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I have a question regarding the Lagrangian in non abelian gauge theory. Say, $G$ is the gauge group and $\mathfrak g$ the associated Lie algebra. The Lagrangian is often written as

$$ \mathcal L=-\frac {1}{4} \text{tr} (F_{\mu \nu} F^{\mu \nu}) + \overline \psi (i \tilde D-m)\psi $$

with

$$ \tilde D=\gamma^\mu D_\mu \qquad D_\mu=\partial_\mu-igA_\mu $$

As far as i understood, in this equation $\psi$ is an element of a representation $V$ of $\mathfrak g$ and $A_\mu \in \text{End}(V)$, right? What's puzzling me now, is that this equation has to be Lorentz-invariant as well, so the $\psi$'s are still Dirac spinors, that is they are elements of the fundamental representation $\Delta$ of the Clifford Algebra $\text{Cliff}(1,3)$. How should i interpret this ambiguity and in particular how should i calculate $\gamma^\mu A_\mu \psi$ in local coordinates?

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  • $\begingroup$ I am not sure to understand correctly what is your problem, but note that $\psi$ is multicomponent: it is an element of $V\otimes\Delta$. $\endgroup$ – Start wearing purple Jul 4 '13 at 21:38
  • $\begingroup$ :-| Of course, thank you!! $\endgroup$ – user83496 Jul 4 '13 at 22:01
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The field $\psi$ is not one Dirac spinor but a collection of those: it belongs to tensor product representation $V\otimes\Delta$, and accordingly has two indices $\alpha$, $a$. In components (I guess this is what you mean by local coordinates) one has $$(\gamma^{\mu}A_{\mu}\psi)_a^{\alpha}=\gamma_{ab}^{\mu}A_{\mu}^{\alpha\beta}\psi^b_{\beta},$$ where $a,b$ are spinor labels and $\alpha,\beta$ are gauge group representation labels.

For example, when $V$ is the fundamental representation of $SU(3)$, the multicomponent field $\psi$ contains three Dirac spinors corresponding to different quark colors.

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