Variation of simple Lagrangian

[bndnchrs]

Hey, I'm doing some examples in QFT and I don't want to go too far with this one:

Doing gauge symmetries, we first introduce the Unitary spacetime-dependent gauge transformation that gives us a gauge potential. With the new gauge added Lagrangian, I want to take its variation to confirm the equations of motion. Here's my question.

When we take $$\frac{d}{d \phi} -m^2\phi^*\phi$$ for example...

how do we handle the complex conjugate through our variation?

 

[CompuChip]

Usually you take it as an independent field, so you take the variation of the lagrangian with respect to phi (giving one equation) and with respect to phi* (giving another equation).

If this construction troubles you, you can decompose phi into two real fields,
$$\phi = \phi_1 + i \phi_2$$
and obtain the two equations by taking the variation with respect to phi_i (i = 1, 2).