- #1

Frostman

- 115

- 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_\mu-iqA_\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
- Euler-Lagrange 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?