Adjoint representation and spinor field valued in the Lie algebra

Question

I'm following the lecture notes by Timo Weigand on QFT.

On page 169, section 6.2 he is briefly touching on the non-abelian gauge symmetry in the SM.

The fundamental representation makes sense to me. For example, for $SU(3)$, we define the object or column vector with three component, suppressing spinor indices, $\psi(x)=(\psi_1, \psi_2, \psi_3)^T$. The fundamental representation, a $3*3$ matrix $V(h)$, acts on this column vector, with each component itself a Dirac spinor that has 4 components of complex numbers. Lagrangian density then is showed to remain invariant when the $\psi$ is multiplied by the $3*3$ matrix $V(h)$.

But the next step puzzles me a little, when we use the adjoint representation. Here instead of a 3 component column vector, we use a $3*3$ matrix $\psi(x)$, whose elements $\psi_{ij}(x)$ are complex numbers based on the definition given in Eqn.(6.32). $$\forall x: \psi(x) \equiv \psi_{i j}(x) \in\left\{\mathbb{C}^{N, N} \mid \psi^{\dagger}(x)=\psi(x), \operatorname{tr} \psi(x)=0\right\}$$

It is a $3*3$ matrix or representation of the $SU(3)$ Lie algebra.

But when in Eqn.(6.34) $$\mathcal{L}=\operatorname{tr} \bar{\psi}(i \partial-m) \psi \equiv \operatorname{tr}\left[\psi_{i j}(i \partial-m) \psi_{j l}\right] \equiv \psi_{i j}(i \partial-m) \psi_{j i} $$ the Lagranian density is defined as $tr(\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi)= \bar{ \psi_{ij}}(i\gamma^\mu\partial_\mu-m)\psi_{ji}$. Here $\psi$ is a $3*3$ matrix whose elements, $\psi_{ji}$, are not complext numbers but Dirac spinors.

In another word, is this what we mean by the spinor indices are suppressed in the adjoint representation? How can we think of $\psi(x)$ both as a $3*3$ matrix with components to be both complex numbers as defined in (6.32) and Dirac spinors as used in (6.34) that have themselves four components, each a complex number.

Answer 1

Perhaps this will help. Let $t^a$ be the generators of the Lie algebra is some representation. For example, this could be the adjoint representation as you're asking about and each $t^a$ will be a $3\times 3$ matrix.

Then any field $\phi$ valued in the Lie algebra may be written generically as $\phi=\phi_a t^a$. This is a common trick employed when dealing with gauge connections.

For the spinors, let $\alpha$ be the spinor index. Then we could write $$ \psi_\alpha=\psi_{\alpha a}t^a. $$ The fields $\psi_i$ would then be valued in the Lie algebra. If we wanted to suppress the mention of $t$, then we could define $$ \psi_{\alpha ij}=\psi_{\alpha a}(t^a)_{ij}. $$ Suppressing the spinor index means writing $\psi_{ij}$ and viewing each component as spinor-valued.