- #1
jameson2
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Homework Statement
Given the Lagrangian density 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})
and gauge transformation \phi^{\mu}\rightarrow \phi^{\mu} + \partial^{\mu}\alpha
(c) Introduce one extra real scalar field \sigma and write some interacting Lagrangian L^{\prime}=L(\phi^{\mu})+L^2(\phi^{\mu},\sigma) which is invariant under the gauge transformation and gives the original L for \sigma=0.
(d) Can we solve (c) with sigma that has a canonical kinetic term -\frac{1}{2}(\partial_{\mu}\sigma)^2
Homework Equations
The Attempt at a Solution
The first parts of the question show that \partial_{\mu}\phi^{\mu}=0 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?