I am reading Tong's lecture note gauge theory. On page 6 in chapter 1 he writes
Instead, the key distinction is the choice of Abelian gauge group. A $U(1)$ gauge group has only integer electric charges and admits magnetic monopoles. In contrast, a
gauge group $R$ can have any irrational charges, but the price you pay is that there are no longer monopoles.
I don't quite understand a gauge group can have any irrational charges. As answered in this question
Also note that in the physics literature, we often identifies charge operators with Lie algebra generators for a Cartan subalgebra (CSA) of the gauge Lie algebra.
Which generator of non-Abelian Lie algebra should be identified with electric charge generator? For example for $SU(3)$ gauge group, if we choose
$$ \lambda^8=\frac{1}{\sqrt{3}}\left(
\begin{array}{cccc}
1&\,\,\,0&\,\,\,0 \\
0&\,\,\,1 &\,\,\,0 \\
0&\,\,\,0&\,\,\,-2
\end{array} \right)$$
Does this mean the charge is $1/\sqrt{3}$?
Best Answer
Looks like you are trying to understand the color charge of the color SU(3) through the flavor electric charge of flavor SU(3), which has nothing-nothing-nothing to do with it, except we use the same hermitian generators, the Gell-Mann matrices halved, for the triplet (quark) representation, to describe them.
In the former case, any flavor of quarks has eight color charges , i = 1, ..., 8,
$$ \int\!\! d^3x~~ \bar q \cdot \tfrac{1}{2}\lambda^i ~\gamma ^0 q, $$ which are strictly conserved, and are each coupled to the eight respective gluons (the color gauge fields) for consistency of the theory. Their actual values, related among themselves, are immaterial, and can be linearly combined and/or absorbed into redefinitions of them and their gauge-fields. Their eigenvalues hardly matter; for the triplet of quarks, for whose labels people use R,G,B, each gluon mutates color differently, and all you'd need is to ensure total color is conserved in an interaction.
In the latter case, for flavor SU(3), the light, u,d,s, quarks, you have eight almost conserved flavor charges that look like the above ones, but are violated by small amounts, isospin and hypercharge breaking. It turns out the electric charge Q is related to this quark triplet by the Gell-Mann–Nishijima formula , $$ Q= \frac{\lambda^3}{2} +\frac{\sqrt{3}}{6} \lambda^8. $$ (I simplified it for your convenience, since it modifies trivially for other hadrons.) You may confirm the electric charges (2/3, -1/3,-1/3) of the (u,d,s) triplet: that was the design!
The normalizations are, indeed, irrational and freaky, but the action of the Cartan subalgebra of these two generators (simultaneously diagonal in this basis) "conspires" to yield related eigenvalues in the root lattice, so