QED: The Strange Theory of Light and Matter

What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school... It is my task to convince you not to turn away because you don't understand it. You see my physics students don't understand it... That is because I don't understand it. Nobody does.

There is a most profound and beautiful question associated with the observed coupling constant, e - the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!

atoms in the air scatter light from the sun and make the sky blue

Mysteries like these repeating cycles make it very interesting to be a theoretical physicist: Nature gives us such wonderful puzzles! Why does She repeat the electron at 206 times and 3,640 times its mass?

Figure 18. As the thickness of a layer increases, the two surfaces produce a partial reflection of monochromatic light whose probability fluctuates in a cycle from 0% to 16%. Since the speed of the imaginary stopwatch hand is different for different colors of light, the cycle repeats itself at different rates. Thus when two colors such as pure red and pure blue are aimed at the layer, a given thickness will reflect only red, only blue, both red and blue in different proportions (which produce various hues of violet), or neither color (black). If the layer is of varying thicknesses, such as a drop of oil spreading out on a mud puddle, all of the combinations will occur. In sunlight, which consists of all colors, all sorts of combinations occur, which produce lots of colors.

Figure 36. A "trick" can be played on Nature by slowing down the light that takes shorter paths: glass of just the right thickness is inserted so that all the paths will take exactly the same time. This causes all of the arrows to point in the same direction, and to produce a whopping final arrow-lots of light! Such a piece of glass made to greatly increase the probability of light getting from a source to a single point is called a focusing lens.

To understand this better, we need to know that the cycle of zero to 16% partial reflection by two surfaces repeats more quickly for blue light than for red light. Thus at certain thicknesses, one or the other or both colors are strongly reflected, while at other thicknesses, reflections of both colors is cancelled out (see Fig. 18). The cycles of reflection repeat at different rates because the stopwatch hand turns around faster when it times a blue photon than it does when timing a red photon. In fact, that's the only difference between a red photon and a blue photon (or a photon of any other color, including radio waves, X-rays, and so on)-the speed of the stopwatch hand.

With the thinnest possible layer of glass, we find that the number of photons arriving at A is nearly always zero-sometimes it's like 1. When we replace the thinnest layer with a slightly thicker one, we find that the amount of light reflected is higher-closer to the expected 8%. After a few more replacements the count of photons arriving at A increases past the 8% mark. As we continue to substitute still "thicker " layers of glass-we're up to about 5 millionths of an inch now-the amount of light reflected by the two surfaces reaches a maximum of 16%, and then goes down, through 8%, back to zero-if the layer of glass is just the right thickness, there is no reflection at all. (Do that with spots!) With gradually thicker and thicker layers of glass, partial reflection again increases to 16% and returns to zero-a cycle that repeats itself again and again(see Fig. 5). Newton discovered these oscillations and did one experiment that could be correctly interpreted only if the oscillations continued for 34,000 cycles! Today, with lasers (which produce a very pure, monochromatic light), we can see this cycle still going strong after more than 100,000,000 repetitions-which corresponds to glass that is more than 50 meters thick. (We don't see this phenomenon every day because the light source is normally not monochromatic.) So it turns out that our prediction of 8% is right as an overall average (since the actual amount varies in a regular pattern from zero to 16%), but it's exactly right only twice each cycle-like a stopped clock (which is right twice a day).

It is hard to believe that nearly all the vast apparent variety in Nature results from the monotony of repeatedly combining just these three basic actions. But it does. I'll outline a bit of how some of this variety arises.

Why are all the theories of physics so similar in their structure? There are a number of possibilities. The first is the limited imagination of physicists: when we see a new phenomenon we try to fit it into the framework we already have-until we have made enough experiments, we don't know that it doesn't work. Another possibility is that it is the same damn thing over and over again-that Nature has only one way of doing things, and She repeats her story from time to time. A third possibility is that things look similar because they are aspects of the same thing- some larger picture underneath, from which things can be broken into parts that look different, like fingers on the same hand. Many physicists are working very hard trying to put together a grand picture that unifies everything into one super-duper model. It's a delightful game, but at the present time none of the speculators agree with any of the other speculators as to what the grand picture is.

Whether the proton decays or not is not known. To prove that it does not decay is very difficult.

Throughout this entire story there remains one especially unsatisfactory feature: the observed masses of the particles, m. There is no theory that adequately explains these numbers. We use the numbers in all our theories, but we don't understand them-what they are, or where they come from. I believe that from a fundamental point of view, this is a very interesting and serious problem.

Now, let's look again at the partial reflection of light by a layer of glass. How does it work? I talked about light reflected from the front surface and the back surface. This idea of surfaces was a simplification I made in order to keep things easy at the beginning. Light is really not affected by surfaces. An incoming photon is scattered by the electrons in the atoms inside the glass, and a new photon comes back up to the detector. It's interesting that instead of adding up all the billions of tiny arrows that represent the amplitude for all the electrons inside the glass to scatter an incoming photon, we can add just two arrows-for the "front surface" and "back surface" reflections-and come out with the same answer. Let's see why.

the price of gaining such an accurate theory has been the erosion of our common sense.

The backwards-moving electron when viewed with time moving forwards appears the same as an ordinary electron, except it's attracted to normal electrons-we say it has a "positive charge." (Had I included the effects of polarization, it would be apparent why the sign of j for the backwards-moving electron appears reversed, making the charge appear positive.) For this reason it's called a "positron." The positron is a sister particle to the electron, and is an example of an "anti-particle." This phenomenon is general. Every particle in Nature has an amplitude to move backwards in time, and therefore has an anti-particle. When a particle and its anti-particle collide, they annihilate each other and form other particles. (For positrons and electrons annihilating, it is usually a photon or two.) And what about photons? Photons look exactly the same in all respects when they travel backwards in time-as we saw earlier-so they are their own anti-particles. You see how clever we are at making an exception part of the rule!

The probability of an event is always represented by a single final arrow-no matter how many arrows were drawn, multiplied, and added to achieve it.

A source of white light-many colors mixed together-emits photons in a chaotic manner: the angle of the amplitude changes abruptly and irregularly in fits and starts. But when we construct a monochromatic source, we are making a device that has been carefully arranged so that the amplitude for a photon to be emitted at a certain time is easily calculated: it changes its angle at a constant speed, like a stopwatch hand. (Actually, this arrow turns at the same speed as the imaginary stopwatch we used before, but in the opposite direction-see Fig. 67.)

Thus, some things that satisfy the rules of algebra can be interesting to mathematicians even though they don't always represent a real situation. Arrows on a plane can be "added" by putting the head of one arrow on the tail of another, or "multiplied" by successive turns and shrinks. Since these arrows obey the same rules of algebra as regular numbers, mathematicians call them numbers. But to distinguish them from ordinary numbers, they're called "complex numbers." For those of you who have studied mathematics enough to have come to complex numbers, I could have said, "the probability of an event is the absolute square of a complex number. When an event can happen in alternative ways, you add the complex numbers; when it can happen only as a succession of steps, you multiply the complex numbers." Although it may sound more impressive that way, I have not said any more than I did before-I just used a different language.

So now, I present to you the three basic actions, from which all the phenomena of light and electrons arise. -ACTION #1: A photon goes from place to place. -ACTION #2: An electron goes from place to place. -ACTION #3: An electron emits or absorbs a photon. Each of these actions has an amplitude-an arrow-that can be calculated according to certain rules.

To summarize, where the time is least is also where the time for the nearby paths is nearly the same; that's where the little arrows point in nearly the same direction and add up to a substantial length; that's where the probability of a photon reflecting off a mirror is determined. And that's why, in approximation, we can get away with the crude picture of the world that says that light only goes where the time is least (and it's easy to rpve that were the time is least, the angle of incidence is equal to the angle of reflection, but I don't have the time to show you). So the theory of quantum electrodynamics gave the right answer-the middle of the mirror is the important part for reflection-but this correct result came out at the expense of believing that light reflects all over the mirror, and having to add a bunch of little arrows together whose sole purpose was to cancel out. All that might seem to you to be a waste of time-some silly game for mathematicians only. After all, it doesn't seem like "real physics" to have something there that only cancels out!

Throughout these lectures I have delighted in showing you that the price of gaining such an accurate theory has been the erosion of our common sense. We must accept some very bizarre behavior: the amplification and suppression of probabilities, light reflecting from all parts of a mirror, light travelling in paths other than a straight line, photons going faster or slower than the conventional speed of light, electrons going backwards in time, photons suddenly disintegrating into a positron-electron pair, and so on. That we must do, in order to appreciate what Nature is really doing underneath nearly all the phenomena we see in the world.

The lay reader only wanted to have the illusion of understanding and to catch a few buzzwords to throw around at cocktail parties.

In this intuitively easy analysis, the "front surface" and "back surface" arrows are mathematical constructions that give us the right answer, whereas the analysis we just did-with the space-time drawing and the arrows forming part of a circle-is a more accurate representation of what is really going on: partial reflection is the scattering of light by electrons inside the glass.

Thus light is something like raindrops-each little lump of light is called a photon-and if the light is all one color, all the "raindrops" are the same size.

Underneath so many of the phenomena we see every day are only three basic actions: one is described by the simple coupling number, j; the other two by functions-P(A to B) and E(A to B)- both of which are closely related. That's all there is to it, and from it all the rest of the laws of physics come.

In this way, great globs of physics theory were synthesized into a simplified theory. The theory of gravitation, on the other hand, was not understandable from the laws of motion, and even today it stands isolated from the other theories. Gravitation is, so far, not understandable in terms of other phenomena.

It is to be emphasized that no matter how many arrows we draw, add, or multiply, our objective is to calculate a single final arrow for the event. Mistakes are often made by physics students at first because they do not keep this important point in mind. They work for so long analyzing events involving a single photon that they begin to think that the arrow is somehow associated with the photon. But these arrows are probability amplitudes, that give, when squared, the probability of a complete event.

Most of the phenomena you are familiar with involve the interaction of light and electrons-all of chemistry and biology, for example. The only phenomena that are not covered by this theory are phenomena of gravitation and nuclear phenomena; everything else is contained in this theory.

When we deal with probabilities under ordinary circumstances, there are the following "rules of composition": 1) if something can happen in alternative ways, we add the probabilities for each of the different ways; 2) i the event occurs as a succession of steps-or depends on a number of things happening "concomitantly" (independently)-then we multiply the probabilities of each of the steps (or things).

The chance that an atom emits a photon is enhanced if some photons (in a state that the atom can emit into) are already present. This phenomenon of "stimulated emission" was discovered by Einstein when he launched the quantum theory proposing the photon model of light. Lasers work on the basis of this phenomenon.