A How Does Gauge Invariance Shape Classical Yang-Mills Theory with Scalar Fields?

[spaghetti3451]

Let us consider a classical field theory with gauge fields $A_{\mu}^{a}$ and a scalar $\phi^{a}$ such that the Lagrangian is gauge-invariant under the transformation of

1. the gauge fields $A_{\mu}^{a}$ in the adjoint representation, with dimension $D_{\bf R}$, of the gauge group $SU(N)$, and
2. the scalar $\phi^{a}$ in the fundamental representation, with dimension $N$, of the gauge group $SU(N).$

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1. Why can we represent $\phi$ as a traceless Hermitian $N \times N$ matrix, so that $\phi = \phi^{a}T^{a}$ where the $T^a$ are the representation matrices in the fundamental representation?

2. Why can we write down the variation of $\phi$ under a gauge transformation with gauge parameters $\theta^{a}$ as

$$\delta\phi = ig[\theta^{a}T^{a},\phi]$$

and the gauge covariant derivative as

$$D_{\mu}\phi = \partial_{\mu}\phi - igA_{\mu}^{a}[T^{a},\phi]?$$

 

[dextercioby]

The only explanation that I know of is the one by the Craiova school: C. Bizdadea, E. M. Cioroianu, M. T. Miauta , I. Negru, and S. O. Saliu.
Lagrangian cohomological couplings among vector fields
and matter fields, Ann. Phys. (Leipzig) 10 (2001) 11––12, 921––934