# Equations of motion for the Yang-Mills $SU(2)$ theory

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.)

• 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? – Prahar May 7 '14 at 13:39
• 1. Yup, i know basis for classical field theory. 2. Nope. – Cheshire Cat May 7 '14 at 13:56

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.

• Well ok, but i was expecting explicitely shown euquations of motion :) – Cheshire Cat May 7 '14 at 19:00
• @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. – Hunter May 7 '14 at 19:03
• @CheshireCat - If I gave you the equations of motion, then I would have solved the problem for you. – Prahar May 7 '14 at 19:30
• 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. – Cheshire Cat May 7 '14 at 19:51
• We don't give out solutions on this forum. See the description of the 'homework' tag for details. – Prahar May 7 '14 at 19:56