Feynman propagator on the cylinder - position space representation

[DrFaustus]

Hi all!

Does anyone know the position space representation of the Feynman propagator on the cylinder? The momentum space representation is the same as in Minkowski 2D space, but the position space representation is different because the integrals over momenta are now sums. Or could someone point me to some specific literature where I could find it?

 

[Sunset]

Hi DrFaustus!

Did you have a look in Kleinert's book on Path Integrals? He did a lot in curvelinear coordinates, maybe you find something.

Best regards

 

[DrFaustus]

Hey Sunset, thanks a lot fo the hint but it doesn't appear to be of any use. I had a browse through the book and it seems to me that he's "only" dealing with quantum mechanical systems and not with fields. Which made me realize I wasn't too precise in my question...

I need the Feynman propagator for a scalar quantum field on the cylinder. Any other suggestions?

 

[Sunset]

Quantum mechanics has the same form as 1+0 dimensional QFT. I found the propagator in cylindrical coordinates in a paper of Grosche
http://www.iop.org/EJ/abstract/0305-4470/30/5/025"
(see also the attachment)

I think one should only add the \int d^3 x in the exponential.

The problem is that it's a quite difficult expression ;)

I'm curious, for what do you need it? Thought about the singularity at the pole r=0?

Martin

 

[peteratcam]

Searching for (screened) coulomb interactions with periodic boundary conditions might help (since 1/(q^2+m^2) is just screened coulomb. The Ewald summation literature might have it as an example.

The 1d version (on a circle) is tractable, I have the answer in some notes somewhere.

 

[DrFaustus]

Sunset -> Thanks for the help... but it again doesn't help :( I know QM is 1+ 0 D QFT, but the one spatial dimension makes a world of difference. The QM propagator in cylindrical coordinates is not the same as the field theoretical propagator on the cylinder. When I say "on the cylinder" I mean that the spacetime I'm living on is a cylinder, i.e. \mathbb{R}\times\mathbb{S}. So it's 2D. It seems to me that the expression in your attachement is the propagator for a problem in QM with axial symmetry in 3 spatial dimension (judging by what I'm guessing are spehrical harmonics in the last line.) Basically what I'm looking for is an explicit expression for

<br /> \int d \omega \sum _{p \in \mathbb{Z}} \frac{e^{i \omega t - i p x}}{- \omega^2 + p^2 + m^2 - i \epsilon}<br />

which is the inverse Fourier transform of the Feynamn propagator in momentum space. (M is a positive constant and \epsilon is a small positive infinitesimal.) In Minkowski space the sum would be another integral and one can find that expression in books. But with the sum, I was not able to find it anywhere... I need it just for "completness" as my supervisor wants to put in into the paper we're workin on.

peteratcam -> I'm not sure if a screened coulomb interaction is actually of any help. Do your notes include a sum/integral as above?

 

[Sunset]

Oh ok, thought you were talking about fields in cylindrical coordinates. Since you're talking about space-time in cylindrical coordinates, my atachement is of course of no use for you.

Best regards

 

[peteratcam]

peteratcam -> I'm not sure if a screened coulomb interaction is actually of any help. Do your notes include a sum/integral as above?

I don't know if screened coulomb is any help either!
The result I remember is from some notes which used to be on MIT OCW, but now I can only find them here:
http://www-math.mit.edu/~etingof/
http://www-math.mit.edu/~etingof/lect.ps
Page 51, Quantum Mechanics on the Circle.
The notation takes a while to get used to, but it essentially gives you the answer to the question:
<br /> \sum_p \frac{e^{ipx}}{p^2+m^2}<br />