It began with the Feynman diagram which I introduced earlier in the context of relativistic kinematics. In Fig. 1 I show the Feynman diagrams for single and double photon exchange in quantum electrodynamics or , for which Richard P. Feynman shared a Nobel Prize. As before, I will draw Feynman diagrams "left to right" instead of the conventional "down to up." The idea of was (and is) that all electromagnetic interactions between charged particles can be described in terms of the exchange of photons created by one particle and destroyed by another. The simplest case is the "first-order" diagram in Fig. 1, where two electrons exchange a single photon. The next (second-order) process is a factor of less important, where is the fine structure constant (not a very mnemonic name any more), which is (sort of) the strength of the "vertex" (the point where the photon begins or ends). Because each successive diagram (single photon exchange, double photon exchange, triple photon exchange ) is a factor of about 19,000 less important than the one before, is a perturbation theory that converges very rapidly. That is, you can get a pretty accurate result with very few diagrams.
Each diagram, you see, is rigourously equivalent
to a big messy integral which is definitely less appealing to
the brain's right hemisphere; but the big integral can be evaluated
to give the correct formula for the interaction of the two
electrons to that order in
, properly taking
into account all the ramifications of quantum field theory.
Let's take another step back
for better perspective.