(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given the Lagrangian density [tex] L(\phi^{\mu})=-\frac{1}{2}(\partial_{\mu}\phi^{\nu})(\partial^{\mu}\phi_{\nu}) + \frac{1}{2}(\partial_{\mu}\phi^{\mu})^2+\frac{m^2}{2}(\phi^{\mu}\phi_{\mu}) [/tex]

and gauge transformation [tex] \phi^{\mu}\rightarrow \phi^{\mu} + \partial^{\mu}\alpha [/tex]

(c) Introduce one extra real scalar field [tex] \sigma [/tex] and write some interacting Lagrangian [tex] L^{\prime}=L(\phi^{\mu})+L^2(\phi^{\mu},\sigma) [/tex] which is invariant under the gauge transformation and gives the original L for [tex] \sigma=0 [/tex].

(d) Can we solve (c) with sigma that has a canonical kinetic term [tex] -\frac{1}{2}(\partial_{\mu}\sigma)^2 [/tex]

2. Relevant equations

3. The attempt at a solution

The first parts of the question show that [tex] \partial_{\mu}\phi^{\mu}=0 [/tex] which simplifies the Lagrangian, and also that the initial Lagrangian is not invariant under the gauge transformation. I got those out.

But parts (c) and (d) seems like the kind of thing you either know or you don't, is there a way of working it out?

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# Homework Help: Quantum Field Theory-Gauge Transformations

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