How Do I Derive the Zeta Function Using Zeta Function Regularization?

[robousy]

...on the off chance anyone knows this, I'm trying to get from:

$$V=\frac{1}{2A}Tr Log(\frac{-\Box}{\mu^2})$$

to

$$V=\frac{(-1)^{\eta-1}}{4\pi^\eta\eta!}\frac{\pi}{L}^{D-1}\zeta'(1-D)$$

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??

:)

 

[Avodyne]

What are $D$, $\eta$, and $L$? On what space is the box operator defined?

 

[robousy]

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.

 

[blechman]

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?

 

[robousy]

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.

 

[blechman]

Incidently I'm reading Radion Effective Potential in Brane World by Garriga, Pujolas and Tanaka.

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.