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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 $1/\sqrt{3}$?

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    $\begingroup$ The eigenvalues of the triplet color representation for $\lambda^8$ you wrote down, are, indeed, $1/\sqrt 3$ or $-2/\sqrt 3$. No electric charges there, if that's what you are talking about. The Gell-Mann--Nishijima electric charge embedded in the ungauged flavor SU(3) is rational, but noninteger by design, and I assume you are not asking about that. Your question is deeply imprecise and confusing as it stands. $\endgroup$ Commented Jan 1, 2022 at 15:48
  • $\begingroup$ Thanks. If we just consider this $SU(3)$ theory. How to define charge? If it is the eigenvalue of generator, can I just rescale generators get any charge I want? How to interprete other non-diagonal generators? $\endgroup$
    – thone
    Commented Jan 1, 2022 at 23:56
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    $\begingroup$ All Hermitean generators are unitarily diagonalizable and have discrete eigenvalues. You may, indeed, normalize them arbitrarily. The eigenvalues, however, are related on a root lattice. $\endgroup$ Commented Jan 2, 2022 at 4:43
  • $\begingroup$ @Cosmas Zachos, Good explaination! Thanks. To define the charge, which of the nine Gell-mann matrices should I use? $\endgroup$
    – thone
    Commented Jan 2, 2022 at 8:57

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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

  • the resulting electric charges are multiples of each other! So the nonabelian flavor group (here SU(3), ungauged)is primed to yield charge quantization at the end of the day.
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  • $\begingroup$ Thanks for your answers. Sorry for my unclear question. I just want to know how to define electric charge for a general gauge theory, for example $SU(N)$. My experience comes from QED, where the charge of gauge symmetry is zero since it is not a physical symmetry, while the electric charge comes from the global one. Here your answer seems in QCD. If I understand correctly, the definition of the electric charge depends matter multiplets I consider. For the dundamental representation quark, the definition is the one you given above. Why should we define it in such way? $\endgroup$
    – thone
    Commented Jan 3, 2022 at 0:15
  • $\begingroup$ Just for matching baryon charge? If so, how to define the electric charge in a general theory, for example $SU(N)$ with fundamental matter? Is this related with gauge anomaly cancelation? Sorry for my stupid question, for I do not know much about particle physics. $\endgroup$
    – thone
    Commented Jan 3, 2022 at 0:19
  • $\begingroup$ Everything is related to anything else, at some level, but you are possibly asking 7 questions, not one, and I don't know which. QCD has nothing to do with electric charge. If a gauged SU(N), like the GUT SU(5), includes electric charge, then, indeed, the EM U(1) is an unbroken subgroup of it, and anomaly cancelations imposes constraints on the multiplets, which have already been accounted for, as needed in all models, right from the start. The way quarks were introduced, heuristically, had nothing to do with gauge groups: they were for flavor SU(3), before ever suspecting gauged color. $\endgroup$ Commented Jan 3, 2022 at 1:00
  • $\begingroup$ ... Gell-Mann & Zweig guessed them, and they worked. Contrary to modern discussions, quarks were heuristic, phenomenological, and hardly inevitable. Electric charge was a neat, easy, visible handle on the sudoku solved. $\endgroup$ Commented Jan 3, 2022 at 1:03
  • $\begingroup$ Could we use more specific names for the "eight color charges" (gauge charge?) to differentiate this concept form the "three color charges" (dimension of fundamental representation of SU(3)?) ? $\endgroup$
    – Davius
    Commented Jul 23, 2022 at 8:14

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