Recently, I have been learning about non-Abelian gauge field theory by myself. Thanks @ACuriousMind very much, as with his help, I have made some progress.
I am trying to extend the Dirac field equation with a coupling to a $SU(2)$ gauge field:
$$(i{\gamma}^{\mu }{D}_{\mu}-m)\psi =0$$
where
$${ D }_{ \mu }=\partial _{ \mu }+ig{ A }_{ a }^{ \mu }{ T }_{ a }$$
the ${ T }_{ a }$ is the $SU(2)$ Lie group generator, with $[{ T }_{ a },{ T }_{ b }]=i{ f }^{ abc }{ T }_{ c }$, and the ${\gamma}^{\mu }$ are the Dirac matrices. When I write explicitly the first part of the Dirac equation, with spinor form $\psi=(\phi,\chi)^T$, I get (spatial part):
$$\begin{pmatrix} 0 & { \sigma }^{ i } \\ -{ \sigma }^{ i } & 0 \end{pmatrix}\partial _{ i }\begin{pmatrix} \begin{matrix} \phi \\ \chi \end{matrix} \end{pmatrix}+ig\begin{pmatrix} 0 & { \sigma }^{ i } \\ -{ \sigma }^{ i } & 0 \end{pmatrix}{ A }_{ a }^{ i }{ T }_{ a }\begin{pmatrix} \begin{matrix} \phi \\ \chi \end{matrix} \end{pmatrix}$$
My problem is: I only known the linear representation of ${ T }_{ a }$ is Pauli spin matrix from text book, but they are the set of 2-dimension matrixes, In above expression, I need to know the 4-dimension matrix of ${ T }_{ a }$ because of the spinor is 4-dimension, I
check some test book, but didn't find the explicitly statement of the 4-D matrix.
So, as mentioned in title, What is the 4-dimension representation of the $SU(2)$ generators, or how can I calculate it?
Best Answer
Comment to the question (v4): OP seems to effectively conflate spacetime symmetries and internal gauge symmetries. They act in different representations, or more precisely as a tensor product of representations.
For instance the fermion $\psi$ carries two types of indices, say $\psi^{\alpha i}$, $\alpha=1,2,3,4,$ and $i=1,2$. The fermion acts
as a $4$-dimensional Dirac spinor representation under Lorentz transformations.
as a $2$-dimensional fundamental representation of the gauge group $SU(2)$ under gauge transformations.
Similarly, the $4\times 4$ Dirac matrices $\gamma^{\mu}$ and the $2\times 2$ $SU(2)$ gauge group generator $T^a$ act on different representations. The product of $\gamma^{\mu}$ and $T^a$ is a tensor product. In particular, the term $\gamma^{\mu}T^a\psi$ in OP's formula again carries two types of indices, and is evaluated as
$$ (\gamma^{\mu}T^a\psi)^{\alpha i}~=~(\gamma^{\mu})^{\alpha}{}_{\beta}~ (T^a)^{i}{}_{j}~\psi^{\beta j}. $$