Talk:Gluon/Numerology draft

Numerology of gluons
Unlike the single photon of QED or the three W and Z bosons of the weak interaction, there are eight independent types of gluon in QCD.

This may be difficult to understand intuitively. Quarks carry three types of color charge; antiquarks carry three types of anticolor. Gluons may be thought of as carrying both color and anticolor, but to correctly understand how they are combined, it is necessary to consider the mathematics of color charge in more detail.

Color charge and superposition
In quantum mechanics, the states of particles may be added according to the principle of superposition; that is, they may have be in a "combined state" with a probability, if some particular quantity is measured, of giving different several different outcomes. A relevant illustration in the case at hand would be a gluon with a color state described by:


 * $$(r\bar{b}+b\bar{r})/\sqrt{2}$$

This is read as "red anti-blue plus blue anti-red." (The factor of the square root of two is required for normalization, a detail which is not crucial to understand in this discussion.) If one were somehow able to make a direct measurement of the color of a gluon in this state, half the time one would see red-antiblue and the other half blue-antired; it does not mean that the red from the first term and anti-red from the second (and the antiblue and blue) sum to zero color charge in any sense.

Color singlet states
It is often said that the stable strongly-interacting particles observed in nature are "colorless," but more precisely they are in a "color singlet" state, which is mathematically analogous to a spin singlet state. Such states allow interaction with other color singlets, but not with other color states; because long-range gluon interactions do not exist, this illustrates that gluons in the singlet state do not exist either.

The color singlet state is :


 * $$(r\bar{r}+b\bar{b}+g\bar{g})/\sqrt{3}$$

In words, if one could measure the color of the state, there would be equal probabilities of it being red-antired, blue anti-blue, or green-antigreen.

Eight gluon colors
There are eight remaining independent color states, which correspond to the "eight types" or "eight colors" of gluons. Because states can be mixed together as discussed above, there are many ways of presenting these states, which are known as the "color octet." One commonly used list is :


 * $$(r\bar{b}+b\bar{r})/\sqrt{2}$$
 * $$-i(r\bar{b}-b\bar{r})/\sqrt{2}$$
 * $$(r\bar{r}-b\bar{b})/\sqrt{2}$$
 * $$(r\bar{g}+g\bar{r})/\sqrt{2}$$
 * $$-i(r\bar{g}+g\bar{r})/\sqrt{2}$$
 * $$(b\bar{g}-g\bar{b})/\sqrt{2}$$
 * $$-i(b\bar{g}-g\bar{b})/\sqrt{2}$$
 * $$(r\bar{r}+b\bar{b}-2g\bar{g})/\sqrt{6}$$

These are equivalent to the Gell-Mann matrices; the translation between the two is that red-antired is the upper-left matrix entry, red anti-blue is the left middle entry, blue anti-green is the bottom middle entry, and so on. The critical feature of these particular eight states is that they are linearly independent, and also independent of the singlet state; there is no way to add any combination of states to produce any other. (It is also impossible to add them to make $$r\bar{r}$$, $$g\bar{g}$$, or $$b\bar{b}$$ ; if it were, then the forbidden singlet state could also be made.) There are many other possible choices, but all are mathematically equivalent, at least equally complex, and give the same physical results.

Group theory details
Technically, QCD is a gauge theory with SU(3) gauge symmetry. Quarks are introduced as spinor fields in Nf flavours, each in the fundamental representation (triplet, denoted 3) of the colour gauge group, SU(3). The gluons are vector fields in the adjoint representation (octets, denoted 8) of colour SU(3). For a general gauge group, the number of force-carriers (like photons or gluons) is always equal to the dimension of the adjoint representation. For the simple case of SU(N), the dimension of this representation is N2&minus;1.

In terms of group theory, the assertion that there are no color singlet gluons is simply the statement that quantum chromodynamics has an SU(3) rather than a U(3) symmetry. There is no known a priori reason for one group to be preferred over the other, but as discussed above, the experimental evidence supports SU(3).