Electromagnetic Lagrangian, EoM, Polarisation States

1. Dec 8, 2017

binbagsss

1. The problem statement, all variables and given/known data

Attached:

2. Relevant equations

Euler-Lagrange equations to find the EoM

3. The attempt at a solution

Solution attached:

I follow, up to where the sum over $\mu$ reduces to sum over $\mu=i$ only, why are there no $\mu=0$ terms? I don't understand at all.

Many thanks

2. Dec 8, 2017

TSny

According to equation (9), what can you say about $F_{00}$? What does this tell you about $F^{00}$?

3. Dec 9, 2017

binbagsss

oh for a antisymmetric tensor all diagonal entries must be zero? $F_{ii}=0$ as is $F_{00}$?

4. Dec 9, 2017

TSny

Yes. The expression for $F_{\mu \nu}$ in (9) shows that $F$ is antisymmetric. In particular, if you let $\mu = 0$ and $\nu = 0$ in the expression for $F_{\mu \nu}$ in (9) then you see that $F_{00} = 0$.