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S6a. Nonperturbative computations in quantum field theory
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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.