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I have an exercise for Yang-Mills theory. I can't find answer anywhere.

Derive equations of motion for the Yang-Mills theory with the gauge group $SU(2)$ interacting with $SU(2)$ doublet of scalar fields.

I don't even know how to derive EOM for Lagrangian here. Any help? Or any source for appropriate handbook? (I've been using Maggiore, it doesn't have EOM for Yang-Mills here.)

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    $\begingroup$ I have two questions - 1. Do you know of the Euler-Lagrange equations and how they are derived? 2. Do you know the Lagrangian of Yang-Mills theory coupled to scalars/fermions, etc? $\endgroup$
    – Prahar
    May 7, 2014 at 13:39
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    $\begingroup$ 1. Yup, i know basis for classical field theory. 2. Nope. $\endgroup$ May 7, 2014 at 13:56

1 Answer 1

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The Lagrangian of Yang-Mills theory coupled to scalars/fermions, etc. takes the form $$ {\cal L}_{YM} = - \frac{1}{2} \text{Tr} F_{\mu\nu} F^{\mu\nu} + (D_\mu \phi) (D^\mu \phi)^* + i {\bar \psi} \gamma^\mu D_\mu \psi + \cdots $$ where the $\cdots$ represents other interactions terms that might be present. Let me explain the notation in the above expression.

  1. $\phi_i$ and $\psi_i$ are multiplets in some representation $R$ of the gauge group $G$. Here, $i = 1, \cdots, \dim R$

  2. The generators in representation $R$ are denoted as $T^a$. These are normalized to satisfy $$ [T^a, T^b] = i f^{ab}{}_c T^c,~~~~ \text{Tr} ( T^a T^b )= \frac{1}{2}\delta^{ab} $$ In other words, the $T_{ij}^a$'s are just some set of matrices satisfying the above properties.

  3. The covariant derivatives acting on the fields is \begin{align} (D_\mu \phi)_i &= \partial_\mu \phi_i - i g T^a_{ij} A_\mu^a \phi_j \\ (D_\mu \psi)_i &= \partial_\mu \psi_i - i g T^a_{ij} A_\mu^a \psi_j \\ \end{align} where $(A_\mu)_{ij} = A_\mu^a T_{ij}^a$.

  4. $$ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + g [A_\mu, A_\nu ] \\ $$ Explicitly $$ F_{\mu\nu}^a = \partial_\mu A^a_\nu - \partial_\nu A_\mu^a + i g f_{bc}{}^a A^b_\mu A_\nu^c $$

This completely specifies the Lagrangian of Yang Mills theory. You can now use the variational principle to determine the equations of motion.

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  • $\begingroup$ Well ok, but i was expecting explicitely shown euquations of motion :) $\endgroup$ May 7, 2014 at 19:00
  • $\begingroup$ @CheshireCat I doubt anybody will actually show you explicitly how to do this; it is too boring and tedious. But you should take the variation of the field as per usual, both the scalar/fermion field and the gauge fields, to get the E.O.M. $\endgroup$
    – Hunter
    May 7, 2014 at 19:03
  • $\begingroup$ @CheshireCat - If I gave you the equations of motion, then I would have solved the problem for you. $\endgroup$
    – Prahar
    May 7, 2014 at 19:30
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    $\begingroup$ That's why i wrote here, i want to see a solution for a problem :) I found something in S. Pokorski, "Gauge field theories", in 1.3 chapter. $\endgroup$ May 7, 2014 at 19:51
  • $\begingroup$ We don't give out solutions on this forum. See the description of the 'homework' tag for details. $\endgroup$
    – Prahar
    May 7, 2014 at 19:56

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