Does each vector in $su(3)$ represent a different kind/type of gluon (infinite kinds/types of gluons); or, are they all considered the same kind/type?

Question

According to Does gluons have names?, it seems that there is no way to give names to gluons because they are not "always the same". So, it seems to imply that the same gluon can change its color charge / state depending on its interactions. Some sources say explicitly that there are 8 types of gluons which might be misleading one consider that such 8 kinds are rather 8 linearly independent vectors in $su(3)$ which allow the definition of infinite color charge combinations in that space for gluons. If a gluon can take any of these infinite combinations of color charges, does this imply that that there are infinite kinds of gluons? Or do they all belong to the same kind but simply have different color charge combinations? For instance, an electron and the positron can have only 1 charge value (-1, in the case of electrons, or +1 in the case of its anti-particle, the positron) and, if that value changed, they'd cease to be electrons or positrons, respectively.

Answer 1

There are two "levels" of charge in gauge theories. There is the representation, which is a vector space, and then there is a vector in that space.

The vectors within a representation are all interrelated by the gauge symmetry, but the different representations are not. Every particle is unambiguously in one representation.

Different electric charges like $+2/3$ and $-1$ are labels of different representations of $U(1)$. Each of those representations is one-dimensional, so the question of a basis is largely moot, but the particles still have infinitely many orientations in the gauge space because there is a scalar multiplier.

The color charge of the quarks is called $\mathbf 3$ and it's 3-dimensional. There is a standard basis for it with vectors named "red", "green" and "blue", although it's meaningless because the SU(3) symmetry is unbroken and no distinguishing properties of these so-called colors can be identified.

The color charge of the gluon field is called $\mathbf 8$ and it's 8-dimensional. There is a standard basis of sorts, the Gell-Mann matrices, but the basis vectors don't have names.

When people say there are 8 gluons, they mean the representation is 8-dimensional (or the symmetry group is, which amounts to the same thing).

People usually don't say that there are 18 quarks (6 of each flavor), though there are by the counting used for the gluons. They never say that there is just 1 gluon, though there is by the counting used for the quarks.

You could make a case that there are infinitely many gluons and quarks, but no one ever says that.

In any case, just as electrons always have the same charge, $-1$, gluons also always have the same charge, $\mathbf 8$. But at another level, electrons have a $U(1)$ orientation that varies, and gluons have (or rather are) a $SU(3)$ orientation that varies.

Answer 2

The gluons do have names. So, the three dimensions of $\mathrm{SU}(3)$ are named red, blue, and green. Their anti-charges are labeled anti-red (or cyan), anti-blue (or yellow), and anti-green (or magenta).

For gluons, they're not in the vector basis, they're in the traceless anti-hermitian matrices that are the generators in the vector basis. So, a complex $3\times 3$ matrix has $2\times 3\times 3 = 18$ real numbers needed to specify it. Limiting that to anti-hermitian matrices requires the real part of the matrix to be anti-symmetric and the imaginary part to be symmetric. That limits the number of degrees of freedom to $3\times(3-1)/2 + 3\times(3+1)/2= 9$. The requirement that it's traceless knocks off one more degree, to leave $8$.

Now, because the matrices have the structure they have, they carry two color charges. Traditionally, we label them with (color)/(anti-color). So that gives: red-cyan, red-yellow, red-magenta, blue-cyan, blue-yellow, blue-magenta, green-cyan, green-yellow, and green-magenta. That gives 9. What's the extra one? Well, the sum of red-cyan + blue-yellow + green-magenta is the trace degree of freedom, so that's the one that gets excluded. Traditionally, that one is called "white" or "colorless".

Constructing the real list of names requires a bit of care to ensure that the diagonal ones are actually orthogonal, and not just linearly independent. The Gell-Mann matrices can be described as (up to overall normalization): $\lambda_1=$ green-cyan $+$ red-magenta, $\lambda_2=$ green-cyan $-$ red-magenta, $\lambda_3=$ red-cyan $-$ green-magenta, $\lambda_4=$ red-yellow $+$ blue-cyan, $\lambda_5=$ blue-cyan $-$ red-yellow, $\lambda_6=$ green-yellow $+$ blue-magenta, $\lambda_7=$ blue-magenta $-$ green-yellow, and $\lambda_8=$ red-cyan $+$ green-magenta $-$ 2 blue-yellow.

And, yes, they are all considered to be different "types" of particles, the same way that red, blue, and green quarks are different, or electrons and positrons are different. The number of different types of gluons affect things like calculations of the heat capacity of the early universe, and the properties of quark-gluon plasmas made in labs. Also, don't forget the number of polarizations, since each polarization is distinct, too.