Matrices and the mapping on SU(3)

Quaternions as 2 x 2 matrices
Quaternions can be represented as 2 x 2 matrices.

a + bi + cj + dk =

a + bi -c + di

c + di a - bi

(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 =

1 0

0 1

i =

i 0

0 -i

j =

0 -1

1 0

k =

0 i

i 0

(9.2)

-1 =

-1 0

0 -1

-i =

-i 0

0 i

-j =

0 1

-1 0

-k =

0 -i

-i 0

(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 - Some essential remarks about SU(2), Pauli matrices and quaternions are 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, here denoted as s1, s2 and s3.

s1 =

0 1

1 0

s2 =

0 i

-i 0

s3 =

1 0

0 -1

(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 0

0 1

-1 =

-1 0

0 -1

see (9.2) and (9.3)

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

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



This scheme shows the first 3 (out of 8) Gell-Mann matrices, lambda 1, lambda 2 and lambda 3 and their products. These products are the matrices I propose as gluons. The red frames are the Pauli matrices. The green frames are the quaternion units, except for -1. The large blue frames together are one subset. Lambda 4 up to 8 (not shown) are not to be used.      (9.15)

I proposed to take as 8th gluon

      (9.16)

Prof. Mulder replied that the matrices I proposed indeed are symmetries, but they don't obey the commutation relation [A,B] = C no more, where C is a Lie-algebra. The group decays in a number of subsets that cannot represent all of QCD no more and you have to perform quite some acrobatics to repair this. This doesn't make things simpler nor easier, he said.

Finally I asked him whether it should be possible to take the matrices of i, j, k, -i, -j and -k (the large blue frames in figure 9.11) as one group and the two remaining gluons 1 and -1 as a second group. In my theory 1 and -1 are the time axis and space as well as color space is different from time. So it might not be too disastrous if the 6 mentioned gluons were one force and 1 and -1 would be a different force. But prof Mulders did not reply.

   NDER long term ONSTRUCTION


Resemblances and differences of QQD versus QCD
What makes me think the quaternions of SU(2) can mimic the Gell-Mann matrices of SU(3)? A summary of resemblances (green numbers) and differences (red numbers) between QCD gluons (part a) and QQD quaternions (part b).

1a) There are 3 colors that together give something that has no color no more.

1b) There are 3 imaginary quaternions that when multiplied with each other, give something that has no imaginarity no more: ijk = -1 en kji = 1.

2a) The 3 colors together give white, a kind of “7th color that isn't a color”.

2b) The 3 imaginary quaternion units, when multiplied with each other, give 1 or -1. When 1 is called white then -1 is black . Black and white, instead of white only.

3a) Red * green = green * red = antiblue, or yellow: * = * = . The order of multiplication is not important.

3b) In quaternions the order of multiplication is important:
i * j = k en j * i = -k.
There is little other way than to accept there are often two possible outcomes, when colors meet. The possible outcomes superpose. They can cancel each other. Or double their value in the wavefunction.

4a) Every color has one anticolor.
and (or red and cyan),
and (or green and magenta),
and (or blue and yellow).

4b) Each of the imaginary quaternions has one opposite:
i and -i,
j and -j,
k and -k.

5a) A color and the appropriate anticolor give something that has no color no more and that is white.

5b) An imaginary quaternion unit times its opposite gives something that isn't imaginary no more and that is 1.
i * -i = 1
j * -j = 1
k * -k = 1

6a) One can order the 6 colors in a 2 dimensional colorcircle. White, an eventual 7th color, can be thought of being placed in the middle of the circle.



6b) The imaginary quaternion units form 6 points in a 3 dimensional coordinate system. The 7th unit 1 and the 8th unit -1 are quaternion units on the real axis. The real axis is a 4th dimension then (not shown). Does this mean SU(3) is replaced by a kind of SU(2) X O(1), with 2 subsets [i, j, k, -i, -j, -k] and [1, -1]?



7a) Red can transform to green by a red-to-green gluon.

7b) A color like i can transform to the color j by multiplication with the gluon j/i. Just like in complex numbers dividing by i is the same as multiplying by -i. However, j has to be LEFT-MULTIPLIED by -i:
j/i = -i * j = -k, see page 1 of this storyline, eq. (1.8)
When a quark of color i transforms into a quark of color j by the gluon j/i, the quark color has to be RIGHT-MULTIPLIED by the gluon:
i * j/i = i * -i * j = j.
Both in gluon absorption as well as in gluon emission, the color always has to be right-multiplied by the gluon, see Page 2 of this storyline, eq. (2.3).

Will the b parts make things easier and simpler than the parts a? When so, maybe things can be calculated that in SU(3) are very difficult and doubtful, if not impossible.


Dark multiplication rules
In backward time evolving vacuum applies dark multiplication rules, see also paragraph Dark multiplication rules at page 2 of EXPANSION OF THE UNIVERSE.

DARK DARK   DARK DARK   DARK DARK

-1 * -1 -1 * 1   -i * -i -i * i   1 * i 1 * -i

= 1 * 1 = 1 * -1   = i * i = i * -i = -1 * -i = -1 * i

= -1 = 1   = 1 = -1   = -i = i

(9.17)

DARK

1 0

0 1

1 0

0 1

=

-1 0

0 -1

-1 0

0 -1

=

-1 0

0 -1

(9.18)

				s1	s2	s3
		s1		-1	-i	j
		s2		i	-1	-k
DARK		s3		-j	k	-1		(9.19)

(9.19) means DARK s1 * s2 = -i, DARK s1 * s3 = j, and so on.


References
https://en.wikipedia.org/wiki/special_unitary_group for the matrix of SU(2) and the Gell-Mann matrices.

https://en.wikipedia.org/wiki/quaternion for the matrix of the quaternions.

https://en.wikipedia.org/wiki/pauli_matrices for the Pauli matrices.


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