Branch-cut singularity and the multiparticle contribution to the full propagator

[gremezd]

I have been reading Chapter 7 of Peskin & Schroeder about full propagator, the Kallen Lehman spectral representation, and got stuck at the branch cut singularities and at the complex logarithm of negative numbers. I have posted in the Analysis forum (but have not received any answer) the following question:

Does anyone know what a branch-cut singularity is? I have been trying to understand its importance in physics, but I got lost. I would guess that a singularity in physical context should mean that the value of a function should become very big near that singularity. But if we take complex logarithm, we can become big only in two cases, when the argument is either 0 or infinity.
However, people choose the negative part of a real line in a complex plane as a branch cut for a complex logarithm, and say that this branch cut is a weak singulartiy compared to a simple pole. What do they mean by that?

Can anyone comment something about the "weakness of branch-cut singularities" in QFT or overall in physics.

 

[strangerep]

Does anyone know what a branch-cut singularity is?

From the sound of your question, I'm guessing you haven't looked at the
Wiki page on this subject? I.e.,

http://en.wikipedia.org/wiki/Branch_cut

I have been trying to understand its importance in physics, but I got lost. I would guess that a singularity in physical context should mean that the value of a function should become very big near that singularity.

More generally, "singularity" can mean that the function becomes ill-defined at that point.
In the case of a branch cut, the function ceases to be single-valued there, which is
not quite as nasty as diverging to infinity at a pole singularity.

 

[Meir Achuz]

A branch cut is a curve on the complex plane, across which the function is discontinuous.

 

[gremezd]

Yes, I have read wikepedia both on a branch point and complex logarithm.
But how do we interpret singulartities: do they constitute the essence of physical quantities or, as you say, they make these physical quantities ill-defined? and have to be got ridden of?

I found it interesting that the logarithm of negative numbers can be thought of as integrating over a continuum of poles. But it's quite difficult for me to put in one place the images of these different but closely related concepts: branch-cut singularity, continuum of poles, complex logarithm of negative numbers, discontinuity of logarithm at the branch cut; and the meanging of this in a physical context..

 

[alxm]

This is why you should learn the math before you try to learn the physics that uses it.
I'd suggest a book/course on complex analysis.

There's no physical significance of any mathematical thing in itself. Its significance is whatever you decided to give it when you created a mathematical model.

 

[alalaika]

well well, maybe I can help you out.
branch-cut singularities are continuous singularities, which means, you have endless singularities from one point on which is the branch-CUT. Now two particles start to exist from p^2=(2m)^2 on. If they have additional energy (see equation for relativistic particles: p^2=E^2+m^2) their momentum p increases, and as the energy can increase continuously you have continuous possible values for p. And with that you have continuous values for M forming poles and these are the possible singularities and obviously this will be a continuum. You can actually look at figure 7.2 in Peskin and it'll become all clear...