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S7c. Bound states in relativistic quantum field theory
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Bound states are supposed to be poles of the S-matrix, and 
Bethe-Salpeter equations for the bound state dynamics can be 
obtained approximately from resumming infinite families of 
Feynman diagrams. See Chapter 14 of Weinberg's QFT I. But...

Perturbative QED, even in the fully rigorous treatment in 
    G. Scharf,                                                     
    Finite Quantum Electrodynamics: The Causal Approach, 2nd ed.   
    New York: Springer-Verlag, 1995.                               
has nothing at all to say about how to model bound states. 
Bound states don't exist perturbatively: The poles in the S-matrix 
can arise only by summing infinitely many Feynman diagrams.
(Sum the geometric series 1+x+x^2+... to see how poles arise by 
summation.)

I haven't seen a single rigorous treatment of such an issue in
quantum field theory. 

Weinberg states in his QFT book (Vol. I) repeatedly that bound state 
problems (and this includes the Lamb shift) are still very poorly 
understood (though the Lamb shift is one of the most accurately 
predicted physical quantity). On p.564 he says, 
   'These problems are those inbolving bound states [...] 
   such problems necessarily involve a breakdown of ordinary 
   perturbation theory. [...] The pole therefore can only arise 
   from a divergence of the sum of all diagrams [...]'
On p.560, he writes,
   'It must be said that the theory of relativistic effects 
   and radiative corrections in bound states is not yet in an 
   entirely satisfactory shape.'
This remark suggests that he seems to think that, in contrast, 
for scattering problems, the theory is in an entirely satisfactory 
state, as given in the rest of his book. Thus 'satisfactory' 
does not mean 'mathematically rigorous', but only 
'well understood from a physical, approximate point of view'.

There are, of course, methods for approximating bound state problems,
based on Bethe-Salpeter equations, Schwinger-Dyson equations, and
some other approaches. See, e.g., the review 
    H. Grotch and D.A. Owen,
    Foundations of Physics 32 (2002), 1419-1457.
or hep-ph/0308280.
But all of this is done in completely uncontolled approximations,
and to get numerically consistent results is currently more an
art than a science.

This leaves plenty of scope for interesting (but hard)
new work on bound states on both the physical and mathematical side.