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Feynman Diagrams

Richard Feynman was the physicist who developed the method still used today to calculate rates for electromagnetic and weak interaction particle processes. The diagrams he introduced provide a convenient shorthand for the calculations. They are a code physicists use to talk to one another about their calculations.

In Feynman diagrams (time ordered form):

• Left-to-right in the diagram represents time; a process begins on the left and ends on the right.
• Every line in the diagram represents a particle; the three types of particles in the simplest theory (QED) are:

• Up and down (vertical) displacement in a diagram indicates particle motion, but no attempt is made to show direction or speed, except schematically.
• Any vertex (point where three lines meet) represents an electromagnetic interaction; possible vertices are (and notice that all six of these processes are just different orientations of the same three elements):

• Any diagram which can be built using these parts is a possible process provided:
  1. Conservation of energy and momentum is required at every vertex
  2. Lines entering or leaving the diagram represent real particles and must have E²=p²c²+m²c⁴.
  3. Lines in intermediate stages in the diagram represent "virtual particles," which do not need to have the right relationship between E, p, and m, but which can never be observed if they do not!

  The first thing to realize is that no single vertex diagram represents a possible process - no matter how you try, you cannot satisfy rules (1) and (2) above at the same time for such a process.

The simplest process we can consider is a two particle collision or "scattering" event. Let us start and end the process with one electron and one positron-- only their momenta and energies change in the process:

Feynman tells us to draw all possible diagrams. First, lets add one intermediate photon line. We find three time-ordered diagrams:

(a)	(b)	(c)
The first two figures (a and b) are just different orientations (time-orderings) of the same event.  We use the figure (d)  below as a shortcut to show both orientations.		This third diagram (c) is really quite a different process -- it is an intermediate  stage with only  a photon (a virtual photon) present.
(d)
Notice that this diagram does not have time orderings, just a start and stop.

We can also draw more complicated diagrams with more photons, for example:

In fact, we could have any number of photons!

What makes the diagrams useful is that each diagram has a definite complex number quantity -- called an amplitude -- related to it by a set of rules (the Feynman rules). One part of these rules is that there is a multiplication factor of  for each photon, so the amplitudes for diagrams with many photons are small, compared to those with only one. The quantity "e" here is the electromagnetic coupling or electric charge.

Technically, the Feynman rules give the rate as a power series expansion in the coupling parameter. The technique is only useful when this parameter is small, that is, for electromagnetic or weak interactions but not for strong interactions except at very high energies.

Calculations in QED keeping up to four photons have been made for certain quantities. They give a result that matches experimental data up to the twelfth decimal place!

Real and Virtual Particles

Because Feynman diagrams represent terms in a quantum calculation, the intermediate stages in any diagram cannot be observed. Physicists call the particles that appear in intermediate, unobservable, stages of a process "virtual particles". Only the initial and final particles in the diagram represent observable objects, and these are called "real particles."

Feynman Rules

The Feynman Rules for a theory are very simple, but lead to increasingly complicated mathematical expressions as increasingly complicated diagrams are constructed.

The rules for any process are:

• Draw all possible diagrams (up to some number of photons, depending on the accuracy desired). Different time-orderings of a given process are represented by the same diagram.
• Given the initial momentum and energy, define how momentum and energy flow for each line in the diagram. Where each diagram has a closed loop, there is an arbitrary momentum and energy flow around the loop and we must integrate over all possible choices for these quantities. Each intermediate line in the diagram contributes a factor to the amplitude of  1/(E²-p²c²-m²c⁴) where m is the appropriate mass for the particle type represented by the line. Note that this says that the more "virtual" the particle represented by a line is, the smaller the contribution of the diagram.
• Add the amplitude factors from all possible diagrams to get the total amplitude for the process.

The expected rate for the process can then be calculated -- it is proportional to the absolute value of the total amplitude squared. [Note that this is not the same as the sum of the squares of the absolute values of the individual amplitudes.] For more information on this topic, take a look at the discussion of quantum interference.