Equations of motion for Lagrangian of scalar QED [closed]

Question

I really would appreciate your help with this exercise.

I have the Lagrangian for scalar electrodynamics given by: $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}(x)F^{\mu\nu}(x)+(D_\mu\varphi(x))^*(D^\mu\varphi(x))-V(\varphi^*(x)\varphi(x)) $$ where $F_{\mu\nu}(x)=\partial_\mu A_\nu(x)-\partial_\nu A_\mu(x)$ is the electromagnetic field strength tensor, $D_\mu=\partial_\mu+ieA_\mu$ ist the covariant derivative, $e$ is the electric charge and $V(\varphi^*\varphi)=m^2\varphi^*\varphi+\lambda(\varphi^*\varphi)^2$ is the potential of the scalar field.

I have to determine the equations of motion for both the complex scalar field $\varphi$ and the electromagnetic field $A_\mu$ by using the Euler-Lagrange equations.

Now I know, that because the scalar field is complex it has twice the degrees of freedom so I get two equations of motion (?). They should be given by: $$\frac{\partial \mathcal{L}}{\partial\varphi}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi)}=0$$ and $$\frac{\partial \mathcal{L}}{\partial\varphi^*}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi^*)}=0.$$

For the electromagnetic field $A_\mu$ it should just be: $$\frac{\partial \mathcal{L}}{\partial A_\mu}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\mu)}=0.$$

Now I'm stuck. Surely I'm just complicating things but I'm not sure how to compute those derivatives. Can somebody help me? I appreciate it!

Answer 1

I have to determine the equations of motion for both the complex scalar field $\varphi$ and the electromagnetic field $A_\mu$ by using the Euler-Lagrange equations.

Now I know, that because the scalar field is complex it has twice the degrees of freedom so I get two equations of motion (?). They should be given by: $$\frac{\partial \mathcal{L}}{\partial\varphi}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi)}=0$$

Yes.

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and $$\frac{\partial \mathcal{L}}{\partial\varphi^*}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi^*)}=0.$$

Yes.

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For the electromagnetic field $A_\mu$ it should just be: $$\frac{\partial \mathcal{L}}{\partial A_\mu}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\mu)}=0.$$

No, this one quoted above is wrong. (And you can see this by the fact that the indexing doesn't make sense... there is a summation convention, but three of the indices on the second term have the same symbol...)

You get a separate equation for each of the four $A_\nu$: $$\frac{\partial \mathcal{L}}{\partial A_\nu}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu)}=0.$$