Feynman diagrams
The pictures below show a typical set of Feynman diagrams used to do calculations in QED—in this case for so-called Compton scattering of a photon by an electron. The straight lines in the diagrams represent electrons; the wavy ones photons. At some level each diagram can be thought of as representing a process in which an electron and photon come in from the left, interact in some way, then go out to the right. The incoming and outgoing lines correspond to real particles that propagate to infinity. The lines inside each diagram correspond to virtual particles that in effect propagate only a limited distance, and have a distribution of energy-momentum and polarization properties that can differ from real particles. (Exchanges of virtual photons can be thought of as producing familiar electromagnetic forces; exchanges of virtual electrons as yielding an analog of covalent forces in chemistry.)
To work out the total probability for a process from Feynman diagrams, what one does is to find the expression corresponding to each diagram, then one adds these up, and squares the result. The first two blocks of pictures above show all the diagrams for Compton scattering that involve 2 or 3 photons—and contribute through order α3. Since for QED α≃1/137, one might expect that this would give quite an accurate result—and indeed experiments suggest that it does. But the number of diagrams grows rapidly with order, and in fact the kth order term can be about (-1)k αk (k/2)!, yielding a series that formally diverges. In simpler examples where exact results are known, however, the first few terms typically still seem to give numerically accurate results for small α. (The high-order terms often seem to be associated with asymptotic series for things like Exp[-1/α].)
The most extensive calculation made so far in QED is for the magnetic moment of the electron. Ignoring parts that depend on particle masses the result (derived in successive orders from 1, 1, 7, 72, 891 diagrams) is
2 ( 1 + α/(2 π) + (3 Zeta[3]/4 - 1/2π2 Log[2] + π2/12 + 197/144) (α/π)2 + (83/72 π2 Zeta[3] - 215 Zeta[5]/24 - 239 π4/2160 + 139 Zeta[3]/18 + 25/18 (24 PolyLog[4, 1/2] + Log[2]4 - π2 Log[2]2) - 298/9 π2 Log[2] + 17101 π2/810 + 28259/5184) (α/π)3 - 1.4 (α/π)4+ …),
or roughly
2. + 0.32 α - 0.067 α2 + 0.076 α3 - 0.029 α4 + …
The comparative simplicity of the symbolic forms here (which might get still simpler in terms of suitable generalized polylogarithm functions) may be a hint that methods much more efficient than explicit Feynman diagram evaluation could be used. But it seems likely that there would be limits to this, and that in the end QED will exhibit the kind of computational irreducibility that I discuss in Chapter 12.
Feynman diagrams in QCD work at the formal level very much like those in QED—except that there are usually many more of them, and their numerical results tend to be larger, with expansion parameters often effectively being α π rather than α/π. For processes with large characteristic momentum transfers in which the effective α in QCD is small, remarkably accurate results are obtained with first and perhaps second-order Feynman diagrams. But as soon as the effective α becomes larger, Feynman diagrams as such rapidly seem to stop being useful.