How to interpret Feynman diagram
Elementary particles (photons and electrons) are elementary excitations of the corresponding quantum fields, and are best understood as harmonic modes. Just as a resonating string can sound not at all (no particle), in the basic mode (1 particle = 1 extremum per length), in the first overtone (2 particles = 2 extrema per length), in higher overtones (k particles = k extrema per length) or in a superposition of these (a Fock state - superposition of different particle number states). Indeed, Fock space, which is the mathematical framework in which one commonly defines QED, is defined by the harmonic analysis of a free wave equation, the 4-dimensional analogue of the differential equation describing an idealized string of a guitar.
Two kinds of particles correspond (in this analogy) to two strings (with an associated tensor product state space). E.g., two electrons and a photon = first overtone on first string and basic mode on second string.
Now a guitar is has interactions between its modes (mainly through its resonating body), hence produces complex sounds. Similarly, QED (which has in place of six strings fields with three kinds of particle excitations - electrons positrons and photons) produces a complex behavior of the states of the quantum fields, to which many particle modes may contribute.
Feynman diagrams tell you how a complex state (i.e., sound in the analogy) can be decomposed into elementary interactions - namely the vertices of the diagrams. There are only very few kinds of them, in QED essentially only one, which depending on how it is viewed, takes several possible interpretations:
All this is virtual, just internal book-keeping of the decomposition process by which the physicist creates a picture of what happens in Nature (without all this imagery, which is special to Feynman's approach to QFT). Note that most single diagrams are meaningless (giving rise to so-called infinities or divergences); many such diagrams simultaneously are needed (and must be ''summed'') to describe a real process - though in some import cases a few of them give a reasonable approximation.
Real (which means observable) are only the free ends of the Feynman diagrams - these correspond to real particles (satisfying physical laws such as energy and momentum conservation). As the ends have a direction, this tells you what may come in and what may go out, namely the input and output of real scattering events. And then there is an elaborate machinery that turns the collection of _all_ Feynman diagrams with the same collection of ends into a recipe for calculating highly accurate probabilities for these events.
Thus scattering is not what happens inside a single Feynman
diagram, but what happens when you take all Feynman diagrams with the
same input and output together.
Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ