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This question pertains to the following passage from Weinberg's second volume on QFT. It appears on page 4, section 15.1.

To make the Lagrangian invariant, we need a field $A^\alpha_\mu$, whose transformation rule involves a term $\partial_\mu \epsilon^\mu$, which can be used to cancel the second term in Eq. (15.1.8). Since this field carries an $\alpha$-index, we would expect it also undergo a matrix transformation like Eq. (15.1.1), but with $t_\alpha$ replaced with the adjoint representation matrices (15.1.6).

The bold emphasis is mine, because it is this expectation that I'm not grasping. First though, here are the referenced equations,

$$\delta\left(\partial_{\mu} \psi_{\ell}(x)\right)=i \epsilon^{\alpha}(x)\left(t_{\alpha}\right)_{\ell}^{m}\left(\partial_{\mu} \psi_{m}(x)\right)+i\left(\partial_{\mu} \epsilon^{\alpha}(x)\right)\left(t_{\alpha}\right)_{\ell}^{m} \psi_{m}(x) \tag{15.1.8}$$

This passage is motivating the need for a covariant derivative, since Eq. (15.1.8) shows that derivative terms of the matter fields, $\partial_\mu\psi_l(x)$ do not transform in the same way as the fields themselves: $$\delta \psi_{\ell}(x)=i \epsilon^{\alpha}(x)\left(t_{\alpha}\right)_{\ell}^{m} \psi_{m}(x) \tag{15.1.1}$$

and lastly, the adjoint representation matrices are those defined by the structure constants themselves,

$$\left(t_{\alpha}^{\mathrm{A}}\right)^{\beta}{ }_{\gamma} \equiv-i C_{\gamma \alpha}^{\beta} \tag{15.1.6}$$

Now to my confusion. When Weinberg says:

Since this field carries an $\alpha$-index, we would expect it also undergo a matrix transformation like Eq. (15.1.1)...

Even this is surprising to me, because $\psi_l$ doesn't carry an $\alpha$-index. It's not obvious to me why $A^\alpha_\mu$ carrying this index should mean necessarily that it will undergo a matrix transformation.Furthermore, it's entirely unclear to me why the $t_\alpha$ should be replaced by the adjoint representation matrices. Is that something we should also expect from looking at how $\partial_{\mu} \psi_{l}(x)$ transforms?

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