The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$ By making the minimal substitution $$\partial^\mu\mapsto\hat{D}^\mu=\partial^\mu+iq\hat{A}^\mu$$ we could introduce electromagnetic field into the equation and obtain $$\hat{\mathcal{L}}=\hat{\mathcal{L}}_{\text{KG}}+\hat{\mathcal{L}}_{\text{INT}}$$ The expected results should be $$\hat{\mathcal{L}}_{\text{INT}}=-iq\left(\hat{\phi}^\dagger(\partial^\mu\hat\phi)-(\partial^\mu\hat\phi^\dagger)\hat{\phi}\right)\hat{A}^\mu+q^2\hat{A}^\mu\hat{A}_\mu\hat{\phi}^\dagger\hat{\phi}\tag{1}$$ However, I did the subsitution myself and kept getting $$\hat{\mathcal{L}}_{\text{INT}}=iq\left(\hat{\phi}^\dagger(\partial^\mu\hat\phi)+(\partial^\mu\hat\phi^\dagger)\hat{\phi}\right)\hat{A}^\mu+q^2\hat{A}^\mu\hat{A}_\mu\hat{\phi}^\dagger\hat{\phi}$$ But I could not understand how i got it wrong. Could someone please explain how to get the correct expression equation (1)?
1 Answers
The error that you've made is that the factor $(\partial_{\mu} \phi^{\dagger})$ in the Klein-Gordon Lagrangian should be written as $(\partial_{\mu} \phi)^{\dagger}$. This makes a difference, since the hermitian conjugate is applied to the gradient of the field, $\partial_{\mu}\phi$, not $\phi$. Ordinarily, you could get away with your notation, since $(\partial_{\mu}\phi)^{\dagger}=\partial_{\mu}\phi^{\dagger}$. However, since $(D_{\mu}\phi)^{\dagger}=\partial_{\mu}\phi^{\dagger}-iqA_{\mu}\phi^{\dagger} \ne \partial_{\mu}\phi^{\dagger} +iqA_{\mu}\phi^{\dagger} = D_{\mu}\phi^{\dagger}$, you'll get into trouble. In other words, when you make the substitution $\partial_{\mu} \mapsto D_{\mu}$ apply the hermitian conjugate to $D_{\mu}\phi$, not $\phi$.
I hope this clears things up!
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