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Basic question on scalar filed theory that is getting on my nerves. Say that we have the langrangian density of the free scalar (not hermitian i.e. "complex") field

[tex]L=-1/2 (\partial_{\mu} \phi \partial^{\mu} \phi^* + m^2 \phi \phi^*)[/tex]

Thus the equations of motion are

[tex](\partial_{\mu} \partial^{\mu} - m ) \phi=0 [/tex] the kg equation plus the complex conjugate equation. fine. Now I have been taught to do this calculation by thinking of the scalar field as really a complex function i.e.

[tex] \phi=\phi_1 + i \phi_2[/tex] with phi1 phi2 reals.

This is giving the right results e.g. [tex]\frac{\partial}{\partial \phi^*} \phi \phi^*=2 \phi[/tex] but also in the same way I get

[tex]\frac{\partial}{\partial \phi} {\phi} =2 [/tex]

which is quite crazy. So how should one actually do the differentiation in Lagrange's equations? The functional derivative doesn't really help me here it is the product of phi and it's complex that is giving the problem.

[tex]L=-1/2 (\partial_{\mu} \phi \partial^{\mu} \phi^* + m^2 \phi \phi^*)[/tex]

Thus the equations of motion are

[tex](\partial_{\mu} \partial^{\mu} - m ) \phi=0 [/tex] the kg equation plus the complex conjugate equation. fine. Now I have been taught to do this calculation by thinking of the scalar field as really a complex function i.e.

[tex] \phi=\phi_1 + i \phi_2[/tex] with phi1 phi2 reals.

This is giving the right results e.g. [tex]\frac{\partial}{\partial \phi^*} \phi \phi^*=2 \phi[/tex] but also in the same way I get

[tex]\frac{\partial}{\partial \phi} {\phi} =2 [/tex]

which is quite crazy. So how should one actually do the differentiation in Lagrange's equations? The functional derivative doesn't really help me here it is the product of phi and it's complex that is giving the problem.

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