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In relativistic scattering theory (e.g. in quantum electrodynamics) the existence of the $S$-matrix as well as of Moller operators is postulated as far as I understand (although at some stage it has to be modified due to existence of infrared divergences). On the other hand if one considers the non-relativistic scattering problem of several particles with Coulomb potential, it seems to be very different from the usual scattering theory with rapidly decaying potential and much of it does not work (again, this is my impression, and I am not an expert).

I am wondering why one should expect existence of $S$-matrix in quantum electrodynamics with properties similar to $S$-matrix for non-relativistic scattering theory with rapidly decaying potential (rather than Coulomb potential). Do I miss something on the non-relativistic Coulomb scattering?

I guess that this question might not be appropriate for this site as QFT is not yet a mathematical theory. But this question is a direct continuation of this one on this site where I did get some helpful comments. Moreover I imagine that an answer might be known to some mathematical physicists.

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  • $\begingroup$ From a physical standpoint, higher order corrections in a perturbative solution in QED grants that the Coulomb potential gets screened. This implies that what normally is done to assume a mass term in the Coulomb potential and taking it to go to zero at the end of computation, at the leading order (Born approximation), is justified. $\endgroup$
    – Jon
    Commented Feb 3, 2020 at 15:38
  • $\begingroup$ @Jon: Thanks, although your comment is quite concise for me. I will try to guess what you said. If one computes scattering of two charged particles in the tree approximation (thus no screening) in non-relativistic limit then one recovers the Coulomb low like it would be the Born approximation to non-relativistic scattering (the next order of approximation implies Ueling effect, but let's ignore this case). While one gets this compatibility, I do not understand why it makes sense: does the non-relativistic $S$-matrix makes sense for the Coulomb potential? Does Born approximation makes sense? $\endgroup$
    – asv
    Commented Feb 3, 2020 at 16:01
  • $\begingroup$ also at physics.stackexchange.com/questions/527229/… $\endgroup$
    – Carlo Beenakker
    Commented Feb 3, 2020 at 17:00
  • $\begingroup$ @CarloBeenakker: Thanks, that was me. As you see, that did not help... $\endgroup$
    – asv
    Commented Feb 3, 2020 at 17:18

2 Answers 2

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A rigorous and rather complete description of non-relativistic scattering, including a discussion of long-range potentials, can be found in the following book:

Dereziński, Jan; Gérard, Christian, Scattering theory of classical and quantum $N$-particle systems, Texts and Monographs in Physics. Berlin: Springer. xii,444 p. (1997). ZBL0899.47007.

To handle long-range potentials, the definition of the wave/Møller operators and the corresponding S-matrix must be modified by using different "free dynamics". This idea was introduced by Dollard, as has been mentioned in comments on your previous question. In the quantum 2-body case, the short-range and long-range cases are treated and contrasted respectively in Secs.4.6 and 4.7 of the book. The quantum N-body case with long-range potentials is treated in the last several sections of Ch.6. You may want to read the Introduction section for the whole book, as well as the Introductions of the relevant chapters for a lot of context and for hints which earlier pats of the book to read if you want to quickly jump into chapters 4 or 6.

In relativistic QFT, the scattering problem has a quite different nature, due to the fact that particle number is not conserved. This creates an obstacle to the naive scattering theory in the sense that a state with few incoming particles may give rise to an infinite number of outgoing particles (this is intimately tied in with infrared divergences). These problems appear already at low orders in perturbation theory (a mathematically well-posed setting), so they can be discussed at that level, without the need to go into non-perturbative construction of interacting QFT (which is basically an open problem anyway). Still, there are standard ideas for how to define an S-matrix in this context, in particular for QED, again going back to Dollard's idea of suitably modifying free asymptotic dynamics.

The following paper is a modern mathematical synthesis of these ideas (including historical references), extending them to a larger class of QFTs, which contains QED:

Duch, Paweł, Infrared problem in perturbative quantum field theory, [arXiv:1906.00940].

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Trying to understand scattering for non-relativistic toy models of QED, and in particular in the massless case (i.e. long range interactions) is an active area of research in mathematical physics. You can for instance look at the article "Towards a Construction of Inclusive Collision Cross-Sections in the Massless Nelson Model" in AHP 2012 by Wojciech Dybalski who is an expert in the field.

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