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orentago
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Homework Statement
Show that the complex Klein-Gordon Lagrangian density:
[tex]L=N\left(\partial_\alpha\phi^{\dagger}(x)\partial^\alpha\phi(x)-\mu^2\phi^{\dagger}(x)\phi(x)\right)[/tex]
is invariant under charge conjugation:
[tex]\phi(x)\rightarrow C\phi(x)C^{-1}=\eta_c \phi^\dagger (x)[/tex]
Where [tex]C[/tex] is a unitary operator and [tex]\eta_c[/tex] is a phase factor.
Homework Equations
The Attempt at a Solution
The transformation can also be written as follows: [tex]\phi^\dagger (x) \rightarrow \eta_c^{-1} \phi(x)[/tex]
Hence performing the transformations on [tex]\phi(x)[/tex] and [tex]\phi^\dagger (x)[/tex] gives:
[tex]N\left(\partial_\alpha(\eta_c^{-1}\phi(x))\partial^\alpha(\eta_c\phi^\dagger (x))-\mu^2(\eta_c^{-1}\phi(x))(\eta_c \phi^\dagger(x))\right)=N\left(\partial_\alpha\phi(x)\partial^\alpha \phi^\dagger (x)-\mu^2\phi(x) \phi^\dagger(x)\right)=N\left(\partial^\alpha\phi(x)\partial_\alpha \phi^\dagger (x)-\mu^2\phi(x) \phi^\dagger(x)\right)[/tex]
Where the final step can be made fairly easily by raising and lowering indices. I'm a little unsure over my first assumption about how [tex]\phi^\dagger (x)[/tex] transforms, but otherwise I'm fairly confident in the rest of my steps. Is this solution valid?