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Electromagnetic field quantization

  1. Sep 11, 2013 #1
    1. Hey,
    So I have to show this proof: [tex]\int d^{4}x(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu})=\frac{1}{2}\int d^{4}xA^{\mu}(\square n_{\mu\nu}-\partial_{\mu}\partial_{\nu})A^{\nu}[/tex]



    2. Where
    [tex]F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}[/tex]




    3. ok, so I spent forever trying to type all of the above out, so you are going to have to have my solution as an image. I am not sure how right my solution is, and was wondering whether someone could have a look at it please? I'd be very grateful.
     

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    Last edited: Sep 11, 2013
  2. jcsd
  3. Sep 11, 2013 #2
    On the 8th line of the page, 7th line of the derivation, both factors of A appear to the left of the partials. Only one of them should be on the left, the other should be on the right of the partials. Also, there should be an overall minus sign. On the 9th line of the page, 8th line of the derivation, the factors of A are correctly placed, one on the left of the partials and one on the right.

    My question to you is: Why are you allowed to move one of the factors of A to the left but not the other and why should there be an overall minus sign?

    Also, on the 8th line of the derivation, you are very sloppy with the raising of indices and that needs to be fixed.
     
  4. Sep 11, 2013 #3
    Hey,

    I shall have a look in the morning with regards yours questions, however I'm not quite sure as to why there should be an overall minus sign? I took the minus sign from the -1/4 into the brackets and so lost the minus... I'm guessing I've gone wrong somewhere?
     
  5. Sep 11, 2013 #4

    vela

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    Explain what you did going from
    $$\int d^4x\ \frac{1}{4} \eta^{\mu\alpha}\eta^{\nu\beta} (
    -\partial_\mu A_\nu \partial_\alpha A_\beta+\partial_\mu A_\nu \partial_\beta A_\alpha-\partial_\nu A_\mu \partial_\alpha A_\beta+\partial_\nu A_\mu \partial_\beta A_\alpha)$$ to
    $$\int d^4x\ \frac{1}{4} \eta^{\mu\alpha}\eta^{\nu\beta} [2 (
    \partial_\mu A_\nu \partial_\beta A_\alpha-\partial_\mu A_\nu \partial_\alpha A_\beta)].$$ And what were you trying to do when you went from there to
    $$\frac{1}{2} \int d^4x \ \eta^{\mu\alpha}\eta^{\nu\beta} A_\nu(x) A_\mu(x) (
    \partial_\mu \partial_\beta - \partial_\mu \partial_\alpha)?$$
     
  6. Sep 12, 2013 #5

    SO with the first line you have given I thought because everything is just dummy indices, and terms are repeated twice, I could just write it like that. I guess I have done something wrong?

    With the second line you are writing about, I figured I could just take out terms common to both, but according to a previous replier, I can't write it like that...But I'm not sure as to why...
     
  7. Sep 12, 2013 #6
    So as I've said before I'm not quite sure as to why there should be an overall minus sign. I'm using Peskin and Schroeder as a guide and they don't have one in there either. Still trying to figure out the other thing you asked.

    Anyways I made some changes, to try and correct the raising of indices.. not sure if what I have done is right...
     

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    Last edited: Sep 12, 2013
  8. Sep 12, 2013 #7

    vela

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    Oh, okay. [strike]You have[/strike] I made a sign error in the first line I didn't notice earlier.

    Well, what you basically did there was akin to writing
    $$\frac{df}{dx}\frac{dg}{dy} = fg \frac{d}{dx}\frac{d}{dy}.$$ Hopefully, you can see that's totally wrong.

    The usual technique is to integrate by parts to move the derivative from one term onto the other. Integrating by parts is essentially using the product rule. Consider the product rule applied to differentiate fg':
    $$(fg')' = f'g' + fg'' \hspace{1em} \rightarrow \hspace{1em} f'g' = (fg')' - fg''.$$ If you integrate this, you'd have
    $$\int f'g' = \int (fg')' - \int fg'' = (fg')\big|_{-\infty}^\infty -\int fg'' = -\int fg''.$$ The first term, the integral of (fg')', is assumed to vanish, so the upshot is is that you can move the differentiation from one term onto the other at the cost of a minus sign.
     
    Last edited: Sep 12, 2013
  9. Sep 12, 2013 #8
    I'm rather lost, as I didn't realize I needed to do integration by parts....I'm assuming then that the method for acquiring the factor of 2 wasn't incorrect then?..

    Also I think I've still screwed up the raising and lowering of indices
     
  10. Sep 12, 2013 #9

    vela

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    The factor of 2 is fine. I confused myself with a sign error I made.

    I'd take this one step at a time. Try to understand how to turn ##(\partial_\mu A_\nu)(\partial_\alpha A_\beta)## into ##-A_\nu (\partial_\mu \partial_\alpha A_\beta)##. Once you have that down, then worry about the indices.
     
  11. Sep 12, 2013 #10
    So everything is fine up to the 5th line of the derivation? Sorry, I get confused easily, so trying to just make sure I'm ok up to a certain point...
     
  12. Sep 12, 2013 #11
    I don't know how to delete this message. vela has said it all.
     
    Last edited: Sep 12, 2013
  13. Sep 12, 2013 #12
    So I should keep the negative in front of the 1/4 for the moment then? I'm assuming I lose the negative sign later then...
     
  14. Sep 12, 2013 #13

    vela

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    It looks good right up to the line
    $$\frac{1}{2} \int d^4x \ \eta^{\mu\alpha}\eta^{\nu\beta} A_\nu(x) A_\mu(x) (
    \partial_\mu \partial_\beta - \partial_\mu \partial_\alpha).$$ That line doesn't make sense.
     
  15. Sep 12, 2013 #14
    OK< shall have a bash from there then...
     
  16. Sep 12, 2013 #15
    Hmm I think I get it now... So I can move one of the A's to the left of the brackets from both terms because of the integration by parts, but the second A term in both terms has to go on the right as it has not been taken out through integration by parts?
     
    Last edited: Sep 12, 2013
  17. Sep 12, 2013 #16

    vela

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    Right. In the original expression, the A's are both being acted on by operators, the ##\partial##'s, and the results are multiplied together. Integration by parts lets you move one of the operators from one factor onto the other, but in doing so, you pick up the minus sign. So in this particular case, you end up with ##A_\nu## multiplied by the result of applying ##\partial_\mu \partial_\alpha## to ##A_\beta##.
     
  18. Sep 12, 2013 #17
    Hey,

    So I now to sort out my indices... I think I might have it now? ImageUploadedByPhysics Forums1379014298.865437.jpg
     
  19. Sep 12, 2013 #18

    vela

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    You can't pull the ##A_\beta## out of the parentheses like you did. In the first term, it was ##A_\alpha##, not ##A_\beta##.
     
  20. Sep 13, 2013 #19
    So if I switch it to A[itex]_{\alpha}[/itex], and then relabel all the dummy indices at the end...should that be fine?
     
  21. Sep 14, 2013 #20

    vela

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    No, it sounds like you'd just be swapping ##A_\alpha## for ##A_\beta##.
     
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