--------------------------------------------------------- S6a. Nonperturbative computations in quantum field theory --------------------------------------------------------- There is well-defined theory for computing contributions to the S-matrix in quantum electrodynamics (and other renormalizable field theories) by perturbation theory. There is also much more which uses handwaving arguments and appeals to analogy to compute approximations to nonperturbative effects. Examples are: - relating the Coulomb interaction and corrections to scattering amplitudes and then using the nonrelativistic Schroedinger equation, - computing Lamb shift contributions (now usually done in what is called the NRQED expansion), - Bethe-Salpeter and Schwinger-Dyson equations obtained by resumming infinitely many diagrams. The use of 'nonperturbative' and 'expansion' together sounds paradoxical, but is common terminology in QFT. The term 'perturbative' refers to results obtained directly from renormalized Feynman graph evaluations. From such calculations, one can obtain certain information (tree level interactions, form factors, self energies) that can be used together with standard QM techniques to study nonperturbative effects - generally assuming without clear demonstrations that this transition to quantum mechanics is allowed. Of course, although usually called 'nonperturbative', these techniques also use approximations and expansions. The most conspicous high accuracy applications (e.g. the Lamb shift) are highly nonperturbative. But on a rigorous level, so far only the perturbative results (coefficients of the expansion in coupling constants) have any validity. Although the perturbation series in QED are believed to be asymptotic only, one can get highly accurate approximations for quantities like the Lamb shift. However, the Lamb shift is a nonperturbative effect of QED. One uses an expansion in the fine structure constant, in the ratio electron mass/proton mass, and in 1/c (well, different methods differ somewhat). Starting e.g., with Phys. Rev. Lett. 91, 113005 (2003) one should be able to track the literature. Perturbative results are also often improved by partial summation of infinite classes of related diagrams. This is a standard approach to go some way towards a nonperturbative description. Of course, the series diverges (in case of a bound state it _must_ diverge, already in the simplest, nonrelativistic examples!), but the summation is done on a formal level (as everything in QFT) and only the result reinterpreted in a numerical way. In this way one can get in the ladder approximation Schroedinger's equation, and in other approximations Bethe-Salpeter equations, etc.. See Volume 1 of Weinberg's quantum field theory book.