Bound states in relativistic quantum field theory

Just as in QM, a bound state in QFT is a pole of the nonperturbative T-matrix (, i.e., the S-matrix minus 1 stripped from a delta function), and its possible wave functionals are the functionals in the range of the linear operator obtained as the residual of the T-matrix at the pole. (A proof is given in my answer to this question at Physics Overflow.

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.

Approximate Bethe-Salpeter equations for the bound state dynamics can be obtained by resumming infinite families of Feynman diagrams. See, e.g., Chapter 14 of Weinberg's QFT I.

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.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)

A theoretical physics FAQ