Symmetry model E9CS

affine Group E9 and symmetries of its Coxeterelement

by Mag.rer.nat. Kronberger Reinhard
Email: support@kro4pro.com
16.02.2017

Motivation 1:

Why do we consider the E9 group (more specifically the Coxeter element of this group)?
1) E9 is an affine group and thus has something to do with extension.
2) The extension is flat as the universe.
3) The action of the Coxeter elements of the group produces symmetries involving our current standard model.

The fundamentals here:

https://en.wikipedia.org/wiki/Coxeter_group
https://de.wikipedia.org/wiki/Wurzelsystem
http://home.mathematik.uni-freiburg.de/soergel/Skripten/XXSPIEG.pdf

                                                               
                    
                     Symmetries which arise from the Coxeter element of the E9.

              E9CS = SU(5) x SU(3) x SU(2) x U(1) x U(1)
      (SU(n) = Special unitary group, U(1) unitary group)
                    
                     Pronounced E9Coxeter-Symmetry  
                    
                evidently SU(5) x SU(3) x SU(2) x U(1) x U(1)    Obermenge  SU(3)c x SU(2)L x U(1)Y (Color charge, isospin, Hyper charge)

                     Write the symmetry in order to:  

              E9CS =
SU(5)S x U(1)Y2
x U(1)Y1 x SU(2)L x SU(3)C 
(=Expansion x actual Standard Model)



           
Dynkin Diagram E9 (affine one point extension of group E8):


Dynkin

Derivative of the symmetries of E9CS from the invariants of the Coxeter elements E9:


A Coxeter element is a product of the generating reflections of E9.


For example: Coxeterelement = e1.e2.e3.e4.e5.e6.e7.e8.e9

The Coxeterpolynom is the characteristic polynomial of Coxeter elements  and has the form:


 Coxeter

E9(x) is a polynom with terms of cyclotomic factors zyklische Faktorenfor n>1 and (x-1) for n= 1.
The cyclotomic factors are the characteristical polynom of the An-1 (which is the Dynkin diagram for the SU(n) Liegroup.See more here: https://en.wikipedia.org/wiki/Special_unitary_group).
So finally the symmetry space by the actions of the Coxeterelements is SU(5) x SU(3) x SU(2) x U(1) x U(1)
 
               

Motivation 2:

What bring us the additional symmetries?

(1) These have the potential to describe new particles.

(2) These have the potential to describe the space and time.

(3) These have the potential to describe gravity.


Wish to analogously represent Graviton to the photon as a blend (Weinberg angle see <8>).


<1> The Idea

Light and gravitation just like photon and graviton have something in common.

Both are massless and propagate with the speed of light.

 

We know that light by the
Symmetry breaking 1: SU(2)xU(1)--> U(1)e is described as a mixture.

So light is a part of the electro-weak interactions.

 

we consider analog gravity as a result of a further symmetry breaking

Symmetry breaking 2: SU(5) x U(1) x U(1) --> U(1)g

Our extended standard model allows us this.


We will now like to assign our relevant SU(n)'s to division algebras  (real numbers, complex numbers, ...).

Körper

This 4 divison algebras (real numbers, complex numbers, quaternions and octonions) develop through the doubling process

see more at https://de.wikipedia.org/wiki/Verdopplungsverfahren

Considering the rank of the SU(2) = 1,SU(3)= 2, SU(5) = 4 then this is double as well.

 

There appears to be a connection between the division algebras and the SU(n)’s (n = 2,3,5)..

I think the trivial connections are the rank (maximal torus) of the SU(n) and the orthogonal complex subspaces of
the divison algebra.
For example the quaternions have two orthogonal complex subspaces a+b.i1  and c.i2+d.i3 (a,b,c,d real).
The SU(3) has also 2 neutral elements (toris).


With this assignment we can create backgroundfields to the SU(3) and SU(5) like it is the higgsfield for SU(2).


higgsfield

 

Therefore, we rely analogously on the Higgsfield (2 x complex = doublet)

Higgs

<2> the Octoquintenfield  (5 x Octonions= Quintet).


Oktoquintenfeld

This provides 40 degrees of freedom.

24 of which will be "spent" for our SU(5) tensor bosons for the 5th longitudinal spin degree of freedom (24 Goldstone bosons swallowed over gauge transformation) thus remain 16 left.

 

The S, F, R, G and H charges are the 5 charges of the SU (5) analogous to the 3 color charges of SU (3) and the 2 charges (+ .-) of SU (2).

The letters stand for S = See, F = feeling, R=smelling G = Taste and H = Hear

Calling therefore the charges of the SU (5) sense charges.

Note: These charges have (such as the color charges of quarks with color) nothing to do with the senses, but to give a name to the child for reference only.

We now want to look at the 16 (40-24 = 16) remaining degrees of freedom.

 

Make the following division for the 40 field components of the Octoquinten field as a physical approach:
Take care that the division is not unique because for the left half 4 gray fields we can use 4.3.2.1= 24 Permutations of them in the orange area.
And for the left 4 charges we have five over 4 = 5 Permutations.
So at all we have 5 x 24 = 120 possible permutations.
On the Higgsfield we have 2 x 1= 2 permutations.

Oktoquinten



Analogeous to the Higgspotential we declare a Potential on the Octoquintenfield

<3>  Potential over the Octoquintenfield

higgs potential

golden ratio


Oktoquintenpotential


Einstein form

goldenratio

tachyonen

curvature

golden ratio

<3.2>
Getting a new particle by the Octoquintenfield of m ~ 28 GeV mass.

Planck

Planck

Higgs

Higgs
With this assumption we can calculate the higgsmass.See Assumption 1 at the end.
Higgsmasse

A second particle can be maybe found by the first term of the Octoquintenpotential.
Hint:The Higgsmass as calculated above comes from the second term of the Octoquintenpotential.

28 GeV

Higgs

<3.3>

By using Planckunits we get from the Einstein-Form of the potential the
Max Planck



Max Planck

natural units

Max Planck

Max Planck

combinatoric

<3.4>

Oktoquintenpotential the 16-Cell (Coxeter group B4,D4):

16 cell
Quants

Quants
graph


quants


quants
loop string

stringtheory



dark matter

Hintergrundfeld
<3.5>
combinatorial

generating function

normfactor

standardmodel

But the 16-Cell is not the only 4 polytope which fits for the Oktoquintenpotential.
See more APPENDIX III at the end of the document (600-Cell).

particles in universe

<4>  Lagrangedensity of the Octoquintenfield/Octoquintenpotential

Hint:

I do the same steps as shown in this cooking recipe for the Higgsfield.

https://www.lsw.uni-heidelberg.de/users/mcamenzi/HD_Higgs.pdf



Lagrange
  
lagrange

Lagrange oktoquintenfield

Lagrange Oktoquintenfield

dark matter
dark matter


Lagrange

vacuumexpectation


STEP 1:Lorentzinvariant Lagrangedensity for the Octoquintenfield


Lagrange

Lagrange


SU(5) generators


tensorboson


repelions

covariant derivation

w bosons
U(1)

charge

tensorboson



Lagrange mixing
my

Higgsfield

Graviton

massmatrix

lagrange

lagrange

Mixing Matrix

Lagrange Oktoquintenfield

gravitation


Gravitation

Mixing Boson

Boson mass
 
<5> Curvaturetensors by the Octoquintenfield

curvaturetensor

The construction comes from multiplications (symmetric to the diagonal) by 2 degrees of freedome (complex subspaces).
With this construction the tensor is symmetric in the diagonal.

metric tensor
10 independent fields.



plank curvature


Second CURVATURE TENSOR from the Octoquintenfield (generates a spinpotential)

The construction comes from multiplications by 4 degrees of freedome  (quaternionic subspaces).
With this construction the tensor is symmetric in both diagonals.

Spin curvature

tensor

vacuum energy
5 independent fields A,B,C and two in the diagonal (blue and yellow).

So finally we get three derivation- or curvaturetensors of the Octoquintenpotential for twisted spacetime excitation

Riemann curvature


<6>  Extension of the ART by the second curvaturetensor

   

  The Octoquintenpotential has two symmetric curvaturetensors.This motivates us to extend the Einstein Equation.

    vacuum higgs

    I think this shows that the GR (General Relativity) has to be extended by an imaginäry part (spinpart) to be a consistent quantumtheorie.

    So finally we expect something like GR+i.GR° where GR° is the spinpart.

   
Spin Torsion
cosmological constant problem or vacuum catastrophe


    real

more detailed with the two curvaturetensors of the Octoquintenfield:

   Twistor
Clifford Torus


The second curvaturetensor Cspin is determinded by the first curevaturetensor Cem because its components are a mix of the components of Cem.

<6.1> The vacuumpart of the extended Einstein equation then is:

      Twistertheory

vacuum potential

<6.2>
Scalefactor for the accelerated expanding Universe by our assumption

Hubble


<7> Getting a closed form for the Extended General Relativity EGR. 

Roger Penrose  

For that we define the operator for 4 x 4 matrices or tensors:
Tau Operator

from A to A Tau
and a
Operator
operator quantenfeld
general relativity
 
general relativity

equivalence principle

<7.1>Spin and possible proofing of the Extended General Relativity EGR.
 
general relativity

special relativity

extended general relativity

Fermions

Majorana
Weyl

Dirac

general relativity
Dirac

Zitterbewegung

curvaturetensor

<8> Some important points of the Octoquintenpotential

Weinbergangle
golden mean

Zeropoints


oktoquintenpotential

oktoquintenpotential

weibergangle

Oktoquintenpotential

Minimum

Geometric interpretation of the roots (zeropoints) in cubic equations with 3 real zeropoints


Zeropoints

kubic equation
Hint:On our special OQP (Octoquintenpotential) the zeropoints (spheres) comes from a pentagon.

pentagon


<9>  Candidates for the dark matter in the universe.

Dark Energy comes by definition from the Octoquintenpotential (the second term in the potential).
Dark Matter could be the W , Z Bosons and the particles  by the SU(5) Symmetry (adjoint and fundamental presentation).



APPENDIX I

1) The 16-Cell


On the 16-Cell each vertizes is connected by an edge to all other vertizes except the opposite one!
We have 4 disjunct such pairs which are not connected by an edge.
16 Cell
Now exchanging this points which are not connected is an automorphism (for example p1 with p1-line) because
p1 has the same connections as p1-line.
So at all we generate 2^4=16 automorphisms by this actions because we can say for all 4 pairs 0 means pair IS NOT exchanged and 1 for pair IS exchanged.
So every binary code like (0,1,0,0) is an automorphism.

automorphism
But this are not all automophism.Independend from that we can permutate p1,p2,p3,p4 when we simultaneously permutate their opposite points.
For example:
permutations
This give us 4!=24 automorphisms independend from the 16 automorphisms before.
So at all we get 16.24=384 automorphisms.

With this we can divide the Aut(C16) into AutOpposite(C16) and AutFakt(C16) so  that Aut(C16) = AutOpposite(C16) x AutFakt(C16) = 16 x 4!.

AutOpposite(C16) are the Automorphisms of the 16-Cell C16 which comes from exchanging the opposite vertizes (points).
AutFakt(C16) are the Automorphisms of the 16-Cell C16 which comes from permutating p1,p2,p3,p4 vertizes (points) as described above.
AutFakt(C16) is simply the permutation-group Sym(4) = S4.

Embedding (projection) the 16-Cell into the quartenionic subgroups of the Octoquintenfield:

See also Quaterniongroup Q8

16 Cell



APPENDIX II

We have seen in <3.4> that the 16-Cell fits for the combinatorial form of the Oktoquintenpotential.
But also the 600-Cell fits in the same way.

icosahedron 600 Cell

convex regular 4-polytope

polytopes

sphere

Demitesseract

tesseract

coxeter plane
Together with the symmetries which comes from the pentagon of the zeropoints (see above) we have the symmetries of the D5 demipenteract.
The order of the group = 10.192=1920 =16.5!

demipenteract

See also 10 regular Clebsch Graph.

Assumption 1

Calculating backward from the Combinatorial- to the Planckform of the Oktoquintenpotential (see <3.5>) we get the formular:

higgs

higgs

higgs

higgs

higgsmass

But i have to say that the assumption in <3.2>  is not absolute sure!

Assumption 2

possible combinatorial massrelation of electron,myon and tauon.

generation electron

Hint: there are other subgroups of S8 which have also order 12 or 192!