve an index r that is not color-neutral:
xy xz xt xk xw xr yz yt yk yw yr zt zk zw zr tk tw tr kw kr wr
For a second reason, you want the group to have a natural action on the 4-dim spacetime that is physical at low energies:
  • 6-dim Spin(4) acts naturally on 4-dim spacetime because it is just Lorentz boosts and rotations on 4-dim spacetime;
  • 10-dim Spin(5) acts naturally on 4-dim spacetime because it acts globally on the symmetric space Spin(5) / Spin(4) = S4, and also, there is an isomorphism Spin(5) = Sp(2), so that Spin(5) globally on the symmetric space Sp(2) / Sp(1)xSp(1) = QP1 (where QP1 is quaternionic projective 1-space). Note that Sp(1) = Spin(3) = SU(2) = S3, and that Sp(1)xSp(1) = Spin(4), and that QP1 = S4.
  • 15-dim Spin(6) acts naturally on 4-dim spacetime because there is an isomorphism Spin(6) = SU(4), so that Spin(6) acts naturally on the 4-dim fundamental repesentation space of SU(4).
  • However, 21-dim Spin(7) has no isomorphism that allows it to act naturally on 4-dim spacetime.
    • It acts naturally on the 7-dim vector space of which it is the covering of the rotation group, and
    • it acts naturally on the 7-dim symmetric space Spin(7) / G2, and
    • it acts naturally on the 6-dim symmetric space Spin(7) / Spin(6),

    but Spin(7) does not have a natural full action on a 4-dim spacetime.


Nevertheless, the 13 left-over generators
xr xbyr ybzr zbtr tbkr kbwr wb rb
DO have a nice physcial interpretation.

To see this, notice that they have a natural interpretation interms of symmetric spaces, as thecoset space

Spin(8) / U(4) = Spin(8) / Spin(6)xU(1) = Spin(8) /Spin(6)xSpin(2)

is 28 - 16 = 12-dimensional. If you let the 67 (orrb) correspond to the U(1), then theother 12 of the 13 left-over generators form the 12-dim symmetriccoset space Spin(8)/Spin(6)xU(1). It is only a coset space, so itdoes NOT directly inherit the commutator relations of Spin(8) anddoes NOT directly form a Lie subalgebra of Spin(8).

As DavidFinkelstein has said: "... They cannot do this [form a Liesubalgebra of Spin(8)] exactly. For example, no one of the 13commutes with all the rest, as the generator of U(1) does. Nor do thelast 13 commute with the first 15 conformal generators as the SU(3) xSU(2) x U(1) generators do. ...".

The way to get the 12 to represent the Standard Model is to assigncommutation relations to the 12 that are inherited from the Weylgroup symmetry of Spin(8).

To see how this works, begin by breaking the symmetry of the 8-dimspacetime by splitting it into two separate 4-dim spaces as describedabove, so that

(0,1,2,3,4,5,6,7) =(x,y,z,t,k,w,r,b)

breaks down to

(0,1,2,3) plus (4,5,6,7) = (x,y,z,t) plus(k,w,r,b)

What is the physics of this breakdown, at our low energieswhere it has occurred ?

The Lorentz, deSitter, and conformal structures with indices0,1,2,3,4,5 work as described above.

Now, look at the 13 generators left over, those with at least oneindex 6 or 7.


Only one of them, 67 or rb, involvesONLY the dimensions that are NOT involved in the 15-dim Conformalgroup.

Let 67 = rb represent the U(1) phaseof particle propagators.


That leaves 12 generators:

06 0716 1726 2736 3746 4756 57

or

xr xbyr ybzr zbtr tbkr kbwr wb

Note that these 12 correspond to the two octahedra

that remain after deleting the central cuboctahedron from the24-cell root vector polytope of Spin(8).

Set the common axis of the two octahedra such that the 4 verticeson it are the 4 vertices involving k andw:

The remaining 8 vertices

xr xbyr ybzr zbtr tb

correspond to the 8 gluons of the SU(3) color force. They eachhave one color dimension and one spacetime dimension, so that theyrepresent rotation in two dimensions, one color and one spacetime,thus mixing color and spacetime.

Why don't we see gluons flying around in our spacetimechanging colors?

Because gluons are confined within protons,pions, and other particlescontaining them, such effects can only be seen inside protons,pions, etc. The correspondingquark/hadron phase transition may be a tzimtzumRekiah.

How do the 8 vertices get the commutation relations ofSU(3)?

The 8 vertices that are off the common axis of the twooctahedra

can be considered as the vertices of a cube:

tb----xb |\ | \ | zb----yb | | | | yr-|--zr | \| \| xr----tr

Now look at the cube along itstb-trdiagonal axis, and project all 8 vertices onto a plane perpendicularto thetb-tr axis,giving the diagram

yb xb zb tb tr zr xr yr

with two central points surrounded by two interpenetratingtriangles, which is the root vectordiagram of SU(3), from which commutationrelations of SU(3) can be constructed for{tb,tr,xb,yb,zb,xr,yr,zr).

Since each gluon links 4-dim spacetimeto color internal symmetry space, the gaugegroup SU(3) acts globally on CP2 InternalSymmetry Space, as can be seen by the fibration

CP2 = SU(3) / U(2)


Now go back and look at the 4 vertices on the common axis of thetwo octahedra,

that is, the 4 vertices involving kand w.

kr kbwr wb

They correspond to the 3 weak bosons and the electromagneticphoton. They each have one color dimension and one spacetimedimension, so that they represent rotation in two dimensions, both ofwhich are in internal symmetry space, so there is no mixing ofspacetime and internal symmetry space, and they can be treated asconventional gauge bosons that are the generators of electroweak U(2)= SU(2) x U(1).

How do the 4 vertices get the commutation relations ofU(2)?

All 4 of them are on a line (the Z-axis of the (X,Y,Z) space ofthe two octahedra). Put the two involving w at the ends of a linesegment, with the two involving k at the origin center of a linesegment, thus getting the root vector diagramof U(2) = SU(2) x U(1), which is the sum of the rootvector diagrams of A1=B1=C1 and A0=D1.

kr kbwr wb

A correspondence with the electroweak gauge bosons is then:

Note that each of the neutral bosons (Z0 plus photon) correspondsto a rotation in the colorless color-neutral k-space part of internalsymmetry space that corresponds to translations.

Also note that each of the electric-charged W+ and W- bosonscorresponds to a rotation in the colorful color-neutral w-space partof internal symmetry space that corresponds to conformaltransformations.

These correspondences may be related to:

 

Since each weak boson and photonacts entirely within color internal symmetryspace, the gauge group U(2) acts locally on CP2Internal Symmetry Space, as can be seen by the fibration

CP2 = SU(3) / U(2)

( Global Group Structure of the Standard Model is discussedhere.)


If you combine the 15 Conformal generatorswith the 67 = rb generator of the U(1)particlepropagator phase, you get SU(2,2) x U(1) = U(2,2), so that

we have decomposed the 28 generators of Spin(8) into:

16 generators of U(2,2) = U(1)xSU(2,2) = U(1)xSpin(2,4) for propagator phase, Gravity, and the Higgs Mechanism, acting on 4-dimensional Spacetime.

8 generators of SU(3) for the Color Force, acting on 4-dimensional Internal Symmetry Space.

4 generators of U(2) = U(1)xSU(2) for Electromagnetism and the Weak Force, acting on 4-dimensional Internal Symmetry Space.

 ( Global Group Structure of the Standard Model isdiscussed here.)


The D5 of VoDou D4-D5-E6-E7 Physicsis related to SU(5) Grand Unification.

( Global Group Structure of the Standard Model is discussedhere.)

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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