- #1

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I'd be grateful for a bit of help with the following problems:

Consider the Lagrangian:

[tex] \displaystyle \mathcal{L} = (\partial_{\mu} \phi) (\partial^{\mu} \phi^{\dagger}) - m^2 \phi^{\dagger} \phi [/tex]

where [tex] \phi = \phi(x^{\mu}) [/tex]

Now making a U(1) gauge transformation:

[tex] \displaystyle \phi \longmapsto e^{i \Lambda(x^{\mu})} \phi [/tex]

does the Lagrangian become:

[tex] \displaystyle \mathcal{L} = (\partial_{\mu} \phi) \cdot (\partial^{\mu} \phi^{\dagger}) - m^2 \phi^{\dagger} \phi + \phi \phi^{\dagger} (\partial_{\mu} \Lambda) \cdot (\partial^{\mu} \Lambda) + i \partial_{\mu} \Lambda \cdot (\phi \partial^{\mu} \phi^{\dagger} - \phi^{\dagger} \partial^{\mu} \phi) [/tex] ?

I realize you can add in another field to counteract the gauge transformation so the Lagrangian becomes gauge invariant, but how exactly would you determine the field to "add in" by inspection?

Thanks for any replies