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

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Discussion Overview

The discussion revolves around deriving the zeta function using zeta function regularization, specifically transitioning from a trace logarithm expression involving the box operator to a zeta function derivative expression. The context includes theoretical aspects related to quantum field theory and the calculation of Casimir energy in higher-dimensional models.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant seeks to derive a specific expression involving the zeta function from a trace logarithm of the box operator, indicating a lack of clarity in the paper they are reading.
  • Another participant asks for clarification on the variables D, η, and L, as well as the space on which the box operator is defined.
  • A participant provides definitions for D, η, and the box operator, specifying that D is the dimension of spacetime, η is related to D, and the box operator is defined in flat 5D space.
  • One participant suggests a reference text, Pierre Ramond's "Field Theory: A Modern Primer," for zeta-function evaluation of determinants, noting that the derivation is lengthy.
  • A later reply mentions the specific paper being read by the original poster, which is related to radion effective potential in brane world scenarios, and suggests additional resources for further reading.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the derivation process, and multiple viewpoints and references are presented without resolving the initial query.

Contextual Notes

The discussion includes references to specific texts and papers, indicating a reliance on external resources for understanding the derivation process, but does not resolve the mathematical steps or assumptions involved in the derivation.

Who May Find This Useful

Readers interested in quantum field theory, zeta function regularization, and higher-dimensional models may find the discussion and references beneficial.

robousy
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...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:
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What are [itex]D[/itex], [itex]\eta[/itex], and [itex]L[/itex]? On what space is the box operator defined?
 
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.
 
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?
 
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.
 
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.
 

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