spaghetti3451
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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
$$D_{\mu}\phi = \partial_{\mu}\phi - igA_{\mu}^{a}[T^{a},\phi]?$$
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
$$D_{\mu}\phi = \partial_{\mu}\phi - igA_{\mu}^{a}[T^{a},\phi]?$$