 #1
Frostman
 114
 17
 Homework Statement:

Consider the following Lagrangian density for the electromagnetic field ##A_\mu## coupled to a scalar field ##\phi## (complex)
$$L=\frac14F_{\mu\nu}F^{\mu\nu}+(D_\mu\phi)^*D^\mu\phi$$
Where ##D_\mu = \partial_\muiqA_\mu## and ## F_{\mu\nu}=\partial_\mu A_\nu\partial_\nu A_\mu##
1. Write the equations of motion for the two fields.
2. Applying the Noether theorem following the invariance for phase transformation of the field ##\phi##, write the conserved current.
3. Verify explicitly that the current written in this way is preserved.
 Relevant Equations:

EulerLagrange equations
Noether theorem
It is the first time that I am faced with a complex field, I would not want to be wrong about how to solve this type of problem.
Usually to solve the equations of motion I apply the Euler Lagrange equations.
$$\partial_\mu\frac{\partial L}{\partial \phi/_\mu}\frac{\partial L}{\partial \phi}=0$$
But since ##\phi## and ##\phi^*## are not independent of each other I will have to follow another path and the only one that comes to mind is the principle of least action. Should I use this approach?
What I get next will be two equations of motion (##\phi## and ##\phi^*##) plus that of the electromagnetic field (##A_\mu##): so I will have 3 EOM not 2. Or the EOM for ##\phi## and ##\phi^*## can be consider as one?
Usually to solve the equations of motion I apply the Euler Lagrange equations.
$$\partial_\mu\frac{\partial L}{\partial \phi/_\mu}\frac{\partial L}{\partial \phi}=0$$
But since ##\phi## and ##\phi^*## are not independent of each other I will have to follow another path and the only one that comes to mind is the principle of least action. Should I use this approach?
What I get next will be two equations of motion (##\phi## and ##\phi^*##) plus that of the electromagnetic field (##A_\mu##): so I will have 3 EOM not 2. Or the EOM for ##\phi## and ##\phi^*## can be consider as one?