Electromagnetic Lagrangian, EoM, Polarisation States

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Homework Help Overview

The discussion revolves around the electromagnetic Lagrangian and the equations of motion derived from it, specifically focusing on the properties of the electromagnetic field tensor and its components.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand why certain terms in the sum over indices do not appear in their calculations, specifically questioning the absence of terms for ##\mu=0##.
  • Participants explore the implications of the antisymmetry of the tensor, particularly regarding the diagonal components and their values.

Discussion Status

Participants are actively questioning the properties of the electromagnetic field tensor and discussing its implications. Some guidance has been provided regarding the antisymmetry of the tensor, but multiple interpretations of the implications are still being explored.

Contextual Notes

There is an emphasis on understanding the properties of the electromagnetic field tensor as defined in the context of the Euler-Lagrange equations, with specific reference to equation (9) and its implications for the components of the tensor.

binbagsss
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Homework Statement



Attached:

queey.png

Homework Equations



Euler-Lagrange equations to find the EoM

The Attempt at a Solution


[/B]
Solution attached:

solly.png


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
 

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

oh for a antisymmetric tensor all diagonal entries must be zero? ##F_{ii}=0 ## as is ##F_{00} ##?
 
binbagsss said:
oh for a antisymmetric tensor all diagonal entries must be zero? ##F_{ii}=0 ## as is ##F_{00} ##?
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##.
 

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