**Symmetry
model E9CS**

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://de.wikipedia.org/wiki/Wurzelsystem

http://home.mathematik.uni-freiburg.de/soergel/Skripten/XXSPIEG.pdf

E9CS = SU(5) x SU(3) x SU(2) x U(1) x U(1)

Pronounced E9Coxeter-Symmetry

Write the symmetry in order to:

E9CS =

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

**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.

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

E9(x) is a polynom with terms of cyclotomic factors for 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

**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.

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.

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)..

the divison algebra.

For example the

The

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

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.

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

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 =

On the Higgsfield we have 2 x 1= 2 permutations.

Analogeous to the Higgspotential we declare a Potential on the Octoquintenfield

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

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.

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

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).

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

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.

The construction comes from multiplications by 4 degrees of freedome (quaternionic subspaces).

With this construction the tensor is symmetric in both diagonals.

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

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

more detailed with the two curvaturetensors of the Octoquintenfield:

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

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

and a

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

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.

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.

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:

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

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.

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!

See also 10 regular Clebsch Graph.

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

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

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