QED Explained

Feynman's graphical method provides a way of visualizing each term in the sum over histories. Those pictures, called Feynman diagrams, are one of the most important tools of modern physics. In QED the sum over all possible histories can be represented as a sum over Feynman diagrams like those below, which represent some of the ways it is possible for two electrons to scatter off each other through the electromagnetic force. In these diagrams the solid lines represent the electrons and the wavy lines represent photons. Time is understood as progressing from bottom to top. and places where lines join correspond to photons being emitted or absorbed by an electron. Diagram (A) represents the two electrons approaching each other, exchanging a photon, and then continuing on their way. That is the simplest way in which two electrons can interact electromagnetically, but we must consider all possible histories. Hence we must also include diagrams like (B). That diagram also pictures two lines coming in—the approaching electrons—and two lines going out—the scattered ones—but in this diagram the electrons exchange two photons before flying off The diagrams pictured are only a few of the possibilities; in fact, there are an infinite number of diagrams, which must be mathematically accounted for.

Feynman diagrams aren't just a neat way of picturing and categorizing how interactions can occur. Feynman diagrams come with rules that allow you to read off, from the lines and vertices in each diagram, a mathematical expression. The probability, say. that the incoming electrons, with some given initial momentum. win end up flying off with some particular final momentum is then obtained by summing the contributions from each Feynman diagram. That can take some work, because, as we've said, there are an infinite number of them. Moreover, although the incoming and outgoing electrons are assigned a definite energy and momentum, the particles in the closed loops in the interior of the diagram can have any energy and momentum. That is important because in forming the Feynman sum one must sum not only over all diagrams but also over all those values of energy and momentum.

Feynman diagrams provided physicists with enormous help in visualizing and calculating the probabilities of the processes described by QED. But they did not cure one important ailment suffered by the theory: When you add the contributions from the infinite number of different histories, you get an infinite result. (If the successive terms in an infinite sum decrease fast enough, it is possible for the sum to be finite, but that, unfortunately, doesn't happen here.) In particular, when the Feynman diagrams are added up, the answer seems to imply that the electron has an infinite mass and charge. This is absurd, because we can measure the mass and charge and they are finite. To deal with these infinities, a procedure called renormalization was developed.

The process of renormalization involves subtracting quantities that are defined to be infinite and negative in such a way that, with careful mathematical accounting, the sum of the negative infinite values and the positive infinite values that arise in the theory almost cancel out, leaving a small remainder, the finite observed values of mass and charge. These manipulations might sound like the sort of things that get you a flunking grade on a school math exam, and renormalization is indeed, as it sounds, mathematically dubious. One consequence is that the values obtained by this method for the mass and charge of the electron can be any finite number. That has the advantage that physicists may choose the negative infinities in a way that gives the right answer, but the disadvantage that the mass and charge of the electron therefore cannot be predicted from the theory. But once we have fixed the mass and charge of the electron in this manner, we can employ QED to make many other very precise predictions, which all agree extremely closely with observation, so renormalization is one of the essential ingredients of QED. An early triumph of QED, for example, was the correct prediction of the so-called Lamb shift, a small change in the energy of one of the states of the hydrogen atom discovered in 1947.

The success of renormalization in QED encouraged attempts to look for quantum field theories describing the other three. forces of nature. But the division of natural forces into four classes is probably artificial and a consequence of our lack of understanding. People have therefore sought a theory of everything that will unify the four classes into a single law that is compatible with quantum theory. This would be the holy grail of physics.

A good description of the Feynman Diagrams from QED and how the infinite possibilities must be accounted for in mathematics dealing with them.