Quantum Electrodynamics in Pictures

Quantum electrodynamics (hereinafter referred to as QED), the beloved but capricious child of non-relativistic quantum mechanics and the special theory of relativity, is a very complex physical theory with a furious mathematical apparatus. But, unlike many other complex theories, in its toolkit there is one small and relatively isolated part that allows for a primitive but clear interpretation. I mean the so-called “Feynman diagrams”. They look something like this:

and today it is not easy to write an article on quantum field theory without drawing several similar pictures, and in some works they appear more often than signs of elementary arithmetic. As the name suggests, these diagrams were invented by the outstanding American physicist Richard Feynman. He did this in the late 1940s to graphically describe some of the mathematical expressions that arise in QED. Let me make a reservation right away: of course, calculating Feynman diagrams, with the exception of a few of the most primitive ones, is very difficult. Behind each of them there is a strictly defined mathematical expression, usually very complex. But at the same time, they have a remarkable property – they allow a simple qualitative verbal interpretation and help to understand some of the fundamental ideas underlying modern quantum theory. These ideas are more unusual than complex and, it seems to me, even a high-achieving student can understand them at a basic level.

I want to talk about these diagrams and the ideas they contain. In this case, there will be no formulas in the text at all – only words and pictures. We will move forward “by the method of successive approximations” – this, as we will see below, is quite in the spirit of QED. In some cases, I will not tell you the whole truth first, and then make clarifications. However, of course, I won’t tell you the full truth – it simply won’t fit into the format of a popular text.

In addition, some information that is not important for a basic understanding of the text (but may still be interesting) I will highlight in italics.

So, here we go. Feynman diagrams, which describe the processes occurring in quantum electrodynamics, are very simple. They are constructed from three elements.

The first element is the electron, a particle with non-zero electric charge and mass, which is denoted by a solid straight line with an arrow to the right:

In QED, it is common to understand the term “electron” in a more general way than usual: “electron”, if this does not lead to confusion, is sometimes used to refer to any fermion with an electric charge and mass: for example, a proton, a muon, etc. (To understand what follows, you do not need to know what a “muon” or even a “proton” is.) It is also convenient to think of the electron's charge -e as negative, and the positive quantity +e as the so-called “elementary electric charge”, a quantity of which the charges of all free elementary particles are multiples.

Traditionally, it is believed that time flows from left to right in diagrams. The arrow on the line thus says that the electron is “moving from the past to the future.”

We can change this direction by drawing a line like this with a left arrow:

This line represents a positron, that is, an antielectron – a particle with a charge of +e, equal in magnitude and opposite in sign to the charge of the electron. Sometimes they put it nicely: a positron is an electron flying back in time.

Thus, the first element of the constructor is a solid straight line with an arrow. Below, if there is no need to specifically clarify whether this line symbolizes an electron or a positron, we will call it a “fermion line.”

Fermions are a special large class of elementary particles that make up matter. These include the electron and positron, proton and antiproton, as well as many other, more exotic particles.

The second element of Feynman diagrams is a photon, a quantum of light, a carrier of electromagnetic interactions, a particle without an electric charge and with strictly zero mass. It is indicated by a wavy line:

Since there is no arrow on it, changing its direction is pointless, you will get the same photon. Physicists say about it like this: a photon — absolutely neutral particle, it has no antiparticles.

The third element is the so-called “interaction vertex”: the point at which exactly one photon is attached to the fermion line:

The fermion and photon lines described above are associated with some non-trivial mathematical expressions, which we will not discuss, but the expression for the vertex is very simple – it is a constant equal to the elementary electric charge e. And This fact will be used below.

The constructor is ready. Any diagram consisting of fermion lines, photon lines and vertices corresponds to some real process. (Well, not exactly any, but more on that later.) The remarkable liberalism of quantum theory is manifested here: any process that is not directly prohibited by some law can occur in it.

It seems to me that even a preschooler can cope with creating such diagrams if he is provided with three types of parts with holes for connection.

So, let's start playing with the resulting constructor. Let's draw a diagram like this:

What happened here? Two free (that is, non-interacting) electrons were flying from the left (remember, time is directed from left to right). Then an interaction occurred between them: one of them (for QED it doesn’t matter which one) emitted a photon, and the other absorbed it, after which the electrons, forgetting about each other, freely flew on. This process of interaction between two particles is called “scattering”. In this case, they say that a “single-photon exchange” has occurred between the particles. A couple more important terms: lines, both ends of which are attached to points, correspond to “virtual” particles, while lines whose ends stick out correspond to “real” or “physical” particles. In our diagram, electrons are real, they fly into and out of the interaction zone, and the photon is virtual, it is born and dies inside the diagram.

Electrons can also exchange several virtual photons, for example, like this:

Or like this:

Or even like this (note: the photon lines here pass through each other without a point, which means there is no real intersection!):

An important parameter of a chart is the number of points (vertices) it contains, which is called the “chart order” N. Above I said that the top is just a constant e. All vertices included in one diagram are multiplied. Therefore, we know for sure that the mathematical expression corresponding to the diagram (maybe very complex) is proportional e^N. It turns out that physical quantities that are calculated in QED are expressed through the squares of the absolute values ​​of the quantities to which the diagrams correspond. Then the order diagram N corresponds to a contribution to this value proportional to (e^N)² = α^N. Here α = e² — the famous “fine structure constant” or simply “alpha”, which is numerically approximately equal to 1/137, that is, much less than one.

There is a small omission above. In fact, in the CGSE system α = e²/ħc, where ħ is the reduced Planck constant, and c is the speed of light. But in QED, the so-called “relativistic system of units” is almost universally used, in which ħ=1 and c=1.

The fact that α is quite small is a great stroke of luck for theorists! Indeed, as we have seen, to describe the interaction of two electrons we must take into account the exchange of one, two, three (see diagrams above) and so on, ad infinitum, photons. But the first diagram is of order 2, that is, it gives a contribution to the quantities of interest to us proportional to α². And the second order is 4, its contribution to 137²that is, approximately 20,000 times smaller. Thus, any physical quantity that we will calculate in QED is the sum of successively decreasing contributions (mathematicians will call this sum an asymptotic series)and we can cut off this summation, having achieved the accuracy we need. This is the “method of successive approximations” that I mentioned above.

Theorists call theories of this kind, which involve expansion in a small parameter, “perturbation theories.”

Let's continue our games. We can flip the diagram for a single-photon exchange between electrons by 90 degrees:

Another process is happening here. Two fermions are flying from the left, one of them is “tail first”. That is, by our agreement, an electron and a positron. Then they disappear (“annihilate”) with the formation of a virtual photon. Finally, this photon again disintegrates into an electron and a positron. Of course, this is also scattering, but, as we see, the internal mechanism of an electron and a positron is completely different from that of two electrons.

At this point, the attentive reader might wonder why I didn't start with even simpler diagrams. For example, with this one:

Or with this one:

Or even with this one:

Alas, when I said above that any process whose diagram can be drawn is allowed in QED, I was slightly lying. In fact, there is an important additional requirement: conservation laws must also be observed. There are two and a half basic laws: conservation of electric charge, energy, and momentum (from the point of view of the theory of relativity, the last two are combined into one – the law of conservation of energy-momentum). (There are also nuances associated with the law of conservation of angular momentum, but we will not deal with them here.) With charge conservation in Feynman diagrams, everything is fine: after a little thought, you can understand that the charge flows along fermion lines, and, following the described rules, it is simply impossible to draw a diagram in which the sums of the particle charges before and after the interaction are not equal. But with energy-momentum conservation, things are worse: it must be checked separately each time.

I promised not to write formulas, so I’ll try to explain in words. Let's imagine some particle that wants to decay into several other particles. From the point of view of conservation laws, can she always do this? No! Let M — mass of the original particle, A m is the sum of the masses of the particles into which it decays. Let's move to a reference system in which the original particle is at rest (Einstein's first postulate allows us to do this). Then its total energy is equal to its mass M. And the energy of decay products is no less m (and most likely more, because they are probably moving, and their energies consist of “rest energy” equal to mass, and kinetic energy.) Thus, decay is possible only if M > m (the equality of these masses is a special case that we will not examine).

Let us now look at the three upper diagrams (Fig. 10-12). The first (Fig. 10) describes the decay of a photon (which has zero mass) into two particles of nonzero mass. The law of conservation of energy does not allow this. The third process (Fig. 12) – the emission of a photon by a free fermion – is forbidden for the same reason. And the second process (Fig. 11) – the annihilation of an electron-positron pair into a photon – is simply the inverse process of the first, and therefore it is also forbidden.

Just in case, I will clarify: the law of conservation of energy-momentum prohibits only some real processes, it has nothing against the blocks from Fig. 10-12 being used as components of other, more complex diagrams. If this law is violated “slightly and briefly” by virtual particles, then quantum theory turns a blind eye to such a violation.

By the way, the law of conservation of energy does not object to the annihilation of an electron-positron pair into two photons; it is quite possible:

Could this be possible?

Answer: yes and no. Such processes are possible, but theorists are almost never interested in them. In fact, what is shown in Fig. 14? An electron flies, emits a virtual photon, and then absorbs it itself. Or it can be more complicated, for example:

or

That is, the electron seems to move in a “coat” of virtual photons generated by it. It can be shown that such processes lead to only one thing: they change the mass of the fermion. And since we can determine this mass experimentally, we can use it everywhere from the very beginning, and simply ignore the diagrams describing the photon “coat.”

By the way, for the same reasons, there is no need to take into account any diagrams with a “fermion in a fur coat”, for example, this one:

These diagrams are already taken into account by the change in the electron mass in the simpler diagram of Fig. 5.

Let's continue our tour of the diagram zoo:

Two photons flew, turned into virtual electrons and positrons, which then collapsed back into photons. In other words, they interacted. Perhaps you know that in classical optics this is impossible. From the point of view of 19th century physics, you can set up two spotlights whose beams will intersect and not affect each other in any way. It turns out that in QED this is still possible, light really does scatter on light. It's just that this process is very weak and barely noticeable, because the corresponding diagram has the order of 4.

What about a diagram like this?

The law of conservation of energy does not directly prohibit it (because the masses of all particles participating in the reaction are zero). Nevertheless, such a diagram can be ignored. And this is what the last rule of the Feynman constructor tells us, which I kept silent about above. Namely: any diagram that contains an electron loop with an odd number of photons coming out of it can be ignored.

In fact, the situation here is more interesting. Formally, the diagram with the insert in Fig. 19 is not equal to zero. But to it we must add another, in which the electron in the loop moves in the opposite direction:

And it, as can be proven, has the same magnitude and opposite sign.

This fact is called “Farri's theorem”. It is connected, by the way, with the fact that electrons are fermions, that is, they obey Fermi-Dirac statistics.

Thus, for any diagram containing the structure from Fig. 19, there is an alternative diagram with the structure from Fig. 20, and their sum is always strictly equal to zero.

Otherwise, we can indulge ourselves in everything we want. Let's say, a diagram like this is possible (I'm giving free rein to my imagination):

It should be noted that the number of possible order diagrams N and the complexity of calculating each of them grows very quickly with growth N. If, for example, the electron scattering diagram mentioned above (of order 2)

any theoretical student should actually calculate orally, then with the calculation of some such (about 4)

Even a fifth-year student may not be able to cope, but the monster of the 14th order, which I came up with above (Fig. 21), most likely, can be calculated by a professional team of theorists for several years (and it is not a fact that it will be calculated). By the way, even just drawing all the diagrams that, along with the diagram in Fig. 21, describe 14th order contributions to electron-electron scattering – this is a very difficult problem (without going into details, I will only say that this number grows, roughly speaking, like the factorial N, although there are considerations that make this growth not so fast). But to calculate the full contribution of this order, you must not only draw them, without missing a single one, but also calculate them…

By the way, QED is historically the first quantum field theory, but far from the only one. For example, there is quantum chromodynamics – a theory that describes the forces that hold particles in the nucleus of an atom, which also makes full use of Feynman diagrams – only there are more components, they are different, and the rules are more complicated. Another big problem of quantum chromodynamics is that the interaction constant (analogous to “alpha” in QED) is approximately equal to unity, so it is difficult to build a perturbation theory for it: all diagrams make approximately equal contributions to physical quantities.

Finally, the experts won't forgive me if I don't make a few things clear. As I said at the beginning, Feynman diagrams are very visual. This, in addition to the obvious advantages, has its disadvantages. They are so visual that there is a temptation to take them literally — as “photographs” of electrons, which, like balls, are thrown around by photons.

What I mean is this. Interacting charged particles actually draw tracks in the bubble chamber that are a bit like our diagrams. These tracks look something like this:

If it weren't for the magnetic field that bends the trajectories of charged particles, and the absence of tracks of uncharged particles, the similarity would be even greater. However, don't rely too much on this similarity in your reasoning – this tempting but naive analogy can lead you in a completely wrong direction!

And one more, more subtle point. Let's say you're studying electron-electron scattering. As described above, this process can be described by a set of diagrams:

and so on. Their sum (more precisely, as I said above, the square of the absolute value of the sum) has a physical meaning – the corresponding value can be measured experimentally. But is it possible to see or study a process corresponding to a particular diagram – for example, a two-photon diagram:

The answer is no! All processes described by diagrams occur, in some sense, simultaneously (quantum mechanics allows this and calls it superposition). But we can only mentally extract a separate diagram from a process and look at it. In other words, nature knows nothing about any Feynman diagrams. Something like this happens in nature:

Here the shaded circle represents the sum of all possible virtual processes. And the separate diagram is a visualization invented by man, facilitating calculations. But the visualization is very beautiful and convenient.

So, I tried to illustrate some ideas that stand behind Feynman diagrams. I have no doubt that this text will cause two types of complaints. Some are like “I don't understand anything at all.” Others are like “how could you talk about Feynman diagrams but not use the term “propagator” even once?” [и ещё сотню терминов] and not to mention “sweeping infinities under the rug” [и ещё о сотне вещей]?!” Unfortunately, even if I wanted to, I could not write a text that would satisfy both types of claims. I hope that the compromise I have chosen between rigor and popularity of presentation is quite reasonable, and that none of the simplifications I have used have led to a complete loss of meaning.

PS Thanks to Thorsten Ohl for creating the FeynMF LaTeX package, which is indispensable for drawing Feynman diagrams.