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For a gauge group $SU(3)_{C}$ we can construct its principal and associated bundles; we can introduce spinor fields via spin structures and spinor bundles and so on, arriving in a lagrangian theory that describes eventually the QCD. For mathematical definitions and didatic introduction I would recommend $[1]$.

Studing the mathemetics behind gauge theories, at least three structures are paramount: the spacetime, the clifford algebra and the lie algebra. One way to "see" its manifestations are by the indexes that I shall use inside the lagrangian density. So consider the fermionic part of the lagrangian of QCD.

$$\mathcal{L} = \bar{\psi}\Big[i\gamma^{\mu}\Big(\partial_{\mu}+ig\rho_{*}(\mathcal{A_{s}})\Big)-m\Big]\psi$$

where $\rho_{*}(\mathcal{A}_{s})$ is the representation of the local connection 1-form.

But the form of $(1)$ is condensed and it supress other indexes. My question is precisely on the "total" form of lagrangian $(1)$. So, my question is:

The QCD lagrangian, written with all the other indexes, is the equation $(2)$ in the following?

$$\mathcal{L} = (\bar{\psi^{a}})^{i}\Big[i\gamma^{\mu b}_{a}\Big(\partial_{\mu}+igA_{\mu}^{c}\lambda_{c}\Big)-m\delta^{b}_{a}\Big]^{j}_{i}(\psi_{b})_{j} = (\bar{\psi^{a}})^{i}\Big[i\gamma^{\mu b}_{a}\Big(\partial_{\mu}+igA_{\mu}^{c}\lambda_{c}\Big)\Big]^{j}_{i}(\psi_{b})_{j}-(\bar{\psi^{a}})^{i}\Big[m\delta^{b}_{a}\Big]^{j}_{i}(\psi_{b})_{j} \implies $$ $$\mathcal{L} = (\bar{\psi^{a}})^{i}i\gamma^{\mu b}_{a}\delta^{j}_{i}\partial_{\mu}(\psi_{b})_{j} - g(\bar{\psi^{a}})^{i} \gamma^{\mu b}_{a}\Big[A_{\mu}^{c}\lambda_{c}\Big]^{j}_{i}(\psi_{b})_{j} -(\bar{\psi^{a}})^{i}m\delta^{b}_{a}\delta^{j}_{i}(\psi_{b})_{j} \tag{2}$$

Where, $a$ is the index for the Clifford algebra, $\mu$ the index for spacetime and $j$-$i$-$c$ are the indexes of the lie algebra representation.


$[1]$ M. Hamilton. Mathematical Gauge Theory. 2015.

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  • $\begingroup$ Aren’t there three implicit indices on the spinors (one for spin, one for color, one for flavor)? Or are you assuming only one kind of quark? You’ve only got one $m$, so it looks like it. $\endgroup$
    – Ghoster
    Commented Mar 16, 2023 at 22:55
  • $\begingroup$ Maybe I wrote some things wrong, like the mass. But the whole point is: what are all the indexes of these type of equations? $\endgroup$
    – M.N.Raia
    Commented Mar 17, 2023 at 13:24

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