by Steve Bryson
The simplest quantum field theory used today is the theory of the interaction of electromagnetically charged particles with light. This theory is called Quantum Electrodynamics, or QED. It is usually used to describe the interaction of electrons with light, as that is the most common charged particle that is observed (When you work with the proton, you run into difficulties because of the strong nuclear force, but QED applies here also). QED is a gauge theory (see next week's handout), and the force described by QED (the electromagnetic force) is said to be carried by a particle called a photon. This photon is also to be considered the particle aspect of light. As described in the handout on gauge theories, the photon is, from the perspective of gauge theories, related to how the direction of the charged particle's quantum wave changes from point to point. For the purposes of this handout, however, we will be treating the photon as an independent particle.
Even though the word 'particle' is used to describe both the charged particle (i.e. the electron) and the photon, the formulation of QED is entirely in terms of the quantum waves of both the charged particle and the photon. Basically, QED is a theory of how the photon and the charged particle quantum waves interact. It is actually rather easy to state the equations telling everything about this interaction (once you know the language, of course). These equations only tell you the nature of the interaction, however. In order to tell just what a particular interaction (such as two electrons repelling each other) looks like, you need to solve the equations of QED. This turns out to be very difficult.
Basically the problem is that no one knows how to solve the equations of QED exactly in any but the most trivial cases. What is done is approximate the solution using a technique known as perturbation theory. This technique starts with some exactly known solution to the equations of QED which, hopefully, describe a situation not too different from the situation that we are actually trying to solve. Then this first guess is refined by taking the next most complicated situation and adding that to the first guess. This is done for as many stages as you wish to do calculations. To my knowledge the farthest that this has been so far carried out in QED is four steps, as the calculation gets a lot more complicated with each step.
This technique only works if the contribution from each more complicated situation gets smaller with each more complicated step. In QED this turns out to be the case, as each step gets divided by a factor of 137*137 = 18769 for each level of complication. In other words, the second step adds a contribution to the answer that is 1/18769 as large as the first step, the third step adds a contribution to the answer that is 1/18769 as large as the second step and so on. Thus as you get to later and later steps to your approximation the refinements get smaller and smaller. Once you have gone three, or at most four steps you know that the difference between the real answer and you approximation are so small as to be unobservable.
Now it is true that you still do not have the exact answer. This does not mean that QED is not an exact theory. Presumably (at the distances with which QED is concerned) QED as formulated is an exact theory. It is only that we do not have the mathematical oomph needed to write down exact solutions of QED. It is only the numbers describing a particular case that is approximately calculated.
There is one technical point involved with the calculations in QED (and in perturbation calculations in most quantum field theories). If you simply calculate each step in the most straightforward way, you find that at the second and third step your refinements are infinitely large. This is bad, as when you do an actual experimental observation of the interaction that you are trying to describe there are no infinite results observed. One must resort to a technical trick called renormalization in order to get finite numbers, and when you do you find that the numbers that you calculate are in exact agreement with your observations (if you go enough steps). In a little more detail, here is a schematic version of what you find at the second step of a perturbation calculation in QED:
First step = some finite number
Second step = another finite number + infinity
so
Two step approximation = First step + Second step
= a finite number + infinity
The finite number in the last line is the sum of the finite numbers from the first step and the second step. The really surprising thing is that even though the two step approximation is really an infinite number (and therefore technically wrong), the finite number is exactly the value that you find in an observation. In other words, if we can get rid of that infinite part in the two-step answer then we would have an answer that agrees with experiment.
It turns out that by very cleverly redefining the constants in the original equations of QED one can make the infinity disappear. The constants that must be changed are such things as the mass and electric charge of the charged particle that is being acted on by the photon. You see, originally the mass and electric charge used in QED were the ones that we observe. It turns out that that was the wrong mass and charge to use. If we choose our starting mass and charge just right, then the actual observed mass and charged are consequences of the interactions between the charged particle and light. This is called renormalization. It may seem that this is all some kind of parlor trick, and in fact that is all it was when it was first invented (many physicists still feel this way). In the last decade, however, a much clearer understanding of renormalization has emerged. In particular it seems that the infinities (and therefore the need to renormalize) arise from the perturbation technique and not from the fundamental structure of QED. In fact simple model quantum field theories that exhibit the infinities in pertubation solutions have been shown to be completely finite when solved exactly.
When using pertubation techniques to solve quantum field theories, there is a very convenient pictorial method of constructing what the refinements at each step of the approximation are due to Richard Feynman. Let us consider the case of the QED interaction between light and electrons. One draws the electron as a solid line with an arrow pointed to the right (or up):
An anti-electron is drawn as a solid line with an arrow pointed to the left (or down):
and the photon as a wiggly line:
Then the interaction between a photon and an electron is drawn as a photon line attached to two electron lines:
Note that one can think of this as an electron moving along and interacting (either emitting or absorbing) a photon and changing direction. The fact that this diagram is allowed is determined by the equations of QED. A diagram that is not allowed is one like that above with the roles of photon and electron interchanged. This is why we say that the photon acts on the electron and not vice versa.
This diagram connecting two electron lines with a photon line is called a vertex. This is the only vertex allowed in QED. Note that the arrow on one of the electron lines must point towards the vertex and the other arrow point away. A vertex diagram with, for example, three electron or three photon lines meeting at a point is not allowed. Other than three lines of any kind meeting at a point is also not allowed. Once you know how to draw your allowed diagrams, there are rules for translating the diagrams into mathematical terms in your pertubation calculations. In particular, each vertex gives that term in your calculation a factor of 1/137 in QED.
Now you are ready to roll. To calculate each step in your pertubation expansion, you draw allowed diagrams with increasing number of vertices. Each step in the calculation is defined as the mathematical term given by all possible diagrams with a particular number of vertices.
Let us look at an example of the interaction of two electrons. Other interactions would have diagrams different from the following.
For technical reasons, it turns out that diagrams with an odd number of vertices give approximation terms that are exactly equal to zero. Thus the simplest diagram, and therefore the first step in our approximation, is given by all the diagrams with two vertices. There is only one of these:
The next term is given by all possible diagrams with four vertices, which give terms that are 137*137 = 18769 times smaller than the first term. There are actually many of these, represented by the following:
It is with these diagrams that the troubles begin. In particular, the diagrams with a loop in them give the infinities that must be removed by renormalization. The lines in the loops are called 'virtual particles'. Thus one can have both virtual photons and virtual electrons. This is where the concept of virtual particle arose.
When you allow six vertices (third step in the approximation), things get much more difficult. Here an example:
A really fun thing about Feynman diagrams is that if you have one diagram that is allowed, then rotating that diagram by 90 degrees on the paper gives another allowed diagram. For example, if you have the original vertex diagram from above describing the interaction of an electron with a photon:
and you turn this 90 degrees clockwise, you get a diagram describing an electron and an anti-electron coming together and turning into light:
Now this diagram will actually vanish because it has only one vertex, but the following diagram showing an electron and an anti-electron turning into two photons via an intermediate virtual electron:
This diagram, turned 180 degrees turns into the following:
This describes two photons coming together and turning into an electron-anti-electron pair again via an intermediate virtual electron. Though this is allowed by QED, the chances of two photons coming together like this is extremely low.
The next simplest theory is Quantum Chromodynamics or QCD. QCD is formulated mathematically exactly like QED except that instead of one charge as in QED, QCD describes particles with three charges called red, blue, and green. The three charges are generically referred to as color (hence the chromo in quantum chromodynamics). Particles with color are, by definition, quarks. Thus all the difficulties of QED are present with a host of other complications.
Not the least of these complications is the fact that when one does a pertubation solution of QCD each of the refinement terms at each step is not smaller than the previous step. In fact they get bigger. This means that to get the correct answer one needs to calculate all the possible diagrams, because if you stopped at some step, the remaining steps are at least as significant as what you calculated. But there are an infinite number of diagrams! Thus answers in QCD cannot be calculated using pertubation techniques. This stopped progress in QCD for some time.
There is a way out, however. It turns out that if you use QCD at very small distances (= very high energies), then the terms in your pertubation calculation do get smaller with each step and you can make calculations in QCD just like in QED (including renormalizing, of course). Thus so long as you are looking at situations that are involving distances that are small enough (about the size of a proton), you can do calculations. If you wish to know about the behavior of QCD over larger distances (calculating the mass of the proton, for example) you are out of luck until somebody invents better techniques (one such technique, called 'lattice field theory', approximates spacetime as being a discrete set of points and does numerical calculations of QCD on computers). This unusual behavior of diagrams in QCD comes from the fact that the QCD force (the strong nuclear force) gets weaker with shorter distance.
QCD has its own set of Feynman diagrams. First, the particles involved are quarks and gluons (gluons are the analog of photons and carry the QCD force). The quarks and gluons are drawn as follows:
quark
anti-quark
gluon
Because there are three charges in QCD (as opposed to one in QED), there are more allowed diagrams. There is the basic vertex diagram of a gluon interacting with a quark:
There are also new vertex diagrams that show the interactions of gluons with themselves. The allowed ones are three and four gluons coming together at a point:
This, needless to say, makes calculations in QCD a good deal more complicated than QED even when we can use pertubation theory. The main thing that one finds that is of interest to us is that bound states of three quarks together and quark-anti-quark pairs are to be found in nature. This explains why we see the particles we see. Examples are to be found in the glossary in the first handout.