# Why the final equation of motion of Yang Mills Lagrangian looks like that?

This question is related to a previous question:

How to calculate Noether current for Yang Mills theory

Actually there I calculated the three Eular-Lagrange equations of motions of the Lagrangian:

\begin{align} \mathcal{L}=-\frac{1}{4} F^a_{\mu\nu}~ F_a^{\mu\nu} +\bar{\psi}(i\gamma^{\mu} D_{\mu})\psi- m~ \psi \bar{\psi}, \end{align} with, \begin{align} D_\mu \psi = \partial_\mu \psi - i g~ t^a_{ij}~ A_\mu^a~ \psi\\ F_{\mu\nu}^a = \partial_\mu A^a_\nu - \partial_\nu A_\mu^a + i g f^{abc} A^b_\mu A_\nu^c, \end{align}

due to the three field $\psi,\bar{\psi}, A_\mu$ as:

$\frac{\partial \mathcal{L}}{\partial\psi} - \partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu\psi)} = 0 ~~~~~~~~~~(1)\\ \frac{\partial \mathcal{L}}{\partial\bar{\psi}} - \partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu\bar{\psi})} = 0~~~~~~~~~(2) \\ \frac{\partial \mathcal{L}}{\partial A_\mu} - \partial_\nu \frac{\partial \mathcal{L}}{\partial(\partial_\nu A_\mu)} = 0 ~~~~~~~(3)\\$

yield:

$g \bar{\psi} \gamma^\mu A_\mu^a - m \bar{\psi} -i \partial_\mu \bar{\psi} \gamma^\mu = 0~~~~~~~~(4)\\ g \gamma^\mu A_\mu^a \psi - m \psi +i \partial_\mu \gamma^\mu \psi= 0~~~~~~~~~~~(5)\\ g\bar{\psi} \gamma^\mu t^a_{ij} \psi_j + g f^{abc} A^b_\nu F^{c~\mu\nu}= \partial_\nu F^{a~ \nu \mu}~~~~~~~~(6)$

But still don't know why Equ.(6) is the right equation of motion of the Yang Mills theory, see for instance Peskin's book Equation(15.51) . While (4) and (5) haven't been added ?

in this case they subtracted to vanish each other ?

• How is this different than your previous two questions? also, eq 5 is wrong, change $-i$ to $+i$. Jan 8 '17 at 17:55
• @ AccidentalFourierTransform, the difference is that i didn't find an answer for the previous question, i only knew i should add the third term for (5) .. but now i'm still don't understand why not (4) added to (5) to (6) to have the final equation of motion ? even if -i instead of +i in (5) , still $2 g \bar{\psi} \gamma^\mu A_\mu^a \psi - 2 m \bar{\psi} \psi$ terms remain ..
– S.S.
Jan 8 '17 at 22:07
• @S.S. 1) please change your nickname, we don't encourage racism (joking, of course); 2) Your equation (6) was obtained through the least-action principle. Why should it be wrong? Please provide the mentioned equation from Peskin's book (we're too lazy to find it by ourselves). Jan 10 '17 at 14:10
• @Solenodon Paradoxus. #Please provide the mentioned equation from Peskin's book # : It's the same Equ (6) . I mean in any literature takes about Yang Mills theory, it's mentioned (6) is the equation of motion of the Lagrangian, and as example Peskin. So as i mentioned in my previous comment what about (4) and (5) ? I hope this is not also so naive, but i don't understand how they not added to the equation of motion as well ..
– S.S.
Jan 10 '17 at 15:05
• @S.S. these are equations of motion for fermionic matter. The latter (6) is the equation of motion for the gauge field. All three of them matter, of course. Simply (6) is more important in some applications, while (4) and (5) are more important for another. Jan 10 '17 at 15:15