A contour integral frequenctly encountered

[hneder]

Homework Statement

Need some help here on a frequently encountered integral in Green's function formalism.

Homework Equations

I have an integral/summation as a product of a retarded and advanced Green's functions, looks simply like
[itex]\sum_{p^{\prime}}\frac{1}{p^{2}-{p^{\prime}}^{2}-i\epsilon}\frac{1}{p^{2}-{p^{\prime}}^{2}+i\epsilon}[/itex]
where I have omitted the mass [itex]m[/itex] to make the notation simple. [itex]\epsilon[/itex] is a positive infinitesimal.

The Attempt at a Solution

I can convert the summation to an integral over momentum [itex]p[/itex], this is standard. Then I follow the complex analysis and identify 4 poles and further apply the residue theorem, what I get is an expression proportional to [itex]\frac{1}{\epsilon}[/itex]. This means it is divergent.

Can anyone point out to me what have I missed in this calculation? What mistakes did I make? Many many thanks

 

[nrqed]

Homework Statement

Need some help here on a frequently encountered integral in Green's function formalism.

Homework Equations

I have an integral/summation as a product of a retarded and advanced Green's functions, looks simply like
[itex]\sum_{p^{\prime}}\frac{1}{p^{2}-{p^{\prime}}^{2}-i\epsilon}\frac{1}{p^{2}-{p^{\prime}}^{2}+i\epsilon}[/itex]
where I have omitted the mass [itex]m[/itex] to make the notation simple. [itex]\epsilon[/itex] is a positive infinitesimal.

The Attempt at a Solution

I can convert the summation to an integral over momentum [itex]p[/itex], this is standard. Then I follow the complex analysis and identify 4 poles and further apply the residue theorem, what I get is an expression proportional to [itex]\frac{1}{\epsilon}[/itex]. This means it is divergent.

Can anyone point out to me what have I missed in this calculation? What mistakes did I make? Many many thanks

Hi.

Can you provide more details on your calculation? I am assuming you are doing the four-dimensional integral over [itex] d^4 p [/itex]. Then did you do the [itex] p_0 [/itex] integral using contour integration and got the [itex] 1/ \epsilon [/itex] term before doing the remaining 3-dimensional integral?

 

[hneder]

Thanks for the reply. This is simply a product of two Green's functions, one retarded and one advanced. No, it is a regular 3D integral (summation) and momentum [itex]p[/itex] and [itex]p^{\prime}[/itex] here are both 3D vectors. Somehow I know there is a contour integral to be done. But it is not clear to me how should I do it.

What I did was the following. If I perform the [itex]p[/itex] integration using residue theorem, as I mentioned in the post, I end up with four poles [itex]p\pm i\epsilon[/itex], [itex]-p\pm i\epsilon[/itex] and the final result [itex]\sim \frac{1}{\epsilon}[/itex]. The same result comes out if I convert the [itex]p[/itex] integration into one with energy using [itex]E=p^{2}/{2m}[/itex].

 

[vanhees71]

It doesn't make physical sense if the vectors [itex]p[/itex] and [itex]p'[/itex] are Euclidean.

Anyway, as a purely mathematical exercise, it's an interesting problem, because it demonstrates the problem of Pinch singularities that sometimes occur in real-time many-body QFT if one is not careful enough, because, as you seem to have realized, the expressions do not make sense in the weak limit [itex]\epsilon \rightarrow 0^+[/itex]. Fortunately, one can prove rigorously that such pinch singularities are absent when the correct Schwinger-Keldysh Contour techniques are applied. The details are a bit tedious, and one has to use some care to correctly make sense of the distributions involved in the calculation. See my lecture notes on relativistic many-body theory:

http://fias.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf

(the long Sect. 2.2).

 

[hneder]

Thanks, vanhees71, for your explanation. Your notes in the link are a bit too advanced for a junior student. Hopefully I will get there later. Is there any chance to explain in a simplified language? To be more specific, my problem arises from calculating, for example, the inner product of two scattered wave functions built up with Lippmann-Schwinger equation in momentum space (see, for example, in the Sakurai book), where you immediately encounter the product of two propagators described above.