Dick Feynman was a genius of visualization (he was also no slouch with equations): he made a mental picture of anything he was working on. While others were writing blackboard-filling formulas to express the laws of elementary particles, he would just draw a picture and figure out the answer. He was a magician, a showman, and a show-off, but his magic provided the simplest, most intuitive way to formulate the Laws of Physics. Feynman diagrams (see page 38) are literally pictures of the events that take place as elementary particles move through space, collide, and interact. A Feynman diagram can be nothing more than a few lines describing a couple of colliding electrons, or it can be a vast network of interconnected branching, looping trajectories describing all of the particles making up anything from a diamond crystal to a living being or an astronomical body. These diagrams can be reduced to a few basic elements that summarize everything that is known about elementary particles. Of course there is more than just the pictures—there are all the technical details of how they are used to make precise calculations, but that’s less important. For our purposes a picture is worth a thousand equations.
Quantum Electrodynamics
A quantum field theory begins with a cast of characters, namely, a list of elementary particles. Ideally the list would include all elementary particles, but that is not practical: we are fairly certain that we don’t even know the complete list. But not too much is lost by making a partial list. It is like a theater performance: in reality every story involves everyone on earth, past and present, but no sane author would try to write a play with several billion characters. For any particular story some characters are more important than others, and the same is true in elementary-particle physics.
The original story that Feynman set out to tell is called Quantum Electrodynamics, or QED for short, and it involves only two characters: the electron and the photon. Let me introduce them.
The Electron
In 1897 the British physicist J. J. Thomson made the first discovery of an elementary particle. Electricity was well known before that point, but Thomson’s experiments were the first to confirm that electric currents are reducible to the motion of individual charged particles. The moving particles that power toasters, lightbulbs, and computers are, of course, electrons.
For dramatic effects it’s hard to beat electrons. When a giant lightning bolt rips across the sky, electrons flow from one electrified cloud to another. The roar of thunder is due to a shock wave caused by the collision of rapidly accelerated electrons with air molecules blocking their path. The visible lightning bolt consists of electromagnetic radiation that was emitted by agitated electrons. The tiny sparks and crackling noises due to static electricity, on a very dry day, are manifestations of the same physics on a smaller scale. Even ordinary household electricity is the same flow of electrons, tamed by electrically conducting copper wires.
Every electron has exactly the same electric charge as every other electron. The charge of the electron is an incredibly small number. It takes an enormous number of electrons—about 1019 per second—to create a common electric current of one amp. There is an oddity about the charge of the electron that has puzzled and troubled generations of undergraduates studying physics: the electron’s charge is negative. Why is that? Is there something intrinsically negative about the electron? In fact the negativity of the electron charge is not a property of the electron but rather a definition. The trouble dates back to Benjamin Franklin, who was the first physicist to realize that electricity was a flow of charge.11 Franklin, who knew nothing of electrons, had no way of knowing that what he called positive current was actually a flow of electrons in the opposite direction. For this reason we have inherited the confusing convention of a negative electron charge. As a consequence, we physics professors constantly have to remind students that when electric current flows to the left, electrons move to the right. If this boggles your mind, blame it on Ben Franklin and then ignore it.
If all electrons were suddenly to disappear, a great deal more than toasters, lightbulbs, and computers would fail. Electrons play another very profound role in nature. All ordinary matter is made of atoms, which in turn are made of electrons—each electron whirling around the atomic nucleus like a ball on a rope. Atomic electrons determine the chemical properties of all the elements listed in the periodic table. Quantum Electrodynamics is more than the theory of electrons: it is the basis for the theory of all matter.
The Photon
If the electron is the hero of QED, the photon is the sidekick that makes the hero’s deeds possible. The light emitted by a lightning bolt can be traced to microscopic events in which individual electrons shake off photons when they are accelerated. The entire plot of QED revolves around one fundamental process: the emission of a single photon by a single electron.
Photons also play an indispensable role in the atom. In a sense that will become clear, photons are the ropes that tether the electrons to the nucleus. If photons were to be suddenly eliminated from the list of elementary particles, every atom would instantly disintegrate.
The Nucleus
One of the main goals of QED was to understand the detailed properties of simple atoms, especially hydrogen. Why hydrogen? Hydrogen, having only a single electron, is so simple that the equations of quantum mechanics can be solved. More complex atoms with many electrons, all exerting forces on one another, could be studied only with the aid of powerful computers, which didn’t exist when QED was being formulated. But to study any atom, one more ingredient must be added—the nucleus. Nuclei are made of positively charged protons and electrically neutral neutrons. These two particles are very similar to each other, apart from the fact that the neutron has no electric charge. Physicists group these two particles together and give them a common name: the nucleon. A nucleus is essentially a blob of sticky nucleons. The structure of any nucleus, even of hydrogen, is so complicated that physicists like Feynman decided to ignore it. They concentrated instead on the much simpler physics of the electron and photon. But they couldn’t do away with the nucleus altogether. So they introduced it not as an actor, but as a stage prop. Two things made this possible.
First, the nucleus is much heavier than an electron. It is so heavy that it is almost immobile. No big mistake is made if the nucleus is replaced by an immovable point of positive electric charge.
Second, nuclei are very small by comparison with atoms. The electron orbits the nucleus at about 100,000 nuclear diameters and never gets close enough to be affected by the complicated internal nuclear structure.
According to the reductionist view of particle physics, all the phenomena of nature—solids, liquids, gases, living as well as inanimate matter—are reduced to the constant interaction and collision of electrons, photons, and nuclei. That’s the action and the whole plot—actors crashing into one another, bouncing off one another, and here and there, giving birth to new actors out of the collision. It is this banging away of particles by other particles that Feynman diagrams depict.
Feynman Diagrams
“If you come to a fork in the road, take it.”
— YOGI BERRA
We have the actors, we have the script, and now we need a stage. Shakespeare said, “All the world’s a stage,” and as usual, the Bard got it right. The set for our farce is the whole world: for a physicist that means all of ordinary three-dimensional space. Up-down, east-west, and north-south are the three directions near the surface of the earth. But a stage direction involves not only where an action takes place, but also when it takes place. Thus, there is a fourth direction to space-time: past-future. Ever since Einstein’s discovery of the Special Theory of Relativity, physicists have been in the habit of picturing the world as a four-dimensional space-time that encompasses not only the now, but also all of the future and the past. A point in space-time—a where and a when—is called an event.
A sheet of paper or a blackboard can be used to represent space-time. Because the paper or the blackboard has only two dimensions, we’l
l have to cheat a bit. The horizontal direction on the paper will be a stand-in for all three directions of space. We will have to stretch our imagination and pretend that the horizontal axis is really three perpendicular axes. That leaves us with the vertical direction to represent time. The future is usually taken to be up and the past, down (that of course is just as arbitrary as the fact that maps place the northern hemisphere above the southern). A point on the sheet of paper is an event—a where and a when: a space-time point. This is where Feynman began: particles, events, and space-time.
Our first Feynman diagram depicts the simplest of all stage directions: “Electron, go from point a to point b.” To represent this graphically, draw a line on a piece of paper from event a to event b. Feynman also put a little arrow on the line whose purpose will become clear shortly. The line connecting a with b is called a propagator.
The photon also can move from one space-time point to another. To depict the photon’s motion, Feynman drew another line, or propagator. Sometimes the photon propagator is drawn as a wavy line, sometimes as a dashed line. I will use the dashed line.
Propagators are more than just pictures. They are quantum-mechanical instructions for calculating the probability that a particle starting at point a will show up later at point b. Feynman had the radical idea that a particle doesn’t merely move along a particular path: in an odd way it feels out all paths—random zigzag paths as well as straight paths. We saw a bit of this quantum weirdness in the two-slit experiment. Photons don’t go through just the left slit or the right slit: they somehow sample both paths and in the process create the surprising interference patterns where they are detected. According to Feynman’s theory all possible paths contribute to the probability for the particle to go from a to b. In the end a particular mathematical expression representing all possible paths between the two points gives the probability to go from a to b. All of this is implicit in the notion of a propagator.
Nothing of very great interest would ever happen if all that ever took place were the free motion of electrons and photons. But they both take part in one coordinated action that is responsible for everything interesting in nature. Recall what happens when electrons move from one cloud to another during a lightning storm. Night suddenly turns into day. Light emitted by the sudden violent electric current dramatically illuminates the sky for an instant. Where does that light come from? The answer traces back to individual electrons. When the motion of an electron is suddenly disturbed, it may respond by shaking off a photon. The process, called photon emission, is the basic event of Quantum Electrodynamics. Just as all matter is built of particles, all processes are built from the elementary events of emission and absorption. Thus, the electron—while moving through space-time—can suddenly shoot out a single quantum (or photon) of light. All the visible light that we see, as well as radio waves, infrared radiation, and X-rays, is composed of photons that have been emitted by electrons, either in the sun, the filament of a lightbulb, a radio antenna, or an X-ray machine. Thus, Feynman added to the list of particles a second list: a list of elementary events. This brings us to a second kind of Feynman diagram.
The Feynman diagram representing the event of photon emission is called a vertex diagram. A vertex diagram looks like the letter Y, or better yet, a forked road: the original electron comes to the fork and shoots off a photon. Subsequently, the electron takes one path, and the photon, the other. The point where the three lines join—the event that emits the photon—is the vertex.
Here is a way to view a Feynman diagram as a short “movie.” Get a square of cardboard a few inches on a side and make a long thin slit about one sixteenth of an inch wide. Now place the square over the Feynman diagram (first fill in the dashed lines) with the slit oriented in the horizontal direction. The short line segments showing through the slit represent particles. Start the slit at the bottom of the diagram. If you now move the slit up, you will see the particles move, emit, and absorb other particles and do all the things that real particles do.
The vertex diagram can be turned upside down (remember, past is down, and future is up) so that it describes an electron and a photon approaching each other. The photon gets absorbed, leaving only the lone electron.
Antimatter
Feynman had a purpose in mind when he put little arrows on the electron lines. Each type of electrically charged particle, such as the electron and proton, has a twin, namely, its antiparticle. The antiparticle is identical to its twin, with one exception: it has the opposite electric charge. When matter meets antimatter, look out! The particles and antiparticles will combine and disappear (annihilate), but not without leaving over their energy in the form of photons.
The antiparticle twin of the electron is called the positron. It appears to be a new addition to the list of particles, but according to Feynman, the positron is not really a new object: he thought of it as an electron going backward in time! A positron propagator looks exactly like an electron propagator except that the little arrow points downward toward the past instead of upward toward the future.
Whether you think of a positron as an electron going backward in time or an electron as a positron going backward in time is up to you. It’s an arbitrary convention. But with this way of thinking, you can flip the vertex in new ways. For example, you can flip it so that it describes a positron emitting a photon.
You can even turn it on its side so that it shows an electron and a positron annihilating and leaving only a single photon or a photon disappearing and becoming an electron and a positron.
Feynman combined these basic ingredients, propagators and vertices, to make more complex processes. Here is an interesting one.
Can you see what it describes? If you use the cardboard-with-slit to view the diagram, here is what you will see: initially, at the lower part of the diagram, there are only an electron and a photon. Without warning, the photon spontaneously becomes an electron-positron pair. Then the positron moves toward the electron, where it meets its twin, and together they annihilate, leaving a photon. In the end there are a single photon and a single electron.
Feynman had another way to think about such diagrams. He pictured the incoming electron as “turning around in time” and temporarily moving toward the past, then turning around again toward the future. The two ways of thinking—either in terms of positrons and electrons or in terms of electrons moving backward in time—are completely equivalent. Propagators and vertices: that’s all there is to the world. But these basic elements can be combined in an infinite variety of ways to describe all of nature.
But aren’t we missing something important? Objects in nature exert forces on one another. The idea of force is deeply intuitive. It is one of the few concepts in physics that nature has equipped us to understand without consulting a textbook. A man pushing a boulder is exerting a force. The boulder is resisting by pushing back. The gravitational attraction of the earth keeps us from floating away. Magnets exert forces on pieces of iron. Static electricity exerts forces on bits of paper. Bullies shove wimps. The idea of force is so basic to our lives that evolution made sure that we had a concept of force built into our neural circuitry. But much less intuitive is the fact that all forces originate from attraction and repulsion between elementary particles.
Did Feynman have to add a separate set of ingredients to the recipe: specific rules of force between particles? He did not.
All forces in nature derive from special exchange diagrams, in which a particle like a photon is emitted by one particle and absorbed by another. For example, the electric force between electrons comes from a Feynman diagram in which one electron emits a photon, which is subsequently absorbed by the other electron.
The photon jumping across the gap between the electrons is the origin of the electric and magnetic forces between them. If the electrons are at rest, the force is the usual electrostatic force that famously diminishes according to the square of the distance between the charges.12 If the electrons happen to be moving,
there is an additional magnetic force. The origin of both the electric and magnetic force is the same basic Feynman diagram.
Electrons are not the only particles that can emit photons. Any electrically charged particle can, including a proton. This means that photons can hop between two protons or even between a proton and an electron. This fact is of enormous importance to all of science and life in general. The continual exchange of photons between the nucleus and the atomic electrons provides the force that holds the atom together. Without those jumping photons, the atom would fly apart, and all matter would cease to exist.
Tremendously complicated Feynman diagrams—networks of vertices and propagators—represent complex processes involving any number of particles. In this way Feynman’s theory describes all matter from the simplest to the most complicated objects.
Feel free to add arrows to this picture in various ways to make the solid lines into electrons or positrons.