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In a gauge theory, the fields transform under representations of the gauge group. When studying a special unitary group $SU(n)$, I've usually thought of the elements of a representation as being the exponentials $e^{iM}$ of the elements $M$ of the representation of the associated algebra, $su(n)$. The basis of the algebra are the generators of the group, and $M$ is then a sum over the generators $t_a$ in that representation $$M=\sum_{a} c_a t_a$$ where $c_a \in \mathbb{R}$. The group elements are unitary with determinant one, so the generators must be Hermitian and traceless.

Next, in a gauge theory (simplifying to a single gauge field), the covariant derivative takes the form $$D_\mu := \partial_\mu - i g t_a A_\mu^a$$ where $g$ is the coupling constant, $t_a$ are the generators for the representation of the group the field the covariant derivative is acting on transforms under, and $A_\mu^a$ are the components of the gauge field. Although there is freedom in choosing the value of the coupling constant, the possible associated charges for the fields are set (and quantised) by the nature of the possible representations of the generators, e.g. the possible values of the third component of isospin $T_3$ for $SU(2)$.

The confusion I have arises when trying to apply these rules to $U(1)$. The fundamental representation is the set of phases $e^{i\theta}$, which is what all charged fields gather under the group transformation (with different values of $\theta$). By comparison, this would seem to make any non-zero real number, i.e. $1$, the generator for the associated algebra representation. However, charged fields don't all have the same coupling to the EM field - they have different charges. How can this be possible if all non-zero real numbers generate the same group representation?

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  1. Briefly, the different $U(1)$ irreps (over a complex vector space) are classified by their $U(1)$-charge $\in\mathbb{Z}$. [They are all 1-dimensional since the group $U(1)$ is Abelian.]

  2. Concerning reasons for charge quantization, see e.g. this & this Phys.SE post.

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