Colors as Quaternions
In QCD color is a real property. In QCD, despite all attempts so far have failed, it is not forbidden to observe color. In this storyline we replace colors by the imaginary units of quaternions. Maybe a better name then would be QQD, Quantum Quaternion Dynamics, instead of QCD. The units are imaginary and cannot be observed directly as for example the electric charge can.
Colors as imaginary quaternion units
Quaternions are governed by i = j = k = ijk = -1 (1.1) (Hamilton 1843)
Mark this also could have been defined as -i = -j = -k = kji = 1, this is completely the same.
Complex numbers have 1 imaginary axis and 1 real axis. Quaternions have 3 imaginary axes, their units called i, j and k, and 1 real axis, spanning a 4 dimensional world. For a basic summary of quaternions, see pop-up frame QUATERNIONS.
There are three colors of the strong force and together they form something that has no color no more and that is called white. There are three imaginary quaternion units and multiplied together they form something that has no imaginairity any more: ijk = -1 or kji = 1.
In the storyline NET FORCES IN QCD - read the SUMMARY - the color circle is constructed. Six colors and white, a kind of “seventh color”, in the center.
From (1.2) and the colorcircle we replace the anticolors.
It still can happen we talk about colors and depict them as a color, pretending this world isn't colorless. But we know now it are no colors no more, it are quaternion units.
A color and the corresponding anticolor together gives something that has no color no more. An imaginary quaternion unit multiplied by the corresponding opposite unit gives something that no longer is imaginary:
i x -i = -1, j x -j = -1, k x -k = -1.
Instead of the colorcircle we now use the axes system of the imaginary quaternion units.
In this picture the angle of projection of the chosen 3-dimensional axes system of imaginary quaternion units is striking. The projection resembles the color circle a lot.
Gluons when colors are replaced by quaternions
In QQD Feynman diagrams can and will be used as is usual in QCD, for example:
The equation of this diagram is:
Q(A B) * g * Q(B C) * GL(B D) (2.1) (the equation number being ahead is not a mistake)
Put in words this is the displacement of the quark Q from point A to point B times the coupling-constant of the color-interaction g times the displacement of the quark from B to C times the displacement of the gluon GL from B to D.
In equation (2.1) its composing actions, like Q(A B) and g, are used as a factor. With color we do the same and use color as a factor in the equation.
Regard the picture of equation (2.1). We follow arguments analogous to this paragraph in SUMMARY 1. (I mean the paragraph “It is as if the gluon takes away...” until “...As if the anticolor goes backwards in time.”) We continue the time-symmetric representation for gluons, see just below the mentioned paragraph. Although the normal representation is more realistic, we prefer the time-symmetric representation and it would be too confusing to use both representations together.
At point B the magenta quark emits a gluon. The gluon takes away the magenta from the quark and carries it off in its upper color. “Taking away” normally is a kind of subtraction, but in equation (2.1) you can see that applying an action is multiply in the equation. So here “taking away” is dividing by magenta .
Taking away magenta = / = / -j = * -( -j ) = * j (1.6)
Then a cyan-anticyan correlated-pair-of-colors appears, a cyan-red pair that is, see the colorcircle. The cyan is given to the quark to replace the taken-away magenta. Giving is a kind of adding, which is a multiplication in the equation.
Adding cyan = * = * -i (1.7)
The remaining red is dragged away in the lower part of the gluon. In the normal representation it would indeed be depicted as red, but we use the time-symmetric representation where the lower color of the gluon is depicted as its anticolor cyan, .
The gluon now has the potential to add a magenta to an encountering quark (gluon's upper color) and to remove a cyan from it (the gluon's lower color). The potential to remove cyan is the same as the potential to add a red.
Dividing by cyan = / = / -i = * i = *
The gluon then is
= * / = * -j / -i = * i * -j = * *
When dividing a color like -j by another color like -i, it is the same as -j multiplying by i. But mind, you have to left-multiply -j by i. (1.8)
The notation * / = * -j / -i comes closest to depicting gluons as upper and lower color in time-symmetric representation (which we normally prefer), . Magenta in the upper color, cyan as lower color and the border between them as the breakline.
The notation * * = * i * -j comes closest to the normal representation, .
Other equalities follow:
= / = -j / -i = j / i = / =
which can be checked out in the gluon table.
(In this we recognize the statement/conjecture “So we can forget about the colors of the gluon and use its color-shift only” in page 3 of the storyline NET FORCES IN QCD, just above the paragraph “Identifying gluons of the same color-shift”. But mind, colorshift is abandoned. It's quaternions only now.)
Suppose we would state
= * i * -j = * -j * i = * i / j = * / , which is the opposite from * / .
The first gluon goes from a red quark to a green quark, while the latter gluon goes from a green quark to a red quark. Quaternions usually don't allow a swap of multiplication order, quaternion multiplication is not commutative. The bold “=” is wrong.
In both cases the red quark and the green quark swap color, the difference is only in the time order of gluon emission and gluon absorption. Which suggests reversing multiplication order is reversing time order - well, at least in this case.
= / = -j / -i = i * -j = -k =
A gluon color pair can be converted to a single color, see the quaternion gluon table. (The expression vacuum particle in the table refers to a conclusion from the theory of gravitation.)
Only quarks have single color. Quarks have spin 1/2, mass maybe about 20 MeV and charge is e.g. +2/3 for the u-quark, or -1/3 for the d-quark. Gluons have spin 1, no mass and no charge. So the transition from a single-color quark into a double-color gluon is not possible until the differences are settled somehow. But no one has ever observed a color, in doing so determining it. Colors remain virtual. So in the superpositions gluons might be considered as single colors, and so we do.
We treat color reactions as if there is a conservation law of color. In fact there is the “urge for white end state” which says the color sum is a constant and the constant = 0.
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In QCD gluons have an upper and a lower color, determining a shift in color. Colorshift is a concept introduced in the storyline NET FORCES IN QCD but is abandoned again in quaternions.