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

In summary, the conversation is about a mathematical equation that the speaker is trying to understand, specifically how to get from one equation to another using a reference from a paper. They ask for any insights or explanations from anyone who may have experience with this topic. The conversation also mentions some resources to check out for further information.
  • #1
robousy
334
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??

:)
 
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  • #2
What are [itex]D[/itex], [itex]\eta[/itex], and [itex]L[/itex]? On what space is the box operator defined?
 
  • #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.
 
  • #4
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?
 
  • #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.
 
  • #6
robousy said:
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.
 

1. What is Zeta function regularization?

Zeta function regularization is a mathematical technique used to assign a finite value to infinite series or integrals that would otherwise be undefined. It is based on the Riemann zeta function, which is a mathematical function that is defined for all complex numbers except 1.

2. How does Zeta function regularization work?

Zeta function regularization works by assigning values to series or integrals using the analytical continuation of the Riemann zeta function. This involves manipulating the original equation using complex analysis techniques to obtain a finite result.

3. What are the applications of Zeta function regularization?

Zeta function regularization has various applications in physics, particularly in quantum field theory, where it is used to renormalize divergent expressions and obtain finite predictions. It is also used in number theory, harmonic analysis, and other areas of mathematics.

4. What are the advantages of using Zeta function regularization?

The main advantage of using Zeta function regularization is that it provides a systematic and rigorous method for assigning finite values to otherwise divergent expressions. It also has the advantage of being applicable to a wide range of mathematical problems and has been proven to be mathematically consistent.

5. Are there any limitations to Zeta function regularization?

One limitation of Zeta function regularization is that it can only be applied to certain types of series or integrals, specifically those that can be expressed in terms of the Riemann zeta function. It is also important to note that the results obtained through this technique may not always agree with other methods of regularization and should be used with caution.

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