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 key Coxeter element 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:


The Coxeter element is the product of the generating reflections of E9.


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 of the Coxeterelement is SU(5) x SU(3) x SU(2) x U(1) x U(1)
 


Coxeterpolynomial
Eigenraum

                   

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) 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: U(1) x U(1) --> U(1)

Our extended standard model allows us this.

 

We will now like to assign our relevant SU(n)'s to algebras division (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 dimensions 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) which I hope is known in analytic geometry or another area.

I assume this connection warrants as simply as given.

Notes but no clear allocation can be found in this direction at Corinne A. Manogue and Tevian Dray, John Baez, etc.

 

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

Higgs

<2> the Oktoquintenfield  (5 x Oktonions= Quintett).


Oktoquintenfeld

or written otherwise so that the equivalence to the Higgs field is clear (where i4 is pulled from)

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 Oktoquinten 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 Oktoquintenfield

<3>  Potential over the Oktoquintenfield

higgs potential

golden ratio


Oktoquintenpotential


Einstein form

goldenratio

tachyonen

curvature

golden ratio

Getting a new particle by the Oktoquintenfield of m ~ 28 GeV mass.

Planck

Planck

Higgs

Higgs
Higgsmasse

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

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



Max Planck

natural units

Max Planck

Max Planck

combinatoric

Scalefactor for the accelerated expanding Universe by our assumption

Hubble


Quants

Quants
graph

quants


quants
loop string

stringtheory



particles

electron

Isospin

Hintergrundfeld

combinatorial

generating function

normfactor

standardmodel
quarks
dark matter

particles in universe

quarks

decay


<4>  Lagrangedensity of the Oktoquintenfield/Oktoquintenpotential

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

Lagrange

vacuumexpectation

STEP 1:Lorentzinvariant Lagrangedensity for the Oktoquintenfield


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




vacuumexpectation

higgs
 
<5> Curvaturetensors by the Oktoquintenfield

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 Oktoquintenfield (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 Oktoquintenpotential for twisted spacetime excitation

Riemann curvature


<6>  Extension of the ART by the second curvaturetensor

   

  The Oktoquintenpotential 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 oktoquintenfield:

   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


<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


 

<8> Some important points of the Oktoquintenpotential

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

Dark Energy comes by definition from the Oktoquintenpotential (the second term in the potential).
Dark Matter could be the W , Z Bosons  of the SU(5) Symmetry
.