When I started in 2012 to identify nuclear colors with quaternion units I did that on morphological grounds only. When a few years later I stuck on the existence of Pauli matrices and matrices for quaternions, this identification turned out to be much nearer to the core than expected. The 8 Gell-Mann matrices are the 8 gluons. The first three Gell-Mann matrices contain the 3 Pauli matrices as their core. Multiplying two Pauli matrices yield a quaternion unit which is a gluon in my theory. I experienced it as surprising and encouraging.

# Quaternions as 2 x 2 matrices

Quaternions can be represented as 2 x 2 matrices.

a + bi + cj + dk = (9.1)

Mark the upper row in the matrix contains already all necessary variables (a, b, c and d). The second row is “junk”, extra machinery necessary to keep the matrix running. When you take a, b, c and d and set three of them at zero and set the fourth at 1 or -1, then you get the quaternion unit matrices.

1 = i = j = k = (9.2)

-1 =
-i =
-j =
-k =
(9.3)

SU(2) is the following group:

SU(2) = (9.4)

where the overline denotes complex conjugation.

When you substitute alpha = a + bi and beta = c' + di and c' = -c, then you get eq. (9.1) again. SU(2) then is the 3-dim surface of a 4-dim sphere in a 4-dim space with 2 real and 2 imaginary dimensions. While quaternions cover all of a 4-dim space with 1 real and 3 imaginary dimensions. That is not the same. But the matrices are the same and therefore I dare to state:

Quaternions are SU(2) (9.5)

S HIS ORRECT?

*Update november 2020*

SU(2) has 2 real dimensions and 2 imaginary dimensions

is satisfied in (quaternion unit) = (Pauli matrix) x (Pauli matrix)

when you take axes 1 and i from the first Pauli matrix orthogonal to the axes 1 and i from the second Pauli matrix. So IF the Pauli matrices here can be orthogonal THEN quaternion 1ijk space can be SU(2).

In QM orthogonal states exclude each other. It is one or the other, but not both. Are the Pauli matrices states here? In quaternions there is some ambiguity. A quaternion can define a rotation as well as a point in 1ijk space. But these are not quaternions, it are Pauli matrices, a kind of "square roots" from quaternions, a kind of factorization.

What are the Pauli matrices here? How to proceed?

Also in TONE, paragraph 3.4 Quaternion Units as Product of Pauli Matrices, just below (3.4.8).

## Pauli matrices

There are three Pauli matrices: sigma 1, sigma 2 and sigma 3, denoted as s1, s2 and s3.

s1 = s2 = s3 = (9.6)

When two Pauli matrices are multiplied, one always gets a quaternion unit.

s1 | s2 | s3 | ||||

s1 | 1 | i | -j | |||

s2 | -i | 1 | k | |||

s3 | j | -k | 1 | (9.7) |

(9.7) means s1 * s2 = i, s1 * s3 = -j, and so on. Seven quaternion units 1, i, j, k, -i, -j and -k are formed, but -1 isn't. -1 times a Pauli matrix like s1 just yields -s1. It doesn't yield a quaternion unit. So I don't place the matrix of -1 next to s1, s2 and s3.

1 = -1 = see (9.2) and (9.3)

1 | -1 | ||||

1 | 1 | -1 | |||

-1 | -1 | 1 | (9.8) |

This means that everywhere in this website you can replace every quaternion unit by the appropriate Pauli matrix product. After replacement there is no quaternion no more anywhere in this site. (9.9)

Nevertheless, the quaternion calculation formalism remains intact as long as you always and everywhere keep the Pauli matrices of one product together. (9.10)

A Pauli matrix usually means a particle, e.g. the electron. Two Pauli matrices in multiplication usually mean two particles in interaction, e.g. a collision. Two Pauli matrices *staying* in multiplication then mean two particles in continuous interaction, like in a composite.

When gluon colors are quaternion units AND my theory of gravitation applies to single particles THEN the gluon might be composed of a quark and an antiquark massless coinciding within their time borders. A gluon then is a tasteless, electrically neutral meson from which the composing two quarks had approached each other within their time borders. To let these wild propositions make sense, see the storyline NEKG describing gravitation, especially page 3, The Higgs field - Part 1. AND see the storyline THE EXPANSION OF THE UNIVERSE, page 2 *Dark Mechanics*, describing backward time evolving gravitation, especially paragraphs The calculation of the time border and the paragraph Massless coinciding. Now it is tempting to identify the two quarks with two Pauli matrices which product yields a quaternion unit, a color of the strong force - a gluon. But each of these two quarks has color too and each color consists of two Pauli matrices. So this is not clear yet.

O NVESTIGATE

The gluon as quark antiquark composite is worked out in page 5 of the storyline NET FORCES IN QCD, especially Meson exchange and Four quarks in the shell.

Remark (9.10) then means that the gluon remains intact as long as the constituting quarks remain together, within their time borders. This is obvious since when the quarks are separated you have to provide their mass and that showed to be impossible yet. (9.11)

Also the color i of a quark consists of two Pauli matrices s1 * s2. What does this mean?

ijk = -1. From (9.7) we know

i = s1 * s2,

j = s3 * s1 and

k = s2 * s3. So

ijk = s1 * s2 * s3 * s1 * s2 * s3 = -1

So -1 can be formed from 6 Pauli matrices in multiplication. (9.12)

-1 turns out to be a *glu3on*, see page 7 in the storyline NET FORCES IN QCD. The same page shows that 3 gluons merge easier to 1 gluon than 2 gluons do. From the 3-gluon composite two of the gluons might be absorbed in the *shell* of the third, see Four quarks in the shell at page 5 of the same storyline. Then a single -1 gluon should remain. (9.13)

i = j = k = -1, so

s1 * s2 * s1 * s2 = s3 * s1 * s3 * s1 = s2 * s3 * s2 * s3 = -1

So -1 can be formed from 4 Pauli matrices too, two gluons that are. But as said, 2 gluons merge less easy than 3 do. (9.14)

## Gell-Mann matrices

When I contacted prof. Piet Mulders from VU University, Amsterdam, I found him be willing to read my paper (proposal to a paper) about Gluons as Quaternions. Besides other remarks he judged the chance of publication to be zero because the mapping on SU(3) was missing.

I showed him that the product of two Pauli matrices always yields a quaternion unit, but he stipulated that *really* eight 3x3 matrices are needed and that I *really* should show the connection with the Gell-Mann matrices.

As a reply I came upon with the following scheme.