EM Field Theory (Action Symmetries)

[Raz91]

Homework Statement

I uploaded a picture with the question

Homework Equations

my problem is :

How should I find all the symmetries of the action ?
Is there an easy way to recognize those symmetries or should I try all the symmetries I know and see if the action doesn't change?

Even some tricks to identify symmetries like : lorentz , translation, rotation will help me :)

Thank You!

The Attempt at a Solution

 

[Nate810]

The easiest way to identify the symmetries of an action is to consider the types of symmetries that are possible. For example, if the action is invariant under translations, then it has translational symmetry. Similarly, if the action is invariant under rotations, then it has rotational symmetry. If the action is invariant under Lorentz transformations, then it has Lorentz symmetry. In addition, there may be other types of symmetries that are not as obvious. For example, if the action is invariant under scale transformations, then it has scale symmetry. If the action is invariant under reflections, then it has reflection symmetry. In general, any transformation that leaves the action invariant can be considered a symmetry. Once you have identified the possible types of symmetries, you can then try to determine if the action is actually invariant under those transformations. This can be done by performing the transformation on the action and checking to see if the resulting action is the same as the original action. If it is, then the action is said to be invariant under that transformation and therefore has that type of symmetry.