**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 key Coxeter element 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:**

The Coxeter element is the 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 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) 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.

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.

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

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 Oktoquintenfield

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 Oktoquintenpotential 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 oktoquintenfield:

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