Quaternion applications - Part 3

# Application 5

The first line of the number table below shows quaternion unit 1 and the 8 ways how 1 can be composed as a multiplication of 2 quaternion units. The subsequent 3 lines do the same for quaternion units i, j and k. The ways to compose -1, -i, -j and -k in 2 quaternion units are similar and are not shown.

1 = 1 * 1 = -1 * -1 = -i * i = i * -i = -j * j = j * -j = -k * k = k * -k

i = i * 1 = -i * -1 = 1 * i = -1 * -i = -k * j = k * -j = j * k = -j * -k

j = j * 1 = -j * -1 = k * i = -k * -i = 1 * j = -1 * -j = -i * k = i * -k

k = k * 1 = -k * -1 = -j * i = j * -i = i * j = -i * -j = 1 * k = -1 * -k

Each of the 8 quaternion units 1, -1, i, -i, j, -j, k, -k has 8 ways to be composed as a multiplication of 2 quaternion units. From page 2 we learned we can see all first factors as quark colors and all second factors (if present) as gluons that are emitted or absorbed by the quark. Let's repeat the number scheme above in colors. We use: 1 = , -1 = and the gluon table

In the color table below regard e.g. the cell at 2nd row, 7th column, the first column has column number zero.

 1 2 3 4 = 5

The cell is made of two lines. The upper line means a green quark 1 had just emitted a blue gluon 2. Just before the emission the quark was red, the color shown in column zero of the same line. In the lower line is a second quark 3, a cyan one, that absorbs the blue gluon 4 (which is the same gluon as nr 2), yielding a green quark 5 again.

The multiplication of the two colors in the upper line of a cell always yields the color at the same line in column zero. The first color of the lower line just repeats the color in the same line of column zero.

In column zero in every row the two colors, one above the other, have opposite color. They can be the colors of the two quarks in a meson. Or the opposite colors of a quark-antiquark pair emerging in the outer rims of a quark, shielding the naked color of the quark, see QCD, page 4, paragraph Meson Exchange.