Higher Dimensional Green Function

[robousy]

Hey folks!

I'm starting with the Lagrangian of a massive scalar field and have found an expression for the expectation value of the energy-momentum tensor.

$$<T_{\mu \nu}>=(\partial_\mu \partial_\nu-\frac{1}{2}(g_{\mu \nu}(\partial_\mu \partial_\nu+m^2))G(x-x')$$


let say I have some Green Function G(x-x') and then I compactify the dimension into a circle or radius R, then can someone explain why we write the GF as:

$$G(x-x')=\sum_{n=1}^\infty G_\infty(x-x'+2\pi R n \hat{y})$$

And explain the phrase: the Casimir energy can be easily obtained by summing over the infinite volume Green Function over all the images.

What are the images here?

Any help appreciated!

 

[BenTheMan]

Hmm... I once knew an R. Obousy.

mu and nu in the original expression---do they run over five directions or four?

 

[BenTheMan]

Ok dude. I'm still thinking. Isn't there a zero mode, i.e. n = 0? Or does the sum really run from 1 to infinity?

 

[BenTheMan]

That's what's been bugging me: your stress tensor is wrong. Do you mean

$$<T_{\mu \nu}>=(\partial_\mu \partial_\nu-\frac{1}{2}(g_{\mu \alpha}g_{\nu \beta}\partial^{\alpha} \partial^\beta+m^2))G(x-x')$$

?

Either way, I'm pretty sure that the $$2 \pi R n y$$ means you're summing over modes, like Kaluza-Klein states, or something. Think of it this way, you have one dimension which is a circle, and the fields living on that circle must be periodic---the wave function in the fifth dimension has to be periodic, so it has to be an integer multiple of the circumference of the dimension.

As far as images goes, I think that this is image in the mathematical sense---it sounds like a mathematician wrote the article you're reading, and is just using fancy words. Because the Green's function is periodic in the fifth direction, it has an infinite tower of images---i.e. a whole tower of functions which all give the same value at y = 0.

Again, I'm not too sure about the context in which you have read this, so don't take me too seriously.

 

[robousy]

Hey Ben The Man!

Well spotted on the metric indices. You are right.

I've been thinking a little more about this actually. The images are in fact the virtual photons (I'm calculating the casimir energy...who of thought?) and the green function will ultimately relate to the vacuum energy. I'm reading an Arkani-Hamed paper, http://arxiv.org/PS_cache/hep-th/pdf/0703/0703067v1.pdf (appendix A) where he derives the vacuum energy for a massive scalar in d dims with one compact dimension.

Actually, I've been thinking that maybe the process of compactifying opens up new paths for the virtual photon to get from x to x'. e.g picture the computer sceen you are looking at right now, to get from the middle to the bottom of the screen you have to move DOWN, but if you compactified the screen you could move UP the screen instead, to get to the lower portion.

I suppose its all in the interpretatation of the what the GF is in this case.

 

[BenTheMan]

The images are in fact the virtual photons

I think I agree. This is like saying that the field is quantized in the direction between the parallel plates. In the fifth dimension, the field is a sine wave running around a circle, and you sum over all possible field configurations.

I've been thinking that maybe the process of compactifying opens up new paths for the virtual photon to get from x to x'.

I've been working with a similar problem, and I don't know if this is the case. When you do a calculation, for example, you have to do it in four dimensions if you're at energies below the compactification radius. This is why we can do QFT calculations of electron positron scattering as opposed to doing a string theory calculation---the dynamics of the compact dimension are essentially hidden from low energy physics.

So below your compact dimension radius, the propogator is exactly the same as the four dimensional scalar propogator.

 

[robousy]

Cool, thanks for your insights.

Are you working on anything interesting at the moment.

I'm looking at the casimir energy and force in the RS models, seeing how the bulk gravition C.E effects our brane. I was hoping to be the first, but there are about five papers out that do just that, so I need to find a new spin.

 

[BenTheMan]

Well, right now I am trying to figure out how gauge coupling evolution works in a specific class of string models that give good low energy phenomenology. Hopefully this project will be finished soon, because I am getting pretty tired of it :)