In some Yang-Mills theory with gauge group ##G##, the gauge fields ##A_{\mu}^{a}## transform as(adsbygoogle = window.adsbygoogle || []).push({});

$$A_{\mu}^{a}

\to A_{\mu}^{a} \pm \partial_{\mu}\theta^{a} \pm f^{abc}A_{\mu}^{b}\theta^{c}$$

$$A_{\mu}^{a}

\to A_{\mu}^{a} \pm \left(\partial_{\mu}\theta^{a}-A_{\mu}^{b}f^{bac}\theta^{c}\right)$$

$$A_{\mu}^{a}

\to A_{\mu}^{a} \pm \left(\partial_{\mu}\theta^{a}-iA_{\mu}^{b}(T^{b}_{\text{adj}})^{ac}\theta^{c}\right),$$

where ##T^{a}_{\text{adj}}## is the adjoint representation of the gauge group ##G## and the gauge parameters ##\theta^{a}## are seen to transform in the adjoint representation of the gauge group ##G##.

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Why does this mean that the gauge fields ##A_{\mu}^{a}## transform in the adjoint representation?

Should the transformation of the gauge fields ##A_{\mu}^{a}## in the adjoint representation not be given by

$$A_{\mu}^{a} \to A_{\mu}^{a} \pm i\theta^{b}(T^{b}_{\text{adj}})^{ac}A_{\mu}^{c}?$$

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# A Gauge transformation of gauge fields in the adjoint representation

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