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Zeta function regularization

  1. Oct 24, 2007 #1
    ...on the off chance anyone knows this, I'm trying to get from:

    [tex]V=\frac{1}{2A}Tr Log(\frac{-\Box}{\mu^2})[/tex]

    to

    [tex]V=\frac{(-1)^{\eta-1}}{4\pi^\eta\eta!}\frac{\pi}{L}^{D-1}\zeta'(1-D)[/tex]

    I know this is a shot in the dark, but in case anyone has experience.

    The paper I'm reading explains 'it is easy to show that' to get to the seconds equation !! I hate that. The paper also has a reference...the reference is Birrel and Davies, great I thought, I have that book, the reference is for p340, which takes me to the Index !!! lol.

    Anyway, I guess the key is figuring how the trace of the log of the box operator gives me the derivative of the zeta function.

    Anyone??

    :)
     
    Last edited: Oct 24, 2007
  2. jcsd
  3. Oct 24, 2007 #2

    Avodyne

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    What are [itex]D[/itex], [itex]\eta[/itex], and [itex]L[/itex]? On what space is the box operator defined?
     
  4. Oct 24, 2007 #3
    D is the dimension of spacetime (5), eta is (D-1)/2 and the box operator is the flat space 5D operator. This is a paper where the Casimir energy is calculated in the bulk of the RS model.
     
  5. Oct 25, 2007 #4

    blechman

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    Zeta-function evaluation of determinants is described quite well in Pierre Ramond's text, "Field Theory: A Modern Primer", Chapter 3. The derivation is long, but I think you'll find what you need there.

    Hope that helps!

    Out of curiosity, what paper are you trying to read?
     
  6. Oct 31, 2007 #5
    Hey, Sorry for delayed repy blechman.
    Thanks for the suggestion of ramond, I managed to borrow a copy from my supervisor.

    Incidently I'm reading Radion Effective Potential in Brane World by Garriga, Pujolas and Tanaka.
     
  7. Nov 1, 2007 #6

    blechman

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    You can check out TASI-2002 articles by C. Csaki; also M. Quiros has a bunch of good reviews out there on effective potenitals of higher dim fields. R. Sundrum has a TASI-2004 review that's pretty nice too.
     
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