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How do Maxwells equations result from the field tensor? 
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#1
Oct2812, 10:13 AM

P: 36

Hi,
I've been trying to solve problem 2.1 a in Peskin and schroeder, an introduction to QFT. The problem is to derive Maxwells equations for free space, which I have almost managed to do, using the Euler lagrange euqation And the definition of the field tensor as [tex] F_{μv} = d_μ A_v  d_v A_μ [/tex] So I have managed to get to; [tex] 0=d_μ F^{μv} [/tex] But I am unable to see how this shows Maxwells equations. Any points would be appreciated. Thanks. 


#2
Oct2812, 10:27 AM

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PF Gold
P: 16,461

In what variables are Maxwell's equations usually expressed?



#3
Oct2812, 11:18 AM

P: 36

My apologies for a less than comprehensive post at first;
Well, usually in the vector form of E and B or the 4 vector A, where; [tex] A= ( \phi, \vec{A} ) [/tex] where[tex] \vec{A} [/tex]is the magnetic vector [tex]\phi[/tex] is the scalar electric potential. [tex] E = ∇\phi  \frac{\partial\vec{A}}{\partial t} [/tex] And, [tex] B= ∇ X \vec{A} [/tex] It's clear to me that 2 of maxwells equations result directly from this definition; [tex] ∇.B = ∇. ( ∇ X \vec{A} ) = 0 [/tex] and [tex] ∇ X E = ∇ X ∇\phi  ∇ X \frac{\partial\vec{A}}{\partial t} [/tex][tex] ∇ X E = ∇(∇ X \phi)  \frac{\partial (∇ X\vec{A} )}{\partial t} [/tex][tex] ∇ X E = 0  \frac{\partial ( B )}{\partial t} [/tex][tex] ∇ X E =  \frac{\partial B}{\partial t} [/tex] Which leaves [tex] ∇.E = 0 [/tex] and [tex] ∇ X B = \frac{\partial E}{\partial t} [/tex] to be found from [tex] 0=\partial_μ F^{μv}[/tex] So I have tried putting in [tex] F_{μv} = \partial_μ A_v  \partial_v A_μ [/tex] To give; [tex] 0=\partial_μ ( \partial_μ A_v  \partial_v A_μ )[/tex] [tex] 0=\partial_μ \partial_μ A_v  \partial_μ \partial_v A_μ [/tex] so I then tried expanding this out, hoping that some terms would cancel and that I would recognize others and perhaps then they would be close to the E and B field formulation that I am more familier with. This yielded; [tex] \partial_μ \partial_μ A_v= \frac{\partial^2 \phi}{\partial t^2} + \frac{\partial^2 \vec(A)}{\partial t^2} + ∇^2 \phi + ∇^2 \vec{A} [/tex] And [tex] \partial_μ \partial_v A_μ = \partial_v \partial_μ A_μ = \frac{\partial^2 \phi}{\partial t^2}  \frac{\partial \vec(∇A)}{\partial t}  \frac{\partial ∇\phi}{\partial t} + ∇^2 \vec{A} [/tex] which when combined gives; [tex] 0= \frac{\partial^2 \phi}{\partial t^2} + \frac{\partial^2 \vec(A)}{\partial t^2} + ∇^2 \phi + ∇^2 \vec{A}  ( \frac{\partial^2 \phi}{\partial t^2}  \frac{\partial \vec(∇A)}{\partial t}  \frac{\partial ∇\phi}{\partial t} + ∇^2 \vec{A} ) [/tex] [tex] 0= \frac{\partial^2 \vec(A)}{\partial t^2} + ∇^2 \phi + \frac{\partial \vec(∇A)}{\partial t} + \frac{\partial ∇\phi}{\partial t} [/tex] which is where I am scratching my head..... EDIT:replaced d's with [tex]\partial[/tex] as per dextercioby's suggestion. 


#4
Oct2812, 02:47 PM

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How do Maxwells equations result from the field tensor?
I suggest one tiny bit of LaTex: [itex] \partial [/itex], i.e. \partial.



#5
Oct2812, 07:46 PM

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PF Gold
P: 16,461

Wouldn't it make more sense to work with E's and B's if you want equations giving you E's and B's?



#6
Oct2912, 09:24 AM

P: 647

at the end of post #3, like in the last 4 lines, you have vector things added to scalar things, which is no good. And on the left hand side of those lines there is a free index [itex]\nu[/itex] I think... Those are vectors. You can write it in a vector format, but the index is not summed over.



#7
Nov1412, 10:49 AM

P: 36

I see how I've gone wrong on the last few lines.
Thanks for your help vanadium 50 , jfy4 and dextercioby! 


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