zotero/storage/V9C64Z3Z/.zotero-ft-cache

T h e Strange T h eory of' Light and Matter
Richard P. Feynman

THE STRANGE THEORY OF LIGHT AND MATTER
RICHARD P. FEYNMAN
PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS

Contents

Foreword by Leonard Mautner

Vll

Preface by Ralph Leighton

IX

Acknowledgment

XI

I. Introduction

2. Photons: Particles of Light

36

3. Electrons and Their Interactions

77

4. Loose Ends

124

Index

153

Foreword
The Alix G. Mautner Memorial Lectures were conceived in honor of my wife Alix, who died in 1982. Although her career was in English literature, Alix had a long and abiding interest in many scientific fields. Thus it seemed fitting to create a fund in her name that would support an annual lecture series with the objective of communicating to an intelligent and interested public the spirit and achievements of science.
I am delighted that Richard Feynman has agreed to give the first series of lectures. Our friendship goes back fiftyfive years to our childhood in Far Rockaway, New York. Richard knew Alix for about twenty-two years, and she long sought to have him develop an explanation of the physics of small particles that would be understandable to her and to other non-physicists.
As an added note, I would like to express my appreciation to those who contributed to the Alix G. Mautner Fund and thus helped make these lectures possible.
LEONARD MAUTNER
Los Angeles, California May 1983

Preface
Richard Feynman is legendary in the world of physics for the way he looks at the world: taking nothing for granted and always thinking things out for himself, he often attains a new and profound understanding of nature's behaviorwith a refreshing and elegantly simple way to describe it.
He is also known for his enthusiasm in explaining physics to students. After turning down countless offers to give speeches at prestigious societies and organizations, Feynman is a 1,ucker for the student who comes by his office and asks him to talk to the local high school physics club.
This book is a venture that, as far as we know, has never been tried. It is a straightforward, honest explanation of a rather difficult subject-the theory of quantum electrodynamics-for a nontechnical audience. It is designed to give the interested reader an appreciation for the kind of thinking that physicists have resorted to in order to explain how Nature behaves.
If you are planning to study physics (or are already doing so). there is nothing in this book that has to be "unlearned": it is a complete description, accurate in every detail, of a framework onto which more advanced concepts can be attached without modification. For those of you who have already studied physics, it is a revelation of what you were really doing when you were making all those complicated calculations!
As a boy, Richard Feynman was inspired to study calculus from a book that began, "What one fool can do, another

X

Preface

can." He would like to dedicate this book to his readers with similar words: "What one fool can understand, another can."

RALPH LEIGHTON
Pasadena, California February 1985

Acknowledgment
This book purports to be a record of the lectures on quantum electrodynamics I gave at UCLA, transcribed andedited by my good friend Ralph Leighton. Actually, the manuscript has undergone considerable modification. Mr. Leighton's experience in teaching and in writing was of considerable value in this attempt at presenting this central part of physics to a wider audience.
Many "popular" expositions of science achieve apparent simplicity only by describing something different, something considerably distorted from what they claim to be describing. Respect for our subject did not permit us to do this. Through many hours of discussion, we have tried to achieve maximum clarity and simplicity without compromise by distortion of the truth.

1
Introduction
Alix Mautner was very curious about physics and often asked me to explain things to her. I would do all right, just as I do with a group of students at Caltech that come to me for an hour on Thursdays, but eventually I'd fail at what is to me the most interesting part: We would always get hung up on the crazy ideas of quantum mechanics. I told her I couldn't explain these ideas in an hour or an evening-it would take a long time-but I promised her that someday I'd prepare a set of lectures on the subject.
I prepared some lectures, and I went to New Zealand to try them out-because New Zealand is far enough away that if they weren't successful, it would be all right! Well, the people in New Zealand thought they were okay, so I guess they're okay-at least for New Zealand! So here are the lectures I really prepared for Alix, but unfortunately I can't tell them to her directly, now.
What I'd like to talk about is a part of physics that is known, rather than a part that is unknown. People are always asking for the latest developments in the unification of this theory with that theory, and they don't give us a chance to tell them anything about one of the theories that we know pretty well. They always want to know things that we don't know. So, rather than confound you with a lot of half-cooked, partially analyzed theories, I would like to tell you about a subject that has been very thoroughly analyzed.

4

Chapter 1

I love this area of physics and I think it's wonderful: it is called quantum electrodynamics, or QED for short.
My main purpose in these lectures is to describe as accurately as I can the strange theory of light and matteror more specifically, the interaction of light and electrons. It's going to take a long time to explain all the things I want to. However, there are four lectures, so I'm going to take my time, and we will get everything all right.
Physics has a history of synthesizing many phenomena into a few theories. For instance, in the early days there were phenomena of motion and phenomena of heat; there were phenomena of sound, of light, and of gravity. But it was soon discovered, after Sir Isaac Newton explained the laws of motion, that some of these apparently different things were aspects of the same thing. For example, the phenomena of sound could be completely understood as the motion of atoms in the air. So sound was no longer considered something in addition to motion. It was also discovered that heat phenomena are easily understandable from the laws of motion. 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.
After the synthesis of the phenomena of motion, sound, and heat, there was the discovery of a number of phenomena that we call electrical and magnetic. In 1873 these phenomena were synthesized with the phenomena of light and optics into a single theory by James Clerk Maxwell, who proposed that light is an electromagnetic wave. So at that stage, there were the laws of motion, the laws of electricity and magnetism, and the laws of gravity.
Around 1900 a theory was developed to explain what matter was. It was called the electron theory of matter, and

Introduction

5

it said that there were little charged particles inside of atoms. This theory evolved gradually to include a heavy nucleus with electrons going around it.
Attempts to understand the motion of the electrons going around the nucleus by using mechanical laws--analogous to the way Newton used the laws of motion to figure out how the earth went around the sun-were a real failure: all kinds of predictions came out wrong. (Incidentally, the theory of relativity, which you all understand to be a great revolution in physics, was also developed at about that time. But compared to this discovery that Newton's laws of motion were quite wrong in atoms, the theory of relativity was only a minor modification.) Working out another system to replace Newton's laws took a long time because phenomena at the atomic level were quite strange. One had to lose one's common sense in order to perceive what was happening at the atomic level. Finally, in 1926, an "uncommon-sensy" theory was developed to explain the "new type of behavior" of electrons in matter. It looked cockeyed, but in reality it was not: it was called the theory of quantum mechanics. The word "quantum" refers to this peculiar aspect of nature that goes against common sense. It is this aspect that I am going to tell you about.
The theory of quantum mechanics also explained all kinds of details, such as why an oxygen atom combines with two hydrogen atoms to make water, and so on. Quantum mechanics thus supplied the theory behind chemistry. So, fundamental theoretical chemistry is really physics.
Because the theory of quantum mechanics could explain all of chemistry and the various properties of substances, it was a tremendous success. But still there was the problem of the interaction of light and matter. That is, Maxwell's theory of electricity and magnetism had to be changed to be in accord with the new principles of quantum mechanics that had been developed. So a new theory, the quantum

6

Chapter 1

theory of the interaction of light and matter, which is called by the horrible name "quantum electrodynamics," was finally developed by a number of physicists in 1929.
But the theory was troubled. If you calculated something roughly, it would give a reasonable answer. But if you tried to compute it more accurately, you would find that the correction you thought was going to be small (the next term in a series, for example) was in fact very large-in fact, it was infinity! So it turned out you couldn't really compute anything beyond a certain accuracy.
By the way, what I have just outlined is what I call a "physicist's history of physics," which is never correct. What I am telling you is a sort of conventionalized myth-story that the physicists tell to their students, and those students tell to their students, and is not necessarily related to the actual historical development, which I do not really know!
At any rate, to continue with this "history," Paul Dirac, using the theory of relativity, made a relativistic theory of the electron that did not completely take into account all the effects of the electron's interaction with light. Dirac's theory said that an electron had a magnetic moment-something like the force of a little magnet-that had a strength of exactly l in certain units. Then in about 1948 it was discovered in experiments that the actual number was closer to 1.00118 (with an uncertainty of about 3 on the last digit). It was known, of course, that electrons interact with light, so some small correction was expected. It was also expected that this correction would be understandable from the new theory of quantum electrodynamics. But when it was calculated, instead of 1.00118 the result was infinity-which is wrong, experimentally l
Well, this problem of how to calculate things in quantum electrodynamics was straightened out by Julian Schwinger, Sin-Itiro Tomonaga, and myself in about 1948. Schwinger was the first to calculate this correction using a new "shell

Introduction

7

game"; his theoretical value was around l.00116, which was close enough to the experimental number to show that we were on the right track. At last, we had a quantum theory of electricity and magnetism with which we could calculate! This is the theory that I am going to describe to you.
The theory of quantum electrodynamics has now lasted for more than fifty years, and has been tested more and more accurately over a wider and wider range of conditions. At the present time I can proudly say that there is no significant difference between experiment and theory!
Just to give you an idea of how the theory has been put through the wringer, I'll give you some recent numbers: experiments have Dirac's number at l.00115965221 (with an uncertainty of about 4 in the last digit); the theory puts it at 1.00115965246 (with an uncertainty of about five times as much). To give you a feeling for the accuracy of these numbers, it comes out something like this: If you were to measure the distance from Los Angeles to New York to this accuracy, it would be exact to the thickness of a human hair. That's how delicately quantum electrodynamics has, in the past fifty years, been checked-both theoretically and experimentally. By the way, I have chosen only one number to show you. There are other things in quantum electrodynamics that have been measured with comparable accuracy, which also agree very well. Things have been checked at distance scales that range from one hundred times the size of the earth down to one-hundredth the size of an atomic nucleus. These numbers are meant to intimidate you into believing that the theory is probably not too far off! Before we're through, I'll describe how these calculations are made.
I would like to again impress you with the vast range of phenomena that the theory of quantum electrodynamics describes: It's easier to say it backwards: the theory de-

8

Chapter l

scribes all the phenomena of the physical world except the gravitational effect, the thing that holds you in your seats (actually, that's a combination of gravity and politeness, I think), and radioactive phenomena, which involve nuclei shifting in their energy levels. So if we leave out gravity and radioactivity (more properly, nuclear physics), what have we got left? Gasoline burning in automobiles, foam and bubbles, the hardness of salt or copper, the stiffness of steel. In fact, biologists are trying to interpret as much as they can about life in terms of chemistry, and as I already explained, the theory behind chemistry is quantum electrodynamics.
I must clarify something: When I say that all the phenomena of the physical world can be explained by this theory, we don't really know that. Most phenomena we are familiar with involve such tremendous numbers of electrons that it's hard for our poor minds to follow that complexity. In such situations, we can use the theory to figure roughly what ought to happen and that is what happens, roughly, in those circumstances. But if we arrange in the laboratory an experiment involving just a few electrons in simple circumstances, then we can calculate what might happen very accurately, and we can measure it very accurately, too. Whenever we do such experiments, the theory of quantum electrodynamics works very well.
We physicists are always checking to see if there is something the matter with the theory. That's the game, because if there is something the matter, it's interesting! But so far, we have found nothing wrong with the theory of quantum electrodynamics. It is, therefore, I would say, the jewel of physics-our proudest possession.
The theory of quantum electrodynamics is also the prototype for new theories that attempt to explain nuclear phenomena, the things that go on inside the nuclei of atoms. If one were to think of the physical world as a stage,

Introduction

9

then the actors would be not only electrons, which are outside the nucleus in atoms, but also quarks and gluons and so forth-dozens of kinds of particles-inside the nucleus. And though these "actors" appear quite different from one another, they all act in a certain style-a strange and peculiar style-the "quantum'' style. At the end, I'll tell you a little bit about the nuclear particles. In the meantime, I'm only going to tell you about photons-particles of lightand electrons, to keep it simple. Because it's the way they act that is important, and the way they act is very interesting.
So now you know what I'm going to talk about. The next question is, will you understand what I'm going to tell you? Everybody who comes to a scientific lecture knows they are not going to understand it, but maybe the lecturer has a nice, colored tie to look at. Not in this case! (Feynman is not wearing a tie.)
What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school-and you think I'm going to explain it to you so you can understand it? No, you're not going to be able to understand it. Why, then, am I going to bother you with all this? Why are you going to sit here all this time, when you won't be able to understand what I am going to say? 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 either. That is because I don't understand it. Nobody does.
I'd like to talk a little bit about understanding. When we have a lecture, there are many reasons why you might not understand the speaker. One is, his language is bad-he doesn't say what he means to say, or he says it upside down-and it's hard to understand. That's a rather trivial matter, and I'll try my best to avoid too much of my New York accent.

10

Chapter 1

Another possibility, especially if the lecturer is a physicist, is that he uses ordinary words in a funny way. Physicists often use ordinary words such as "work'' or "action" or "energy" or even, as you shall see, "light" for some technical purpose. Thus, when I talk about "work" in physics, I don't mean the same thing as when I talk about "work" on the street. During this lecture I might use one of those words without noticing that it is being used in this unusual way. I'll try my best to catch myself-that's my job-but it is an error that is easy to make.
The next reason that you might think you do not understand what I am telling you is, while I am describing to you how Nature works, you won't understand why Nature works that way. But you see, nobody understands that. I can't explain why Nature behaves in this peculiar way.
Finally, there is this possibility: after I tell you something, you just can't believe it. You can't accept it. You don't like it. A little screen comes down and you don't listen anymore. I'm going to describe to you how Nature is-and if you don't like it, that's going to get in the way of your understanding it. It's a problem that physicists have learned to deal with: They've learned to realize that whether they like a theory or they don't like a theory is not the essential question. Rather, it is whether or not the theory gives predictions that agree with experiment. It is not a question of whether a theory is philosophically delightful, or easy to understand, or perfectly reasonable from the point of view of common sense. The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense. And it agrees fully with experiment. So I hope you can accept Nature as She is-absurd.
I'm going to have fun telling you about this absurdity, because I find it delightful. Please don't turn yourself off because you can't believe Nature is so strange. Just hear me all out, and I hope you'll be as delighted as I am when we're through.

Introduction

11

How am I going to explain to you the things I don't explain to my students until they are third-year graduate students? Let me explain it by analogy. The Maya Indians were interested in the rising and setting of Venus as a morning "star" and as an evening "star"-they were very interested in when it would appear. After some years of observation, they noted that five cycles of Venus were very nearly equal to eight of their "nominal years" of 365 days (they were aware that the true year of seasons was different and they made calculations of that also). To make calculations, the Maya had invented a system of bars and dots to represent numbers (including zero), and had rules by which to calculate and predict not only the risings and settings of Venus, but other celestial phenomena, such as lunar eclipses.
In those days, only a few Maya priests could do such elaborate calculations. Now, suppose we were to ask one of them how to do just one step in the process of predicting when Venus will next rise as a morning star-subtracting two numbers. And let's assume that, unlike today, we had not gone to school and did not know how to subtract. How would the priest explain to us what subtraction is?
He could either teach us the numbers represented by the bars and dots and the rules for "subtracting" them, or he could tell us what he was really doing: "Suppose we want to subtract 236 from 584. First, count out 584 beans and put them in a pot. Then take out 236 beans and put them to one side. Finally, count the beans left in the pot. That number is the result of subtracting 236 from 584."
You might say, "My Quetzalcoatl! What tedium-counting beans, putting them in, taking them out-what a job!"
To which the priest would reply, "That's why we have the rules for the bars and dots. The rules are tricky, but they are a much more efficient way of getting the answer than by counting beans. The important thing is, it makes no difference as far as the answer is concerned: we can

12

Chapter I

predict the appearance of Venus by counting beans (which is slow, but easy to understand) or by using the tricky rules (which is much faster, but you must spend years in school to learn them)."
To understand how subtraction works-as long as you don't have to actually carry it out-is really not so difficult. That's my position: I'm going to explain to you what the physicists are doing when they are predicting how Nature will behave, but I'm not going to teach you any tricks so you can do it efficiently. You will discover that in order to make any reasonable predictions with this new scheme of quantum electrodynamics, you would have to make an awful lot of little arrows on a piece of paper. It takes seven years-four undergraduate and three graduate-to train our physics students to do that in a tricky, efficient way. That's where we are going to skip seven years of education in physics: By explaining quantum electrodynamics to you in terms of what we are really doing, I hope you will be able to understand it better than do some of the students!
Taking the example of the Maya one step further, we could ask the priest why five cycles of Venus nearly equal 2,920 days, or eight years. There would be all kinds of theories about why, such as, "20 is an important number in our counting system, and if you divide 2,920 by 20, you get 146, which is one more than a number that can be represented by the sum of two squares in two different ways," and so forth. But that theory would have nothing to do with Venus, really. In modern times, we have found that theories of this kind are not useful. So again, we are not going to deal with why Nature behaves in the peculiar way that She does; there are no good theories to explain that.
What I have done so far is to get you into the right mood to listen to me. Otherwise, we have no chance. So now we're off, ready to go!

Introduction

13

We begin with light. When Newton started looking at light, the first thing he found was that white light is a mixture of colors. He separated white light with a prism into various colors, but when he put light of one color-red, for instance-through another prism, he found it could not be separated further. So Newton found that white light is a mixture of different colors, each of which is pure in the sense that it can't be separated further.
(In fact, a particular color of light can be split one more time in a different way, according to its so-called "polarization." This aspect of light is not vital to understanding the character of quantum electrodynamics, so for the sake of simplicity I will leave it out-at the expense of not giving you an absolutely complete description of the theory. This slight simplification will not remove, in any way, any real understanding of what I will be talking about. Still, I must be careful to mention all of the things I leave out.)
When I say "light" in these lectures, I don't mean simply the light we can see, from red to blue. It turns out that visible light is just a part of a long scale that's analogous to a musical scale in which there are notes higher than you can hear and other notes lower than you can hear. The scale of light can be described by numbers-called the frequency-and as the numbers get higher, the light goes from red to blue to violet to ultraviolet. We can't see ultraviolet light, but it can affect photographic plates. It's still lightonly the number is different. (We shouldn't be so provincial: what we can detect directly with our own instrument, the eye, isn't the only thing in the world!) If we continue simply to change the number, we go out into Xrays, gamma rays, and so on. If we change the number in the other direction, we go from blue to red to infrared (heat) waves, then television waves, and radio waves. For me, all of that is "light." I'm going to use just red light for most of my examples, but the theory of quantum electro-

14

Chapter 1

dynamics extends over the entire range I have described, and is the theory behind all these various phenomena.
Newton thought that light was made up of particles-he called them "corpuscles"-and he was right (but the reasoning that he used to come to that decision was erroneous). We know that light is made of particles because we can take a very sensitive instrument that makes clicks when light shines on it, and if the light gets dimmer, the clicks remain just as loud-there are just fewer of them. 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.
The human eye is a very good instrument: it takes only about five or six photons to activate a nerve cell and send a message to the brain. If we were evolved a little further so we could see ten times more sensitively, we wouldn't have to have this discussion-we would all have seen very dim light of one color as a series of intermittent little flashes of equal intensity.
You might wonder how it is possible to detect a single photon. One instrument that can do this is called a photomultiplier, and I'll describe briefly how it works: When a photon hits the metal plate A at the bottom (see Figure 1), it causes an electron to break loose from one of the atoms in the plate. The free electron is strongly attracted to plate B (which has a positive charge on it) and hits it with enough force to break loose three or four electrons. Each of the electrons knocked out of plate B is attracted to plate C (which is also charged), and their collision with plate C knocks loose even more electrons. This process is repeated ten or twelve times, until billions of electrons, enough to make a sizable electric current, hit the last plate, L. This current can be amplified by a regular amplifier and sent through a speaker to make audible clicks. Each time

Introduction

15

a photon of a given color hits the photomultiplier, a click of uniform loudness is heard.
If you put a whole lot of photomultipliers around and let some very dim light shine in various directions, the light goes into one multiplier or another and makes a click of full intensity. It is all or nothing: if one photomultiplier

FIGURE 1. A plwtomultiplier can detect a single photon. When a photon strikes plate A, an electron is knocked loose and attracted to positively charged plate B, knocking more electrons loose. This process continues until billions of electrons strike the last plate, L, and produce an electnc current, which z.s amplified fry a regular amplifier. Ifa speaker z.s connected to the amplifier, clicks ofuniform loudness are heard each time a photon of a given color hits plate A.

goes off at a given moment, none of the others goes off at the same moment (except in the rare instance that two photons happened to leave the light source at the same time). There is no splitting of light into "half particles,, that go different places.
I want to emphasize that light comes in this form-particles. It is very important to know that light behaves like particles, especially for those of you who have gone to school, where you were probably told something about light behaving like waves. I'm telling you the way it does behavelike particles.
You might say that it's just the photomultiplier that detects light as particles, but no, every instrument that has been designed to be sensitive enough to detect weak light has always ended up discovering the same thing: light is made of particles.

16

Chapter 1

I am going to assume that you are familiar with the properties of light in everyday circumstances-things like, light goes in straight lines; it bends when it goes into water; when it is reflected from a surface like a mirror, the angle at which the light hits the surface is equal to the angle at which it leaves the surface; light can be separated into colors; you can see beautiful colors on a mud puddle when there is a little bit of oil on it; a lens focuses light, and so on. I am going to use these phenomena that you are familiar with in order to illustrate the truly strange behavior of light; I am going to explain these familiar phenomena in terms of the theory of quantum electrodynamics. I told you about the photomultiplier in order to illustrate an essential phenomenon that you may not have been familiar with-that light is made of particles-but by now, I hope you are familiar with that, too!
Now, I think you are all familiar with the phenomenon that light is partly reflected from some surfaces, such as water. Many are the romantic paintings of moonlight reflecting from a lake (and many are the times you got yourself in trouble because of moonlight reflecting from a lake!). When you look down into water you can see what's below the surface (especially in the daytime), but you can also see a reflection from the surface. Glass is another example: if you have a lamp on in the room and you're looking out through a window during the daytime, you can see things outside through the glass as well as a dim reflection of the lamp in the room. So light is partially reflected from the surface of glass.
Before I go on, I want you to be aware of a simplification I am going to make that I will correct later on: When I talk about the partial reflection of light by glass, I am going to pretend that the light is reflected by only the surface of the glass. In reality, a piece of glass is a terrible monster of complexity-huge numbers of electrons are jiggling about.

Introduction

17

When a photon comes down, it interacts with electrons throughout the glass, not just on the surface. The photon and electrons do some kind of dance, the net result of which is the same as if the photon hit only the surface. So let me make that simplification for a while. Later on, I'll show you what actually happens inside the glass so you can understand why the result is the same.
Now I'd like to describe an experiment, and tell you its surprising results. In this experiment some photons of the same color-let's say, red light-are emitted from a light source (see Fig. 2) down toward a block of glass. A photomultiplier is placed at A, above the glass, to catch any

FIGURE 2. An experiment to measure the partial reflection of light by a single surface ofglass. For every JOO photons that leave the light source, 4 are reflected by the front surface and end up in the photomultiplier at A, while the other 96 are transmitted by the front surf ace and end up in the photomultiplier at B.

photons that are reflected by the front surface. To measure how many photons get past the front surface, another photomultiplier is placed at B, inside the glass. Never mind the obvious difficulties of putting a photomultiplier inside a block of glass; what are the results of this experiment?
For every I00 photons that go straight down toward the glass at 90°, an average of 4 arrive at A and 96 arrive at B. So "partial reflection" in this case means that 4% of the photons are reflected by the front surface of the glass, while the other 96% are transmitted. Already we are in great difficulty: how can light be partly reflected? Each photon ends

18

Chapter 1

up at A or B-how does the photon "make up its mind" whether it should go to A or B? (Audience laughs.) That may sound like a joke, but we can't just laugh; we're going to have to explain that in terms of a theory! Partial reflection is already a deep mystery, and it was a very difficult problem for Newton.
There are several possible theories that you could make up to account for the partial reflection of light by glass. One of them is that 96% of the surface of the glass is "holes" that let the light through, while the other 4% of the surface is covered by small "spots" of reflective material (see Fig. 3). Newton realized that this is not a possible explanation. 1 In just a moment we will encounter a strange feature of partial reflection that will drive you crazy if you try to stick to a theory of"holes and spots"--or to any other reasonable theory!
Another possible theory is that the photons have some kind of internal mechanism-"wheels" and "gears" inside that are turning in some way-so that when a photon is "aimed" just right, it goes through the glass, and when it's not aimed right, it reflects. We can check this theory by trying to filter out the photons that are not aimed right by putting a few extra layers of glass between the source and the first layer of glass. After going through the filters, the photons reaching the glass should all be aimed right, and

1 How did he know? Newton was a very great man: he wrote, "Because I can polish glass." You might wonder, how the heck could he tell that because you can polish glass, it can't be holes and spots? Newton polished his own lenses and mirrors, and he knew what he was doing with polishing:
he was making scratches on the surface of a piece of glass with powders of increasing fineness. As the scratches become finer and finer, the surface of the glass changes its appearance from a dull grey (because the hght is scattered by the large scratches), to a transparent clanty (because the extremely fine scratches let the light through). Thus he saw that it is impossible to accept the proposition that light can be affected by very small irregularities such as scratches or holes and spots; m fact, he found the contrary to be true. The finest scratches and therefore equally small spots do not affect the light. So the holes and spots theory is no good.

Introduction

19

none of them should reflect. The trouble with that theory is, it doesn't agree with experiment: even after going through many layers of glass, 4% of the photons reaching a given surface reflect off it.
Try as we might to invent a reasonable theory that can

GLASS
FIGURE 3. One theory lo expl,a,in partial reflection by a stngle surface involves a surface made up mainly of "holes" that let light through, with a few "spots" that reflect the light.
explain how a photon "makes up its mind" whether to go through glass or bounce back, it is impossible to predict which way a given photon will go. Philosophers have said that if the same circumstances don't always produce the same results, predictions are impossible and science will collapse. Here is a circumstance-identical photons are always coming down in the same direction to the same piece of glass-that produces different results. We cannot predict whether a given photon will arrive at A or B. All we can predict is that out of 100 photons that come down, an average of 4 will be reflected by the front surface. Does this mean that physics, a science of great exactitude, has been reduced to calculating only the probability of an event, and not predicting exactly what will happen? Yes. That's a retreat, but that's the way it is: Nature permits us to calculate only probabilities. Yet science has not collapsed.
While partial reflection by a single surface is a deep mystery and a difficult problem, partial reflection by two or more surfaces is absolutely mind-boggling. Let me

20

Chapter 1

show vou why. We'll do a second experiment, in which we will measure the partial reflection of light by two surfaces. We replace. the block of glass with a very thin sheet of glass-its two surfaces are exactly parallel to each otherand we place the photomultiplier below the sheet of glass, in line with the light source. This time, photons can reflect from either the front surface or the back surface to end up at A; all the others will end up at B (see Fig. 4). We might

FIGURE 4. An experiment to measure the par-

tial reflection of light by two surfaces of glass.

Otol6

Photons can get to the photomultiplier at A by reflecting off either the front surface or the back

surface of the sheet of glass; alternatively, they

could go through both surfaces and end up hilling

the photomultiplier at B. Depending on the thick-

ness of the glass, 0 to 16 photons out ofevery 100

B

100 to 84•·· 1

get to the photomultiplier at A. These results pose difficulties for any reasonable theory, including

the one in Figure 3. It a,ppears that partial re-

flection can be "turned off" or "amplified" by the

presence of an additional surface.

expect the front surface to reflect 4% of the light and the back surface to reflect 4% of the remaining 96%, making a total of about 8%. So we should find that out of every 100 photons that leave the light source, about 8 arrive at A.
What actually happens under these carefully controlled experimental conditions is, the number of photons arriving at A is rarely 8 out of 100. With some sheets of glass, we consistently get a reading of 15 or 16 photons-twice our expected result! With other sheets of glass, we consistently get only I or 2 photons. Other sheets of glass have a partial reflection of l 0%; some eliminate partial reflection altogether! What can account for these crazy results? After checking the various sheets of glass for quality and uniformity, we discover that they differ only slightly in their thickness.

Introduction

21

To test the idea that the amount of light reflected by two surfaces depends on the thickness of the glass, let's do a series of experiments: Starting out with the thinnest possible layer of glass, we'll count how many photons hit the photomultiplier at A each time 100 photons leave the light source. Then we'll replace the layer of glass with a slightly thicker one and make new counts. After repeating this process a few dozen times, what are the results?
With the thinnest possible layer of glass, we find that the number of photons arriving at A is nearly always zero-sometimes it's I. When we replace the thinnest layer with a slightly thicker one, we find that the amount of light reflected is higher~loser 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 isjust 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 at least 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).

22

Chapter 1

How can we explain this strange feature of partial reflection that depends on the thickness of the glass? How can the front surface reflect 4% of the light (as confirmed in our first experiment) when, by putting a second surface at just the right distance below, we can somehow "turn off' the

Percentage of
reflection
: : ~ 16%b Thickness of glass - - -
Fie URE 5. The results ofan experiment carefully measuring the relationship between the thickness of a sheet of glass and partial reflection demonstrate a phenomenon called "interference." As the thickness of the glass increases, partial reflection goes through a repeating cycle of zero to 16%, with no signs of dying out.

reflection? And by placing that second surface at a slightly different depth, we can "amplify" the reflection up to 16% I Can it be that the back surface exerts some kind of influence or effect on the ability of the front surface to reflect light? What if we put in a third surface?
With a third surface, or any number of subsequent surfaces, the amount of partial reflection is again changed. We find ourselves chasing down through surface after surface with this theory, wondering if we have finally reached the last surface. Does a photon have to do that in order to "decide" whether to reflect off the front surface?
Newton made some ingenious arguments concerning this problem,2 but he realized, in the end, that he had not yet developed a satisfactory theory.
2 lt is very fortunate for us that Newton convinced himself that light is "corpuscles," because we can see what a fresh and intelligent mind looking

Introduction

23

For many years after Newton, partial reflection by two surfaces was happily explained by a theory of waves,3 but when experiments were made with very weak light hitting photomultipliers, the wave theory collapsed: as the light got dimmer and dimmer, the photomultipliers kept making

at this phenomenon of partial reflection by two or more surfaces has to go through to try to explain it. (Those who believed that light was waves never had to wrestle with it.) Newton argued as follows: Although light appears to be reflected from the first surface, it cannot be reflected from that surface. If it were, then how could light reflected from the first surface be captured again when the thickness is such that there was supposed to be no reflection at all? Then light must be reflected from the second surface. But to account for the fact that the thickness of the glass determines the amount of partial reflection, Newton proposed this idea: Light striking the first surface sets off a kind of wave or field that travels along with the light and predisposes it to reflect or not reflect off the second surface. He called this process "fits of easy reflection or easy transmission" that occur in cycles, depending on the thickness of the glass.
There are two difficulties with this idea: the first is the effect of additional surfaces---each new surface affects the reflection-which I described in the text. The other problem is that light certainly reflects off a lake, which doesn't have a second surface, so light must be reflecting off the front surface. In the case of single surfaces, Newton said that light had a predisposition to reflect. Can we have a theory in which the light knows what kind of surface it is hitting, and whether it is the only surface?
Newton didn't emphasize these difficulties with his theory of "fits of reflection and transmission," even though it is clear that he knew his theory was not satisfactory. In Newton's time, difficulties with a theory were dealt with briefly and glossed over-a different style from what we are used to in science today, where we point out the places where our own theory doesn't fit the observations of experiment. I'm not trying to say anything against Newton; I just want to say something in favor of how we communicate with each other in science today.
~ This idea made use of the fact that waves can combine or cancel out, and the calculations based on this model matched the results of Newton's experiments, as well as those done for hundreds of years afterwards. But when instruments were developed that were sensitive enough to detect a single photon, the wave theory predicted that the "dicks''of the photomultiplier would get softer and softer, whereas they stayed at full strength-they just occurred less and less often. No reasonable model could explain this fact, so there was a period for a while in which you had to be clever: You had to know which experiment you were analyzing in order to tell if light was waves or particles. This state of confusion was called the "wave-particle duality" of light, and it was jokingly said by someone that light was waves on Mondays, Wednesdays, and Fridays; it was particles on Tuesdays, Thursdays, and Saturdays, and on Sundays, we think about it! It is the purpose of these lectures to tell you how this puzzle was finally "resolved."

24

Chapter 1

full-sized clicks-there were just fewer of them. Light behaved as particles.
The situation today is, we haven't got a good model to explain partial reflection by two surfaces; we just calculate the probability that a particular photomultiplier will be hit by a photon reflected from a sheet of glass. I have chosen this calculation as our first example of the method provided by the theory of quantum electrodynamics. I am going to show you "how we count the beans"-what the physicists do to get the right answer. I am not going to explain how the photons actually "decide" whether to bounce back or go through; that is not known. (Probably the question has no meaning.) I will only show you how to calculate the correct probability that light will be reflected from glass of a given thickness, because that's the only thing physicists know how to do! What we do to get the answer to this problem is analogous to the things we have to do to get the answer to every other problem explained by quantum electrodynamics.
You will have to brace yourselves for this-not because it is difficult to understand, but because it is absolutely ridiculous: All we do is draw little arrows on a piece of paper-that's all!
Now, what does an arrow have to do with the chance that a particular event will happen? According to the rules of "how we count the beans," the probability of an event is equal to the square of the length of the arrow. For example, in our first experiment (when we were measuring partial reflection by the front surface only), the probability that a photon would arrive at the photomultiplier at A was 4%. That corresponds to an arrow whose length is 0.2, because 0.2 squared is 0.04 (see Fig. 6).
In our second experiment (when we were replacing thin sheets of glass with slightly thicker ones), photons bouncing off either the front surface or the back surface arrived at

Introduction

25

A. How do we draw an arrow to represent this situation? The length of the arrow must range from zero to 0.4 to represent probabilities of zero to 16%, depending on the thickness of the glass (see Fig. 7).
We start by considering the various ways that a photon

FIGURE 6. The strange feature ofpartial reflection by two suifaces has forced physicists away from making absolute predictions to merely calculating the probability of an event. Quantum electrodynamics provides a method for doing this-drawing little arrows on a piece of paper.
The probability of an event is represented by the area of the square on an arrow. For example, an arrow representing a probability of 0.04 (4%) has a length of 0.2.

~
~.2

could get from the source to the photomultiplier at A. Since I am making this simplification that the light bounces off either the front surface or the back surface, there are two possible ways a photon could get to A. What we do in this case is to draw two arrows---one for each way the event can happen-and then combine them into a "final arrow"
~
Vz
~ 0.1
9%
0%
()
0 03
FIGURE 7. Arrows representing /wobabilities from 0% to 16% have lengths offrom Oto 0.4.

26

Chapter 1

whose square represents the probability of the event. If there had been three different ways the event could have happened, we would have drawn three separate arrows before combinmg them.
Now, let me show you how we combine arrows. Let's say we want to combine arrow x with arrow y (see Fig. 8). All

1V/.

D
final arrow

FIGURE 8 Arrows that represent each possible
way an event could happen are drawn and then comInned ("added") zn the f ollounng manner At-
tach the head ofone arrow to the tail ofanother-
unthout changing the direction ofeither one--and draw a ''final arrow" from the tail of the first arrow to the head of the last one

we have to do is put the head of x against the tail of y
(without changing the direction of either one), and draw the final arrow from the tail of x to the head of y. That's all there is to it. We can combine any number of arrows in this manner (technically, it's called "adding arrows"). Each arrow tells you how far, and in what direction, to move in a dance. The final arrow tells you what single move to make to end up in the same place (see Fig. 9).
Now, what are the specific rules that determine the length and direction of each arrow that we combine in order to make the final arrow? In this particular case, we will be combining two arrows-one representing the reflection from the front surface of the glass, and the other representing the reflection from the back surface.
Let's take the length first. As we saw in the first experi-

Introduction

27

ment (where we put the photomultiplier inside the glass), the front surface reflects about 4% of the photons that come down. That means the "front reflection" arrow has a length of 0.2. The back surface of the glass also reflects 4%, so the "back reflection" arrow's length is also 0.2.

final orrow
FIGUU: 9. Any nuab,r ofarrows can b, added
a • aanner descri6ttl in Figure 8
To determine the direction of each arrow, let's imagine that we have a stopwatch that can time a photon as it moves. This imaginary stopwatch has a single hand that turns around very, very rapidly. When a photon leaves the source, we start the stopwatch. As long as the photon moves, the stopwatch hand turns (about 36,000 times per inch for red light); when the photon ends up at the photomultiplier, we stop the watch. The hand ends up pomting in a certain direction. That 1s the d1recuon we will draw the arrow.
We need one more rule in order to compute the answer correctly: When we are considering the path of a photon bouncing off the front surface of the glass, we reverse the direction of the arrow. In other words, whereas we draw the back reflection arrow pointing in the same direction as the stopwatch hand, we draw the front reflection arrow in the opposite direction.
Now, let's draw the arrows for the case of light reflecting from an extremely thin layer of glass. To draw the front

28

Chapter 1

reflection arrow, we imagine a photon leaving the light source (the stopwatch hand starts turning}, bouncing off the front surface, and arriving at A (the stopwatch hand stops). We draw a little arrow of length 0.2 in the direction opposite that of the stopwatch hand (see Fig. 10).

E)

FIGURE 10. In an experiment measuring reflection by two surfaces, we can say that a single photon can arrive at A in two

ways-via thefront or back surface.

stopwatch

An arrow of length 0.2 is drawn

for each way, unth its direction de-

00

31..- termined by the hand of a "stop-
front reflection watch" that times the photon as it

arrow

moves. The ''front reflection" arrow

is drawn in the direction opposite

to that of the stopwatch hand when

it stops turning.

To draw the back reflection arrow, we imagine a photon leaving the light source (the stopwatch hand starts turning), going through the front surface and bouncing off the back surface, and arriving at A (the stopwatch hand stops). This time, the stopwatch hand is pointing in almost the same direction, because a photon bouncing off the back surface of the glass takes only slightly longer to get to A-it goes through the extremely thin layer of glass twice. We now draw a little arrow of length 0.2 in the same direction that the stopwatch hand is pointing (see Fig. 11).
Now let's combine the two arrows. Since they are both the same length but pointing in nearly opposite directions, the final arrow has a length of nearly zero, and its square is even closer to zero. Thus, the probability of light reflecting from an infinitesimally thin layer of glass is essentially zero (see Fig. 12).

Introduction
FIGURE 11. A photon bovnczng off the back surface of a thin layer of glass takes slightly longer to get to A. Thus, the stopwatch hand ends up in a slightly different direction than it did when it timed the front reflection photon. The "back reflection" arrow is drawn in the same direction as the stopwatch hand.

29
E)
stopwatch
( 0,2
back reflection arrow

0.2
,0025~
FIGURE 12. The final arrow, whose square represents the probability of reflection by an extremely thin layer of glass, is drawn by adding the front reflection arrow and the back reflection arrow. The result is nearly uro.

When we replace the thinnest layer of glass with a slightly thicker one, the photon bouncing off the back surface takes a little bit longer to get to A than in the first example; the stopwatch hand therefore turns a little bit more before it stops, and the back reflection arrow ends up in a slightly greater angle relative to the front reflection arrow. The final arrow is a little bit longer, and its square is correspondingly larger (see Fig. 13).
As another example, let's look at the case where the glass is just thick enough that the stopwatch hand makes an extra half turn as it times a photon bouncing off the back surface. This time, the back reflection arrow ends up pointing in exactly the same direction as the front reflection arrow. When we combine the two arrows, we get a final arrow whose length is 0.4, and whose square is 0.16, representing a probability of 16% (see Fig. 14).
If we increase the thickness of the glass just enough so

30
E)
stopwatch
~
front ref lect1on arrow

Chapter 1

0 stopwatch
~2 bock reflection
arrow

FIGURE 13. Thefinalarrow for a slightly thu:ker sheet of gl.ass is a little longer, due to the greater relative angle between the front and back reflection arrows This is because a photon boucmg off the back surface takes a little longer to reach A, compared to the previous example

8 stopwatch
----02
front reflection arrow

G stopwatch
~
back reflection arrow

FIGURE 14. Whenthelayerof glass is JU.St thick enough to allow the stopwatch hand timing the back reflecting photon to make an extra halfturn, the front and back reflection arrows end up pointing in the same direction, resulting in afinal arrow oflength O4, which represents a probability of 16%.

Introduction

31

that the stopwatch hand timing the back surface path makes an extra full turn, our two arrows end up pointing in opposite directions again, and the final arrow will be zero (see Fig. 15). This situation occurs over and over, whenever the thickness of the glass is just enough to let the stopwatch hand timing the back surface reflection make another full turn.

FIGURE 15. When the sheet of glass zs JU.St the nght thickness to allow the stopwatch hand timing the back reflecting photon to make one or more extra full turns, the final arrow zs again zero, and there zs no reflection at all.

E) E)

stopwatch
~
ffant reflection
arrow
02

stopwatch
.,0_2 -
bock reflection arrow

If the thickness of the glass is just enough to let the stopwatch hand timing the back surface reflection make an extra ¼ or ¾ of a turn, the two arrows will end up at right angles. The final arrow in this case is the hypoteneuse of a right triangle, and according to Pythagoras, the square on the hypoteneuse is equal to the sum of the squares on the other two sides. Here is the value that's right "twice a day"- 4% + 4% makes 8% (see Fig. 16).
Notice that as we gradually increase the thickness of the glass, the front reflection arrow always points in the same direction, whereas the back reflection arrow gradually changes its direction. The change in the relative direction of the two arrows makes the final arrow go through a re-

32

Chapter 1

FIG URE 16. When the front and back reflection arrows are at nght angles to each other, the final arrow is the hypoteneuse ofa nght tnangle. Thus zts square is the sum of the other two squares-8%.

FIGURE 17. As thin sheets ofglass are replaced by slightly thicker ones, the stopwatch hand timing a photon reflecting off the back surface turns slightly more, and the relative angle between the front and back reflection arrows changes This causes the final arrow to change in length, and its square to change in me from Oto 16% back to 0, over and over

Introduction

33

peating cycle of length zero to 0.4; thus the square on the final arrow goes through the repeating cycle of zero to 16% that we observed in our experiments (see Fig. 17).
I have just shown you how this strange feature of partial reflection can be accurately calculated by drawing some damned little arrows on a piece of paper. The technical word for these arrows is "probability amplitudes," and I feel more dignified when I say we are "computing the probability amplitude for an event." I prefer, though, to be more honest, and say that we are trying to find the arrow whose square represents the probability of something happening.
Before I finish this first lecture, I would like to tell you about the colors you see on soap bubbles. Or better, if your car leaks oil into a mud puddle, when you look at the brownish oil in that dirty mud puddle, you see beautiful colors on the surface. The thin film of oil floating on the mud puddle is something like a very thin sheet of glassit reflects light of one color from zero to a maximum, depending on its thickness. If we shine pure red light on the film of oil, we see splotches of red light separated by narrow bands of black (where there's no reflection) because the oil film's thickness is not exactly uniform. If we shine pure blue light on the oil film, we see splotches of blue light separated by narrow bands of black. If we shine both red and blue light onto the oil, we see areas that have just the right thickness to strongly reflect only red light, other areas of the right thickness to reflect only blue light; still other areas have a thickness that strongly reflects both red and blue light (which our eyes see as violet), while other areas have the exact thickness to cancel out all reflection, and appear black.
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

34

Chapter 1

certain thicknesses, one or the other or both colors are strongly reflected, while at other thicknesses, reflection 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

Percentage of
Reflection
16%

-red -blue

bluish red v1ole1

blue black violet (d,m violel l
Th1ckness-

t VIOiet

black

reddish VIOiet

F1GURE 18. As the thickness of a layer increases, the two surfaces produce a partial reflection of monochromatic light whose probalnhty fluctuates zn a cycle from 0% to 16% Smee 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 zn 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

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.
When we shine red and blue light on a film of oil, patterns of red, blue, and violet appear, separated by borders of black. When sunlight, which contains red, yellow, green,

Introduction

35

and blue light, shines on a mud puddle with oil on it, the areas that strongly reflect each of those colors overlap and produce all kinds of combinations which our eyes see as different colors. As the oil film spreads out and moves over the surface of the water, changing its thickness in various locations, the patterns of color constantly change. (If, on the other hand, you were to look at the same mud puddle at night with one of those sodium streetlights shining on it, you would see only yellowish bands separated by blackbecause those particular streetlights emit light of only one color.)
This phenomenon of colors produced by the partial reflection of white light by two surfaces is called iridescence, and can be found in many places. Perhaps you have wondered how the brilliant colors of hummingbirds and peacocks are produced. Now you know. How those brilliant colors evolved is also an interesting question. When we admire a peacock, we should give credit to the generations of lackluster females for being selective about their mates. (Man got into the act later and streamlined the selection process in peacocks.)
In the next lecture I will show you how this absurd process of combining little arrows computes the right answer for those other phenomena you are familiar with: light travels in straight lines; it reflects off a mirror at the same angle that it came in ("the angle of incidence is equal to the angle of reflection"); a lens focuses light, and so on. This new framework wiU describe everything you know about light.

2
Photons:
Particles of Light
This is the second in a series of lectures about quantum electrodynamics, and since it's clear that none of you were here last time (because I told everyone that they weren't going to understand anything), I'll briefly summarize the first lecture.
We were talking about light. The first important feature about light is that it appears to be particles: when very weak monochromatic light (light of one color) hits a detector, the detector makes equally loud clicks less and less often as the light gets dimmer.
The other important feature about light discussed in the first lecture is partial reflection of monochromatic light. An average of 4% of the photons hitting a single surface of glass is reflected. This is already a deep mystery, since it is impossible to predict which photons will bounce back and which will go through. With a second surface, the results are strange: instead of the expected reflection of 8% by the two surfaces, the partial reflection can be amplified as high as 16% or turned off, depending on the thickness of the glass.
This strange phenomenon of partial reflection by two surfaces can be explained for intense light by a theory of waves, but the wave theory cannot explain how the detector

Photons: Particles of Light

37

makes equally loud clicks as the light gets dimmer. Quantum electrodynamics "resolves" this wave-particle duality by saying that light is made of particles (as Newton originally thought), but the price of this great advancement of science is a retreat by physics to the position of being able to calculate only the probability that a photon will hit a detector, without offering a good model of how it actually happens.
In the first lecture I described how physicists calculate the probability that a particular event will happen. They draw some arrows on a piece of paper according to some rules, which go as follows:
-GRAND PRINCIPLE: The probability of an event is equal to the square of the length of an arrow called the "probability amplitude." An arrow of length 0.4, for example, represents a probability of 0.16, or 16%.
-GENERAL RULE for drawing arrows if an event can happen in alternative ways: Draw an arrow for each way, and then combine the arrows ("add" them) by hooking the head of one to the tail of the next. A "final arrow" is then drawn from the tail of the first arrow to the head of the last one. The final arrow is the one whose square gives the probability of the entire event.
There were also some specific rules for drawing arrows in the case of partial reflection by glass (they can be found on -pages 26 and 27).
All of the preceding is a review of the first lecture. What I would like to do now is show you how this model of the world, which is so utterly different from anything you've ever seen before (that perhaps you hope never to see it again), can explain all the simple properties of light that you know: when light reflects off a mirror, the angle of incidence is equal to the angle of reflection; light bends when it goes from air into water; light goes in straight lines;

38

Chapter 2

light can be focused by a lens, and so on. The theory also describes many other properties of light that you are probably not familiar with. In fact, the greatest difficulty I had in preparing these lectures was to resist the temptation to derive all of the things about light that took you so long to learn about in school-such as the behavior of light as it goes past an edge into a shadow (called diffraction)-but since most of you have not carefully observed such phenomena, I won't bother with them. However, I can guarantee you (otherwise, the examples I'm going to show you would be misleading) that every phenomenon about light that has been observed in detail can be explained by the theory of quantum electrodynamics, even though I'm going to describe only the simplest and most common phenomena.
We start with a mirror, and the problem of determining how light is reflected from it (see Fig. 19). At S we have a source that emits light of one color at very low intensity (let's use red light again). The source emits one photon at a time. At P, we place a photomultiplier to detect photons. Let's put it at the same height as the source-drawing arrows will be easier if everything is symmetrical. We want to calculate the chance that the detector will make a click after a photon has been emitted by the source. Since it is possible that a photon could go straight across to the de-
tector, let's place a screen at Q to prevent that.
Now, we would expect that all the light that reaches the detector reflects off the middle of the mirror, because that's the place where the angle of incidence equals the angle of reflection. And it seems fairly obvious that the parts of the mirror out near the two ends have as much to do with the reflection as with the price of cheese, right?
Although you might think that the parts of the mirror near the two ends have nothing to do with the reflection of the light that goes from the source to the detector, let

Photons: Particles of Light

39

(a}

hght source
s

screen
~
Q

detector ( photomulhplilr)
p

MIRROR

(b)
FIGURE 19. The classical view of the world says that a mirror will reflect light where the angle of znczdence is equal to the angle of reflection, even if the source and the detector are at different levels, as zn (b).
us look at what quantum theory has to say. Rule: The probability that a particular event occurs is the square of a final arrow that is found by drawing an arrow for each way the event could happen, and then combining ("adding") the arrows. In the experiment measuring the partial reflection oflight by two surfaces, there were two ways a photon could get from the source to the detector. In this experiment, there are millions of ways a photon could go: it could go down to the left-hand part of the mirror at A or B (for example) and bounce up to the detector (see Fig. 20); it could bounce off the part where you think it should, at G; or, it could go down to the right-hand part of the mirror at Kor Mand bounce up to the detector. You might think

40

Chapter 2

A B

D

G H

K

FIGURE 20. The quantum view of the world says that light has an equal amplitude to reflect from every part of the mirror, from A to M.

I'm crazy, because for most of the ways I told you a photon could reflect off the mirror, the angles aren't equal. But I'm not crazy, because that's the way light really goes! How can that be?
To make this problem easier to understand, let's suppose that the mirror consists of only a long strip from left to right-it's just as well that we forget, for a moment, that the mirror also sticks out from the paper (see Fig. 21 ). While

.,.......,
f

:
I

zI:z:I%

z I: ? : I

t :

z

I_ z

:z::z: z::
I I I

z-
I

z::
I

j ~

ABC DE FG HI J KL M

FIGURE 21. To calculate more easily where the light goes, we shall temporanly consider only a strip of mirror divided into litt/,e squares, with one path for each square. This simplification in no way detracts from an accurate analysis of the situation.

there are, in reality, millions of places where the light could reflect from this strip of mirror, let's make an approximation by temporarily dividing the mirror into a definite number of little squares, and consider only one path for each square--our calculation gets more accurate (but harder to do) as we make the squares smaller and consider more paths.

Photons: Particles of Light

41

Now, let's draw a little arrow for each way the light could go in this situation. Each little arrow has a certain length and a certain direction. Let's consider the length first. You might think that the arrow we draw to represent the path that goes to the middle of the mirror, at G, is by far the longest (since there seems to be a very high probability that any photon that gets to the detector must go that way), and the arrows for the paths at the ends of the mirror must be very short. No, no; we should not make such an arbitrary rule. The right rule-what actually happens-is much simpler: a photon that reaches the detector has a nearly equal chance of going on any path, so all the little arrows have nearly the same length. (There are, in reality, some very slight variations in length due to the various angles and distances involved, but they are so minor that I am going to ignore them.) So let us say that each little arrow we draw will have an arbitrary standard length-I will make the length very short because there are many of these arrows representing the many ways the light could go (see Fig. 22).

FIGURE 22. Each way the light can go will be represented in our calculation by an arrow of an arbitrary standard length, as shown.

Although it is safe to assume that the length of all the arrows will be nearly the same,-their directions will clearly differ because their timing is different-as you remember from the first lecture, the direction of a particular arrow is determined by the final position of an imaginary stopwatch that times a photon as it moves along that particular path. When a photon goes way off to the left end of the mirror, at A, and then up to the detector, it clearly takes more time than a photon that gets to the detector by reflecting in the middle of the mirror, at G (see Fig. 23). Or,

42

Chapter 2

imagine for a moment that you were in a hurry and had to run from the source over to the mirror and then to the detector. You'd know that it certainly isn't a good idea to go way over to A and then all the way up to the dectector; it would be much faster to touch the mirror somewhere in the middle.

p

A

G

FIGURE 23. While the kngth of each arrow is essentially the same, the direction will be different because the time it takes for a photon to go on each path ,s different. Clearly, it takes longer to go from S to A to P than from S to G to P.

To help us calculate the direction of each arrow, I'm going to draw a graph right underneath my sketch of the mirror (see Fig. 24). Directly below each place on the mirror where the light could reflect, I'm going to show, vertically, how much time it would take if the light went that way. The more time it takes, the higher the point will be on the graph. Starting at the left, the time it takes a photon to go on the path that reflects at A is pretty long, so we plot a point pretty high up on the graph. As we move toward the center of the mirror, the time it takes for a photon to go the particular way we're looking at goes down, so we plot each successive point lower than the previous one. After we pass the center of the mirror, the time it takes a photon to go on each successive path gets longer and longer, so we plot our points correspondingly higher and higher. To aid the eye, let's connect the points: they form a symmetrical

Photons: Particles of Light

43

A8 CDE F GH

J KL M

A 8 CD EFG H

J K LM

FIGURE 24. Each path the light could go (in this simplified situation) is shown at the top, with a point on the graph below it showing the time it takes a photon to go from the source to that point on the mirror, and then to the photomultiplier. Below the graph is the direction of each arrow, and at the bottom is the result of adding all the arrows. It is evident that the major contribution to the final arrow's length is made by arrows E through I, whose directions are nearly the same because the timing of their paths is nearly the same. This also happens to be where the total time is least. It is therefore approximately right to say that light goes where the time is least.
curve that starts high, goes down, and then goes back up agam.
Now, what does that mean for the direction of the little arrows? The direction of a particular arrow corresponds to the amount of time it would take a photon to get from

44

Chapter 2

the source to the detector following that particular path. Let's draw the arrows, starting at the left. Path A takes the most time; its arrow points in some direction (Fig. 24). The arrow for path B points in a different direction because its time is different. At the middle of the mirror, arrows F, G, and H point in nearly the same direction because their times are nearly the same. After passing the center of the mirror, we see that each path on the right side of the mirror corresponds to a path on the left side whose time is exactly the same (this is a consequence of putting the source and the detector at the same height, and path G exactly in the
middle). Thus the arrow for path J, for example, has the
same direction as the arrow for path D. Now, let's add the little arrows (Fig. 24). Starting with
arrow A, we hook the arrows to each other, head to tail. Now, if we were to take a walk using each little arrow as a step, we wouldn't get very far at the beginning, because the direction from one step to the next is so different. But after a while the arrows begin to point in generally the same direction, and we make some progress. Finally, near the end of our walk, the direction from one step to the next is again quite different, so we stagger about some more.
All we have to do now is draw the final arrow. We simply connect the tail of the first little arrow to the head of the last one, and see how much direct progress we made on our walk (Fig. 24). And behold-we get a sizable final arrow! The theory of quantum electrodynamics predicts that light does, indeed, reflect off the mirror!
Now, let's investigate. What determines how long the final arrow is? We notice a number of things. First, the ends of the mirror are not important: there, the little arrows wander around and don't get anywhere. If I chopped off the ends of the mirror-parts that you instinctively knew I was wasting my time fiddling around with-it would hardly affect the length of the final arrow.

Photons: Particles of Light

45

So where is the part of the mirror that gives the final arrow a substantial length? It's the part where the arrows are all pointing in nearly the same direction-because their tzme is almost the same. If you look at the graph showing the time for each path (Fig. 24), you see that the time is nearly the same from one path to the next at the bottom of the curve, where the time is least.
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 tzme is least (and it's easy to prove that where 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!
Let's test the idea that there really zs reflection going on all over the mirror by doing another experiment. First, let's chop off most of the mirror, and leave about a quarter of it, over on the left. We still have a pretty big piece of mirror, but it's in the wrong place. In the previous experiment the arrows on the left side of the mirror were pointing in directions very different from one another because of the large difference in time between neighboring paths (Fig. 24). In this experiment I am going to make a more detailed calculation by taking intervals on that left-hand part of the

46

Chapter 2

mirror that are much closer together-fine enough that there 1s not much difference in time between neighboring paths (see Fig. 25). With this more detailed picture, we see that some of the arrows point more or less to the right; the others point more or less to the left. If we add all the arrows together, we have a bunch of arrows going around in what is essentially a circle, getting nowhere.

A

B

C

0
FIGURE 25. To test the idea that there is really reflection happening at the ends of the mirror (but it is Just cancelling out), we do an experiment with a IArge piece of mirror that ts located zn the wrong place for reflection from S to P This piece of mirror is divided into much smaller sections, so that the timing from one path to the next is not very different When all the arrows are added, they get nowhere they go m a circle and add up to nearly nothing.
But let's suppose we carefully scrape the mirror away in those areas whose arrows have a bias in one direction-let's say, to the left-so that only those places whose arrows point generally the other way remain (see Fig. 26). When we add up only the arrows that point more or less to the right, we get a series of dips and a substantial final arrow-according to the theory, we should now have a strong reflection! And indeed, we do-the theory is correct! Such a mirror is called a diffraction grating, and it works like a charm.
Isn't it wonderful-you can take a piece of mirror where

Photons: Particles of Light

47

A

B

C

J

FIGURE 26. If only the arrows with a bias in a particular direction-such as to the nght-are added, while the others are disregarded (by etching away the mirror in those places), then a substantial amount of light reflects from tlus piece ofmirror located in the wrong place Such an etched mirror is called a diffraction grating
you didn't expect any reflection, scrape away part of it, and it reflects!1
The particular grating that I just showed you was tailormade for red light. It wouldn't work for blue light; we would have to make a new grating with the cut-away strips spaced closer together because, as I told you in the first lecture, the stopwatch hand turns around faster when it times a blue photon compared to a red photon. So the cuts that were especially designed for the "red'' rate of turning don't fall in the right places for blue light; the arrows get kinked up and the grating doesn't work very well. But as a matter of accident, it happens that if we move the photomultiplier down to a somewhat different angle, the grating made for red light now works for blue light. It's just a
1 The areas of the mirror whose arrows pomt generally to the left also make a strong reflection (when the areas whose arrows pomt the other way are erased) It's when both left-biased and nght-b1ased areas reflect together that they cancel out Thts as analogous to the case of partial reflection by two surfaces whale either surface will reflect on tts own, tf the thickness 1s such that the two surfaces contnbute arrows pomtmg m opposite d1recuons, reflection as cancelled out

48

Chapter2

lucky accident, a consequence of the geometry involved (see Fig. 27).
If you shine white light down onto the grating, red light comes out at one place, orange light comes out slightly above it, followed by yellow, green, and blue light-all the

Blue Red

FIG uRE 27. A diffraction grating with grooves
at the right distance for red light al.so works for
other colors, if the detector is in a different place. Thus it is possib/,e to see different colors reflecting
from a grooved surface-such as a phonograph record-depending on the angle.
colors of the rainbow. Where there is a series of grooves close together, you can often see colors-for example, when you hold a phonograph record (or better, a videodisc)under bright light at the correct angles. Perhaps you have seen those wonderful silvery signs (here in sunny California they're often on the backs of cars): when the car moves, you see very bright colors changing from red to blue. Now you know where the colors come from: you're looking at a grating-a mirror that's been scratched in just the right places. The sun is the light source, and your eyes are the detector. I could go on to easily explain how lasers and holograms work, but I know that not everyone has seen these things, and I have too many other things to talk about.2
2 I can't resist telling you about a grating that Nature has made: salt crystals are sodium and chlorine atoms packed in a regular pattern.

Photons: Particles of Light

49

So a grating shows that we can't ignore the parts of a mirror that don't seem to be reflecting; if we do some clever things to the mirror, we can demonstrate the reality of the reflections from all parts of the mirror and produce some striking optical phenomena.

p

FIGURE 28. Nature has made many types of diffraction gratings in the form of crystals. A salt crystal reflects X-rays (light for which the imaginary stopwatch hand moves extremely f ast--perhaps 10,000 times faster than for visible light) at various angles, from which can be determined the exact arrangement and spacings of the individual atoms.
More importantly, demonstrating the reality of reflection from all parts of the mirror shows that there is an amplitude-an arrow-for every way an event can happen. And in order to calculate correctly the probability of an event in different circumstances, we have to add the arrows for every way that the event could happen-not just the ways we think are the important ones!
Their alternating pattern, like our grooved surface, acts like a grating when light of the right color (X-rays, in this case) shines on it. By finding the specific locations where a detector picks up a lot of this special reflection (called diffraction), one can determine exactly how far apart the grooves are, and thus how far apart the atoms are (see Fig. 28). It is a beautiful way of determining the structure of all kinds of crystals as well as confirming that X-rays are the same thing as light. Such experiments were first done in 1914. It was very exciting to see, in detail, for the first time how the atoms are packed together in different substances.

50

Chapter 2

Now, I would like to talk about something more familiar than gratings-about light going from air into water. This time, let's put the photomultiplier underwater-we suppose the experimenter can arrange that! The source of light is in the air at S, and the dectector is underwater, at D (see Fig. 29). Once again, we want to calculate the probability that a photon will get from the light source to the detector. To make this calculation, we should consider all

water surface

FIGURE 29. Quantum theory says that light can go from a source in air
to a detector in water in many ways. If the problem IS simplified as in the case of the mirror, a graph showing the timing of each path can be drawn, with the direction ofe°'h arrow below it. Once again, the major contribution toward
the length of the final arrow comes from those paths whose arrows point in
nearly the same direction because their timing is nearly the same; once again, this is where the time is least.

Photons: Particles of Light

51

the ways the light could go. Each way the light could go contributes a little arrow and, as in the previous example, all the little arrows have nearly the same length. We can again make a graph of the time it takes a photon to go on each possible path. The graph will be a curve very similar to the one we made for light reflecting off a mirror: it starts up high, goes down, and then back up again; the most important contributions come from the places where the arrows point in nearly the same direction (where the time is nearly the same from one path to the next), which is at the bottom of the curve. That is also where the time is the least, so all we have to do is find out where the time is least.
It turns out that light seems to go slower in water than it does in air (I will explain why in the next lecture), which makes the distance through water more "costly," so to speak, than the distance through air. Ir's not hard to figure out which path takes the least time: suppose you're the lifeguard, sitting at S, and the beautiful girl is drowning, at D (Fig. 30). You can run on land faster than you can swim in water. The problem is, where do you enter the water in order to reach the drowning victim the fastest? Do you run down to the water at A, and then swim like

s
air

Path Of certoinJy not least t 1me1

N water

FIGURE 30. Finding the path of least time for light i.s like finding the path of least time for a lifeguard running and then swimming to rescue a drowning victim: the path of least distance has too much water in it; the path of least water has too much land in it; the path of least time is a compromise between the two.

52

Chapter 2

hell? Of course not. But running directly toward the victim
and entering the water at J is not the fastest route, either.
While it would be foolish for a lifeguard to analyze and calculate under the circumstances, there is a computable position at -which the time is minimum: it's a compromise
between taking the direct path, through J, and taking the
path with the least water, through N. And so it is with light-the path of least time enters the water at a point
between J and N, such as L.
Another phenomenon of light that I would like to mention briefly is the mirage. When you're driving along a road that is very hot, you can sometimes see what looks like water on the road. What you're really seeing is the sky, and when you normally see sky on the road, it's because the road has puddles of water on it (partial reflection of light by a single surface). But how can you see sky on the road when there's no water there? What you need to know is that light goes slower through cooler air than through warmer air, and for a mirage to be seen, the observer must be in the cooler air that is above the hot air next to the road surface (see Fig. 31). How it is possible to look down and see the sky can be understood by finding the path of least time. I'll let you play with that one at home-it's fun to think about, and pretty easy to figure out.

SKY
'', ' .....

COOLER AIR
---- _______ .,..,,
WARMER AIR
ROAD

FIGURE 31. Finding the path of least time explains how a mirage works.
Light goes faster through warm air than through cool air. Some of the sky
ap,pears to be on the road because some of the ltght from the sky reaches the
eye by coming up from the road. The only other time sky ap,pears to be on the
road is when water is reflecting it, and thus a mirage appears to be water.

Photons: Particles of Light

53

In the examples I showed you of light reflecting off a mirror and of light going through air and then water, I was making an approximation: for the sake of simplicity, I drew the various ways the light could go as double straight lines-two straight lines that form an angle. But we don't have to assume that light goes in straight lines when it is in a uniform material like air or water; even that is explainable by the general principle of quantum theory: the probability of an event is found by adding arrows for all the ways the event could happen.
So for our next example, I'm going to show you how, by adding little arrows, it can appear that light goes in a straight line. Let's put a source and a photomultiplier at S and P, respectively (see Fig. 32), and look at all the ways

1/
--

1
-

/
-

-

-

-

-

-

-

-

.

A
g~ - = p

S ------ ------ -- _E__.

$ G
FIGURE 32. Quantum theory can be used to show why light appears to travel m straight lines. When all possible paths are considered, each crooked path has a nearby path of considerably less distance and therefore much less tzme (and a substantially different direction for the arrow) Only the paths near the straight-line path at D have arrows pomtzng m nearly the same direction, because their timings are nearly the same. Only such arrows are important, because zt is from them that we accumulate a large final arrow.

54

Chapter 2

the light could go-in all sorts of crooked paths-to get from the source to the detector. Then we draw a little arrow for each path, and we're learning our lesson well!
For each crooked path, such as path A, there's a nearby path that's a little bit straighter and distinctly shorter-that is, it takes much less time. But where the paths become nearly straight-at C, for example-a nearby, straighter path has nearly the same time. That's where the arrows add up rather than cancel out; that's where the light goes.
It is important to note that the single arrow that represents the straight-line path, through D (Fig. 32), is not enough to account for the probability that light gets from the source to the detector. The nearby, nearly straight paths-through C and E, for example-also make important contributions. So light doesn't really travel only in a straight line; it "smells" the neighboring paths around it, and uses a small core of nearby space. (In the same way, a mirror has to have enough size to reflect normally: if the mirror is too small for the core of neighboring paths, the light scatters in many directions, no matter where you put the mirror.)
Let's investigate this core of light more closely by putting a source at S, a photomultiplier at P, and a pair of blocks between them to keep the paths of light from wandering too far away (see Fig. 33). Now, let's put a second photomultiplier at Q, below P, and assume again, for the sake of simplicity, that the light can get from S to Q only by paths of double straight lines. Now, what happens? When the gap between the blocks is wide enough to allow many neighboring paths to P and to Q, the arrows for the paths to P add up (because all the paths to P take nearly the same time), while the paths to Q cancel out (because those paths have a sizable difference in time). Thus the photomultiplier at Q doesn't click.
But as we push the blocks closer together, at a certain

Photons: Particles of Light

55

TIME
L .,,,,,. 11;,1

~

f

s ~«<:::::

r _ TIM:/
..... .,...,..""--1'
0
FIGURE 33. Light travels m not ;ust the straight-line path, but in the nearfly paths as well. When two blocks are separated enough to allow for these nearlry paths, the photons proceed normally to P, and hardly ever go to Q.

point, the detector at Q starts clicking! When the gap is
nearly closed and there are only a few neighboring paths,
the arrows to Q also add up, because there is hardly any
difference in time between them, either (see Fig. 34). Of course, both final arrows are small, so there's not much light either way through such a small hole, but the detector
at Q clicks almost as much as the one at P! So when you
try to squeeze light too much to make sure it's going in only a straight line, it refuses to cooperate and begins to spread out.3
' This is an example of the "uncertainty principle": there is a kmd of "complementarity" between knowledge of where the light goes between the blocks and where it goes afterwards-precise knowledge of both is impossible. I would like to put the uncertainty principle in its h1stoncal place: When the revolutionary ideas of quantum physics were first commg

56

Chapter2

TIME
~

J
FIGURE 34. When light zs restricted so much that only a few paths are possible, the light that zs able to get through the narrow slit goes to Q almost as much as to P, because there are not enough arrows representing the paths to Q to
cancel each other out.
So the idea that light goes in a straight line is a convenient approximation to describe what happens in the world that is familiar to us; it's similar to the crude approximation that says when light reflects off a mirror, the angle of incidence is equal to the angle of reflection.
Just as we were able to do a clever trick to make light reflect off a mirror at many angles, we can do a similar
out, people still tned to understand them m terms of old-fashioned ideas (such as, hght goes in straight Imes) But at a certam pomt the old-fashioned ideas would begm to fad, so a warmng was developed that said, in effect, "Your old-fashioned ideas are no damn good when. ." If you get nd of all the old-fash10ned ideas and instead use the ideas that I'm explainmg m these lectures-addmg arrows for all the ways an event can happen-there 1s no need for an uncertamty principle!

Photons: Particles of Light

57

trick to get light to go from one point to another in many ways.
First, to simplify the situation, I'm going to draw a vertical dashed line (see Fig. 35) between the light source and the detector (the line means nothing; it's just an artificial line)

TIME A

s

p

M

F1 G URE 35. Analym ofall possible paths from S to P is simplified to include only double straight lines (in a single plane). The effect is the same as zn the more complicated, real case: there is a tzme curve with a minimum, where most of the contnbution to the final arrow is made.
and say that the only paths we're going to look at are double straight lines. The graph that shows the time for each path looks the same as in the case of the mirror (but I'll draw it sideways, this time): the curve starts at A, at the top, and then it comes in, because the paths in the middle are shorter and take less time. Finally, the curve goes back out again.
Now, let's have some fun. Let's "fool the light," so that all the paths take exactly the same amount of time. How can we do this? How can we make the shortest path, through M, take exactly the same time as the longest path, through A?
Well, light goes slower in water than it does in air; it also goes slower in glass (which is much easier to handle!). So, if we put in just the right thickness of glass on the shortest path, through M, we can make the time for that path exactly

58

Chapter 2

the same as for the path through A. The paths next to M, which are just a little longer, \\on't need quite as much glass (see Fig. 36). The nearer \\e get to A, the less glass we ha,e to put in to slow up the light. BJ carefulh calculating and putting in just the right thickness of glass to compensate for the time along each path, we can make all the times the same. When we draw the arrows for each way the light

TIME A

M

FIGURE 36. A "tnck" can be played on Nature by slowing down the light
that takes shorter paths glass ofJust the nght thickness zs inserted so that all the paths will take exactly the same tzme. This causes all of the arrows to point in the same direction, and to produce a whopping final arro~lots of light 1 Such a piece of glass made to greatly increase the probability of light getting from a source to a single point zs called a focusing lens
could go, we find we have succeeded in straightening them all out-and there are, in reality, millions of tiny arrowsso the net result is a sensationally large, unexpectedly enormous final arrow! Of course you know what I'm describing; it's a focusing lens. By arranging things so that all the times are equal, we can focus light-we can make the probability very high that light will arrive at a particular point, and very low that it will arrive anywhere else.
I have used these examples to show you how the theory of quantum electrodynamics, which looks at first like an absurd idea with no causality, no mechanism, and nothing

Photons: Particles of Light

59

real to it, produces effects that you are familiar "ith: light bouncing off a mirror, light bending when it goes from air into \\ater, and light focused by a lens. It also produces other effects that you may or may not ha,e seen, such as the diffracuon grating and a number of other things. In fact, the theoq, continues to be successful at explaining every phenomenon of light.
I have shown you with examples how to calculate the probabilit)' of an e\ent that can happen in alternative ways: we draw an arrow for each way the event can happen, and add the arrows. "Adding arrows" means the arrows are placed head to tail and a "final arrow" is drawn. The square of the resulting final arrow represents the probability of the event.
In order to give you a fuller flavor of quantum theory, I would now like to show you how physicists calculate the probability ofcompound events-events that can be broken down into a series of steps, or events that consist of a number of things happening independently.
An example of a compound event can be demonstrated by modifying our first experiment, in which we aimed some red photons at a single surface of glass to measure partial reflection. Instead of putting the photomultiplier at A (see Fig. 37), let's put in a screen with a hole in it to let the photons that reach point A go through. Then let's put in a sheet of glass at B, and place the photomultiplier at C. How do we figure out the probability that a photon will get from the source to C?
We can think of this event as a sequence of two steps. Step I: a photon goes from the source to point A, reflecting off the single surface of glass. Step 2: the photon goes from point A to the photomultiplier at C, reflecting off the sheet of glass at B. Each step has a final arrow-an "amplitude" (I'm going to use the words interchangeably)-that can be calculated according to the rules we know so far. The am-

60

Chapter 2

plitude for the first step has a length of 0.2 (whose square is 0.04, the probability of reflection by a single surface of glass), and is turned at some angle-let's say, 2 o'clock (Fig. 37).
To calculate the amplitude for the second step, we temporarily put the light source at A and aim the photons at the layer of glass above. We draw arrows for the front and back surface reflections and add them-let's say we end up with a final arrow with a length of 0.3, and turned toward 5 o'clock.

GLASS

,t,
I I I
3: I
e1
s:s
:!:: I
Cl
~I
~ .2
Step1: StoA

•t•
I I
~·,,3: I
.o........ l
01
~,Cl I
-~
Step 2: A to C

...
I
I I o3:,I
~Is
~,:=I
cl
.06;,.1
S to C combined

FIGURE 37. A compound event can be analyud as a succession of steps. In this example, the path of a photon going from S to C can be divided into
two steps: I) a photon gets from S to A, and 2) the photon gets from A to C. Each step can be analyzed separately to produce an arrow that can be regarded in a new way: as a unit arrow (an arrow of length I pointed at 12 o'clock) that has gone through a shrink and turn. In this example, the shrink and turn for Step 1 are 0.2 and 2 o'clock; the shrink and tum for St,ep 2 are 0.3 and 5 o'clock. To get the amplitude for the two steps in succession, we shrink and turn in succession: the unit arrow is shrunk and turned to produce an arrow
of length 0.2 turned to 2 o'clock, which itself is shrunk and turned (as if it
were the unit arrow) l,y 0.3 and 5 o'clock to produce an arrow of length 0.06 and turned to 7 o'clock. This process of successive shrinking and turning is
called "multiplying" arrows.

Photons: Particles of Light

61

Now, how do we combine the two arrows to draw the amplitude for the entire event? We look at each arrow in a new way: as instructions for a shrink and turn.
In this example, the first amplitude has a length of 0.2 and is turned toward 2 o'clock. If we begin with a "unit arrow"-an arrow of length 1 pointed straight up--we can shrink this unit arrow from 1 down to 0.2, and turn it from 12 o'clock to 2 o'clock. The amplitude for the second step can be thought of as shrinking the unit arrow from 1 to 0.3 and turning it from 12 o'clock to 5 o'clock.
Now, to combine the amplitudes for both steps, we shrink and turn in succession. First, we shrink the unit arrow from 1 to 0.2 and turn it from 12 to 2 o'clock; then we shrink the arrow further, from 0.2 down to three-tenths of that, and turn it by the amount from 12 to 5-that is, we turn it from 2 o'clock to 7 o'clock. The resulting arrow has a length of 0.06 and is pointed toward 7 o'clock. It represents a probability of 0.06 squared, or 0.0036.
Observing the arrows carefully, we see that the result of shrinking and turning two arrows in succession is the same as adding their angles (2 o'clock + 5 o'clock) and multiplying their lengths (0.2 * 0.3). To understand why we add the angles is easy: the angle of an arrow is determined by the amount of turning by the imaginary stopwatch hand. So the total amount of turning for the two steps in succession is simply the sum of the turning for the first step plus the additional turning for the second step.
Why we call this process "multiplying arrows" takes a bit more explanation, but it's interesting. Let's look at multiplication, for a moment, from the point of view of the Greeks (this has nothing to do with the lecture). The Greeks wanted to use numbers that were not necessarily integers, so they represented numbers with lines. Any number can be expressed as a transformation of the unit line-by expanding it or shrinking it. For example, if Line A is the

62

Chapter 2

unit line (see Fig. 38), then line B represents 2 and line C represents 3.
Now, how do we multiply 3 times 2? We apply the transformations in succession: starting with line A as the unit line, we expand it 2 times and then 3 times (or 3 times and then 2 times-the order doesn't make any difference). The result is line D, whose length represents 6. What about multiplying l/3 times 1/2? Taking line D to be the unit line, now, we shrink it to 1/2 (line C) and then to 1/3 of that. The result is line A, which represents 1/6.

A----
B---~---
C
D
FIGURE 38. We can express any number as a transformation of the unit
line through expansion or shrinkage. If A is the unit line, then B represents 2 (expansion), and C represents 3 (expansion). Multiplying lines is achieved
through successive transformations. For example, multiplying 3 by 2 means
that the unit line is expanded 3 times and then 2 times, producing the answer, an expansion of 6 (line D). If D is the unit line, then line C represents 1/2 (shrinkage), line B represents 1/3 (shrinkage), and multiplying 112 by 1/3 means the unit line D is shrunk to 1/2, and then to I /3 of that, producing the answer, a shrinkage to 1/6 (line A).

Multiplying arrows works the same way (see Fig. 39). We apply transformations to the unit arrow in succession-it just happens that the transformation of an arrow involves two operations, a shrink and turn. To multiply arrow V times arrow W, we shrink and turn the unit arrow by the prescribed amounts for V, and then shrink it and turn it the amounts prescribed for W-again, the order doesn't make any difference. So multiplying arrows follows the

Photons: Particles of Light

63

~

·?-

~

I

I

..0J..;..
0
c
:::,

I ~I 01
_~-,,I
c,
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C
·c
:::,

~ ~

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XYv

- -V- * w = -X-

it.
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0
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FIGURE 39. Mathematicians found that multiplying arrows can also be expressed as successwe transformations (for our purpose~, successive shrinks and turns) of the unit arrow. As in normal multiplication, the order is not important: the answer, arrow X, can be obtained by multiplying arrow V by arrow W or arrow W by arrow V.

4 Mathematicians have tried to find all the objects one could possibly
find that obey the rules of algebra (A + B = B + A, A * B = B * A,
and so on). The rules were originally made for positive integers, used for counting things like apples or people. Numbers were improved with the invention of zero, fractions, irrational numbers-----numbers that cannot be expressed as a ratio of two integers-and negative numbers, and continued to obey the original rules of algebra. Some of the numbers that mathematicians invented posed difficulties for people at first-the idea of half a person was difficult to imagine-but today, there's no difficulty at all: nobody has any moral qualms or discomforting gory feelings when they hear that there is an average of 3.2 people per square mile in some regions. They don't try to imagine the 0.2 people; rather, they know what 3.2 means: if they multiply 3.2 by 10, they get 32. 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.

64

Chapter 2

Let's go back to the first experiment from the first lecture-partial reflection by a single surface-with this idea of successive steps in mind (see Fig. 40). We can divide the path of reflection into three steps: 1) the light goes from the source down to the glass, 2) it is reflected by the glass, and 3) it goes from the glass up to the detector. Each step can be considered as a certain amount of shrinking and turning of the unit arrow.

I
GLASS~

.~..1. ,
o, ~I
:,~

!S
fA
;,; I -::1
Cl :, f
to 2

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-·:: I
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!II
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FIGURE 40. Reflection by a single surface can be divided into three steps, each with a shrink and/or turn of the unit arrow. The net result, an arrow of /,ength 0.2 pointed zn some direction, is the same as before, but our method of analysis is more detailed now.

You'll remember that in the first lecture, we did not consider all of the ways the light could reflect off the glass, which requires drawing and adding lots and lots of little tiny arrows. In order to avoid all that detail, I gave the impression that the light goes down to a particular point on the surface of the glass-that it doesn't spread out. When light goes from one point to another, it does, in reality, spread out (unless it's fooled by a lens), and there is some shrinkage of the unit arrow associated with that. For the moment, however, I would like to stick to the simplified view that light does not spread out, and so it is ap-

Photons: Particles of Light

65

propriate to disregard this shrinkage. It is also appropriate to assume that since the light doesn't spread out, every photon that leaves the source ends up at either A or B.
So: in the first step there is no shrinking, but there is turning-it corresponds to the amount of turning by the imaginary stopwatch hand as it times the photon going from the source to the front surface of the glass. In this example, the arrow for the first step ends up with a length of 1 at some angle-let's say, 5 o'clock.
The second step is the reflection of the photon by the glass. Here, there is a sizable shrink-from 1 to 0.2-and half a turn. (These numbers seem arbitrary now: they depend upon whether the light is reflected by glass or some other material. In the third lecture, I'll explain them, too!) Thus the second step is represented by an amplitude of length 0.2 and a direction of 6 o'clock (half a turn).
The last step is the photon going from the glass up to the detector. Here, as in the first step, there is no shrinking, but there is turning-let's say this distance is slightly shorter than in step 1, and the arrow points toward 4 o'clock.
We now "multiply" arrows 1, 2, and 3 in succession (add the angles, and multiply the lengths). The net effect of the three steps-I) turning, 2) a shrink and half a turn, and 3) turning-is the same as in the first lecture: the turning from steps 1 and 3-(5 o'clock plus 4 o'clock) is the same amount of turning that we got then when we let the stopwatch run for the whole distance (9 o'clock); the extra half turn from step 2 makes the arrow point in the direction opposite the stopwatch hand, as it did in the first lecture, and the shrinking to 0.2 in the second step leaves an arrow whose square represents the 4% partial reflection observed for a single surface.
In this experiment, there is a question we didn't look at in the first lecture: what about the photons that go to Bthe ones that are transmitted by the surface of the glass?

66

Chapter 2

The amplitude for a photon to arrive at B must have a length near 0.98, since 0.98 * 0.98 = 0.9604, which is close enough to 96%. This amplitude can also be analyzed by breaking it down into steps (see Fig. 41).
The first step is the same as for the path to A-the photon goes from the light source down to the glass-the unit arrow is turned toward 5 o'clock.
The second step is the photon passing through the surface of the glass: there is no turning associated with transmission, just a little bit of shrinking-to 0.98.

Step 1 Step 2

Step 3

"e'"I'

6 I

-·c:I

fA\:__ Step 3

,t5tep 2 step 1

FIGURE 41. Transmission by a single surface can also be divided into three
steps, with a shrink and/or turn for each step. An arrow of length 0.98 has a square of about 0.96, representing a probabilty of transmission of 96% (which, combined with the 4% probability of reflection, accounts for 100% of the light).

The third step-the photon going through the interior of the glass-involves additional turning and no shrinking.
The net result is an arrow of length 0.98 turned in some direction, whose square represents the probability that a photon will arrive at B-96%.
Now let's look at partial reflection by two surfaces again. Reflection from the front surface is the same as for a single surface, so the three steps for front surface reflection are the same as we saw a moment ago (Fig. 40).

Photons: Particles of Light

67

Reflection from the back surface can be broken down into seven steps (see Fig. 42). It involves turning equal to the total amount of turning of the stopwatch hand timing a photon over the entire distance (steps I, 3, 5, and 7), shrinking to 0.2 (step 4), and two shrinks to 0.98 (steps 2 and 6). The resulting arrow ends up in the same direction as before, but the length is about 0.192 (0.98 * 0.2 * 0.98), which I approximated as 0.2 in the first lecture.

FIGURE 42. Reflection from the back surface of a layer of glass can be divided into seven steps. Steps I, 3, 5, and 7 involve turning only; steps 2 and 6 involve shrinks to 0 .98, and step 4 involves a shnnk to 0.2. The result is an arrow of length 0.192-which was approximated as 0.2 in the first lecture-turned at an angle that corresponds to the total amount ofturning by the imaginary stopwatch hand.

In summary, here are the rules for reflection and transmission of light by glass: 1) reflection from air back to air (off a front surface) involves a shrink to 0.2 and half a turn; 2) reflection from glass back to glass (off a back surface) also involves a shrink to 0.2, but no turning; and 3) transmission from air to glass or from glass to air involves a shrink to 0.98 and no turning in either case.
Perhaps it is too much of a good thing, but I cannot resist showing you a cute further example of how things work and are analyzed by these rules of successive steps. Let us move the detector to a location below the glass, and consider something we didn't talk about in the first lecturethe probability of transmission by two surfaces of glass (see Fig. 43).
Of course you know the answer: the probability of a

68

Chapter 2

photon to arrive at B is simply l 00% minus the probability to arrive at A, which we worked out beforehand. Thus, if we found the chance to arrive at A is 7%, the chance to arrive at B must be 93%. And as the chance for A varies from zero through 8% to 16% (due to the different thicknesses of glass), the chance for B changes from 100% through 92% to 84%.

FIGURE 43. Transmission by two suifaces
can be broken down into five steps. Step 2 shrinks the unit arrow to 0.98, step 4 shrinks the 0.98 arrow to 0.98 of that (about 0.96); steps I, 3, and 5 involve turning only. The resulting arrow of length 0.96 has a square of about 0.92, representing a probability of
transmission by two surfaces of 92% (which
corresponds to the expected 8% reflection, which is right only "twice a day"). When the thickness ofthe /,ayer is right to produce a probability of 16% reflection, with a 92% probability of transmission, 108% of the light is
accounted for! Something is wrong with this
analysis!

That is the right answer, but we are expecting to calculate all probabilities by squaring a final arrow. How do we calculate the amplitude arrow for transmission by a layer of glass, and how does it manage to vary in length so appropriately as to fit with the length for A in each case, so the probability for A and the probability for B always add up to exactly 100%? Let us look a little into the details.
For a photon to go from the source to the detector below the glass, at B, five steps are involved. Let's shrink and turn the unit arrow as we go along.
The first three steps are the same as in the previous example: the photon goes from the source to the glass (turning, no shrinking); the photon is transmitted by the

Photons: Particles of Light

69

front surface (no turning, shrinking to 0.98); the photon goes through the glass (turning, no shrinking).
The fourth step-the photon passes through the back surface of the glass-is the same as the second step, as far as shrinks and turns go: no turns, but a shrinkage to 0.98 of the 0.98, so the arrow now has a length of 0.96.
Finally, the photon goes through the air again, down to the detector-that means more turning, but no further shrinking. The result is an arrow of length 0.96, pointing in some direction determined by the successive turnings of the stopwatch hand.
An arrow whose length is 0.96 represents a probability of about 92% (0.96 squared), which means an average of 92 photons reach Bout of every 100 that leave the source. That also means that 8% of the photons are reflected by the two surfaces and reach A. But we found out in the first lecture that an 8% reflection by two surfaces is only right sometimes ("twice a day''}--that in reality, the reflection by two surfaces fluctuates in a cycle from zero to 16% as the thickness of the layer steadily increases. What happens when the glass is just the right thickness to make a partial reflection of 16%? For every 100 photons that leave the source, 16 arrive at A and 92 arrive at B, which means 108% of the light has been accounted for-horrifying! Something is wrong.
We neglected to consider all the ways the light could get to B! For instance, it could bounce off the back surface, go up through the glass as if it were going to A, but then reflect off the front surface, back down toward B (see Fig. 44). This path takes nine steps. Let's see what happens successively to the unit arrow as the light goes through each step (don't worry; it only shrinks and turns!).
First step-photon goes through the air-turning; no shrinking. Second step-photon passes through the glassno turning, but shrinking to 0.98. Third step-photon goes

70

Chapter 2

through the glass-turning; no shrinking. Fourth step-reflection off the back surface-no turning, but shrinking to 0.2 of 0.98, or 0.196. Fifth step-photon goes back up through the glass-turning; no shrinking. Sixth step-photon bounces off front surface (it's really a "back" surface, because the photon stays inside the glass)-no turning, but shrinking to 0.2 of 0.196, or 0.0392. Seventh step-photon

FIGURE 44. Another way that light could be transmitted by two surfaces must be consulered m order to make the calculation more accurate. This path involves two shrmk.s of 0.98 (steps 2 and 8) and two shnnk.s of 0.2 (steps 4 and 6), resulting m an arrow oflength 0.0384 (rounded off to 0.04).

goes back down through glass-more turning; no shrinking. Eighth step-photon passes through back surface-no turning, but shrinking to 0.98 of 0.0392, or 0.0384. Finally, the ninth step-photon goes through air to detector-turning; no shrinking.
The result of all this shrinking and turning is an amplitude of length 0.0384-<:all it 0.04, for all practical purposes-and turned at an angle that corresponds to the total amount of turning by the stopwatch as it times the photon going through this longer path. This arrow represents a second way that light can get from the source to B. Now we have two alternatives, so we must add the two arrows-the arrow for the more direct path, whose length is 0.96, and the arrow for the longer way, whose length is 0.04-to make the final arrow.

Photons: Particles of Light

71

The two arrows are usually not in the same direction, because changing the thickness of the glass changes the relative direction of the 0.04 arrow to the 0.96 arrow. But look how nicely things work out: the extra turns made by the stopwatch timing a photon during steps 3 and 5 (on its way to A) are exactly equal to the extra turns it makes timing a photon during steps 5 and 7 (on its way to B). That means when the two reflection arrows are cancelling each other to make a final arrow representing zero reflection, the arrows for transmission are reinforcing each other to make
an arrow oflength 0.96 + 0.04, or I-when the probability
of reflection is zero, the probability of transmission is I00% (see Fig. 45). And when the arrows for reflection are rein-

Ref lect1on

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FIGURE 45. Nature always makes sure JOO% of the light as accounted for. When the thickness is right for the transmission arrows to accumulate, the arrows for reflection oppose each other; when the arrows for reflection accumulate, the arrows for transmission oppose each other.

72

Chapter 2

forcing each other to make an amplitude of0.4, the arrows for transmission are going against each other, making an amplitude of length 0.96 - 0.04, or 0.92-when reflection is calculated to be 16%, transmission is calculated to be 84% (0.92 squared). You see how clever Nature is with Her rules to make sure that we always come out with l 00% of the photons accounted for! 5
Finally, before I go, I would like to tell you that there is an extension to the rule that tells us when to multiply arrows: arrows are to be multiplied not only for an event that consists of a succession of steps, but also for an event that consists of a number of things happening concomitantlyindependently and possibly simultaneously. For example, suppose we have two sources, X and Y, and two detectors, A and B (see Fig. 47), and we want to calculate the prob-

FIGURE 46. Yet other ways the light could reflect should be considered for a more accurate calculation. In this figure, shrinks of 0.98 occur at steps 2 and 10; shrinks of 0.2 occur at steps 4, 6, and 8. The result is an arrow with a length of about 0.008, which zs another alternative for reflection, and should therefore be added to the other arrows which represent reflection (0.2 for the front surface and 0.192 for the back surface).

5 You'll notice that we changed 0.0384 to 0.04 and used 84% as the square of 0.92, in order to make 100% of the light accounted for. But when everything is added together, 0.0384 and 84% don't have to be rounded off-all the little bits and pieces of arrows (representing all the ways the light could go) compensate for each other and keep the answer correct. For those of you who like this sort of thing, here is an example of another way that the light could go from the light source to the detector at A-a series of three reflections (and two transmissions}, resulting in a final arrow of length 0.98 * 0.2 * 0.2 * 0.2 * 0.98, or about 0.008-a very tiny arrow (see Fig. 46}. To make a complete calculation of partial reflection by two surfaces, you would have to add in that small arrow, plus an even smaller one that represents five reflections, and so on.

Photons: Particles of Light

73

ability for the following event: after X and Y each lose a photon, A and Beach gain a photon.
In this example, the photons travel through space to get to the detectors-they are neither reflected nor transmitted-so now is a good time for me to stop disregarding the fact that light spreads out as it goes along. I now present

X---------------A

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FIGURE 47. If one of the ways a particular event can happen depends on a number of things happenmg independently, the amplitude for this way is calculated by multiplying the arrows of the independent things. In this case, the final event is: after sources X and Y each lose a photon, photomultipliers A and B malu a click. One way this event could happen is that a photon could go from X to A and a photon could go from Y to B (two independent things). To calculate the probability for this ''first way," the arrows for each independent thmg-X to A and Y to B-are multiplied to produce the amplitude for this particular way. (Analysis continued m Fig. 48.)

you with the complete rule for monochromatic light travelling from one point to another through space-there is nothing approximate here, and no simplification. This is all there is to know about monochromatic light going through space (disregarding polarization): the angle of the arrow depends on the imaginary stopwatch hand, which rotates a certain number of times per inch (depending on the color of the photon); the length of the arrow is inversely proportional

74

Chapter 2

to the distance the light goes-in other words, the arrow shrinks as the light goes along.6
Let's suppose the arrow for X to A is 0.5 in length and is pointing toward 5 o'clock, as is the arrow for Y to B (Fig. 47). Multiplying one arrow by the other, we get a final arrow of length 0.25, pointed at IO o'clock.

X

A

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/5
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amplitude for
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(final arrow)

FIG u RE 48. The other way the event described in Figure 47 cou/,d happena photon goes from X to B and a photon goes from Y to A-also depends on two independent things happening, so the amplitude for this "second way" is also calculated l,y multiplying the arrows of the independent things. The ''first way" and "second way" arrows are ultimately added together, resulting an the final arrow for the event. 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.

6 This rule checks out with what they teach in school-the amount of light transmitted over a distance varies inversely as the square of the distance-because an arrow that shrinks to half its original size has a square one-fourth as big.

Photons: Particles of Light

75

But wait! There is another way this event could happen: the photon from X could go to B, and the photon from Y could go to A. Each of these subevents has an amplitude, and these arrows must also be drawn and multiplied to produce an amplitude for this particular way the event could happen (see Fig. 48). Since the amount of shrinkage over distance is very small compared to the amount of turning, the arrows from X to B and Y to A have essentially the same length as the other arrows, 0.5, but their turning is quite different: the stopwatch hand rotates 36,000 times per inch for red light, so even a tiny difference in distance results in a substantial difference in timing.
The amplitudes for each way the event could happen are added to produce the final arrow. Since their lengths are essentially the same, it is possible for the arrows to cancel each other out if their directions are opposed to each other. The relative directions of the two arrows can be changed by changing the distance between the sources or the detectors: simply moving the detectors apart or together a little bit can make the probability of the event amplify or completely cancel out, just as in the case of partial reflection by two surfaces:7
In this example, arrows were multiplied and then added to produce a final arrow (the amplitude for the event), whose square is the probability of the event. 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

7 This phenomenon, called the Hanbury-Brown-Twiss effect, has been used to distinguish between a single source and a double source of radio waves in deep space, even when the two sources are extremely close together.

76

Chapter 2

somehow associated with the photon. But these arrows are probability amplitudes, that give, when squared, the probability of a complete event.8
In the next lecture I will begin the process of simplifying and explaining the properties of matter-to explain where the shrinking to 0.2 comes from, why light appears to go slower through glass or water than through air, and so onbecause I have been cheating so far: the photons don't really bounce off the surface of the glass; they interact with the electrons msule the glass. I'll show you how photons do nothing hut go from one electron to another, and how reflection and transmission are really the result of an electron picking up a photon, "scratching its head," so to speak, and emitting a new photon. This simplification of everything we have talked about so far is very pretty.

8 Keepmg this pnnc1p]e m mmd should help the student avmd bemg confused by thmgs such as the "reduction of a wave packet" and similar magic

3
Electrons and Their
Interactions
This is the third of four lectures on a rather difficult subject-the theory of quantum electrodynamics-and since there are obviously more people here tonight than there were before, some of you haven't heard the other two lectures and will find this lecture almost incomprehensible. Those of you who have heard the other two lectures will also find this lecture incomprehensible, but you know that that's all right: as I explained in the first lecture, the way we have to describe Nature is generally incomprehensible to us.
In these lectures I want to tell you about the part of physics that we know best, the interaction of light and electrons. 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.
We found out in the first lecture that we have no satisfactory mechanism to describe even the simplest of phenomena, such as partial reflection of light by glass. We also have no way to predict whether a given photon will be reflected or transmitted by the glass. All we can do is calculate the probability that a particular event will happen-

78

Chapter 3

whether the light will be reflected, in this case. (This is about 4%, when the light shines straight down on a single surface of glass; the probability of reflection increases as the light hits the glass at more of a slant.)
When we deal with probabilities under ordinary circumstances, there are the following "rules of composition": l) if something can happen in alternative ways, we add the probabilities for each of the different ways; 2) if 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).
In the wild and wonderful world of quantum physics, probabilities are calculated as the square of the length of an arrow: where we would have expected to add the probabilities under ordinary circumstances, we find ourselves "adding" arrows; where we normally would have multiplied the probabilities, we "multiply" arrows. The peculiar answers that we get from calculating probabilities in this manner match perfectly the results of experiment. I'm rather delighted that we must resort to such peculiar rules and strange reasoning in order to understand Nature, and I enjoy telling people about it. There are no "wheels and gears" beneath this analysis of Nature; if you want to understand Her, this is what you have to take.
Before I go into the main part of this lecture, I'd like to show you another example of how light behaves. What I would like to talk about is very weak light of one colorone photon at a time-going from a source, at S, to a detector, at D (see Fig. 49). Let's put a screen in between the source and the detector and make two very tiny holes a few millimeters apart from each other, at A and B. (If the source and detector are I00 centimeters apart, the holes have to be smaller than a tenth of a millimeter.) Let's put A in line with S and D, and put B somewhere to the side of A, not in line with Sand D.

Electrons and Their Interactions

79

When we close the hole at B, we get a certain number of clicks at D-which represents the photons that came through A (let's say the detector clicks an average of one
time for every 100 photons that leave S, or 1%). When we
close the hole at A and open the hole at B, we know from the second lecture that we get nearly the same number of clicks, on average, because the holes are so small. (When we "squeeze" light too much, the rules of the ordinary world-such as light goes in straight lines-fall apart.)

FIGURE 49. Two tiny holes (at A and B) zn a screen that is between a source S and a detector D let nearly the same amount of light through (m this case 1%) when one or the other hole is open. When both holes are open, "interference" occurs. the detector clicks from zero to 4% of the time, depending on the separation of A and B-shown m Figure 51 (a).
When we open both holes we get a complicated answer, because interference is present: If the holes are a certain distance apart, we get more clicks than the expected 2% (the maximum is about 4%); if the two holes are a slightly different distance apart, we get no clicks at all.
One would normally think that opening a second hole would always increase the amount of light reaching the detector, but that's not what actually happens. And so saying that the light goes "either one way or the other" is false. I still catch myself saying, "Well, it goes either this way or that way," but when I say that, I have to keep in mind that

80

Chapter 3

I mean in the sense of adding amplitudes: the photon has an amplitude to go one way, and an amplitude to go the other way. If the amplitudes oppose each other, the light won't get there--even though, in this case, both holes are open.
Now, here's an extra twist to the strangeness of Nature that I'd like to tell you about. Suppose we put in some special detectors-one at A and one at B (it is possible to design a detector that can tell whether a photon went through it)-so we can tell through which hole(s) the photon goes when both holes are open (see Fig. 50). Since the

t

S~1~D
* special detectors
FIGURE 50. When special detectors are put in at A and B to tell which way the light went when both holes are open, the expenment has been changed. Because a photon always goes through one hole or the other (when you are checking the holes), there are two dastznguashable final conditions. 1) the detectors at A and D go off, and 2) the detectors at B and D go off The probability
of either event happening as about 1%. The probabilities of the two events are added in the normal way, which accounts for a 2 %probability that the detector
at D goes off-shown in Figure 5l(b).
probability that a single photon will get from S to D is affected only by the distance between the holes, there must be some sneaky way that the photon divides in two and then comes back together again, right? According to this hypothesis, the detectors at A and B should always go off together {at half strength, perhaps?), while the detector at D should go off with a probability of from zero to 4%, depending on the distance between A and B.

Electrons and Their Interactions

81

Here's what actually happens: the detectors at A and B never go off together--either A or B goes off. The photon does not divide in two; it goes one way or the other.
Furthermore, under such conditions the detector at D goes off 2% of the time-the simple sum of the probabilities
for A and B (1 % + l %). The 2% is not affected by the
spacing between A and B; the interference disappears when detectors are put in at A and B!
Nature has got it cooked up so we'll never be able to figure out how She does it: if we put instruments in to find out which way the light goes, we can find out, all right, but the wonderful interference effects disappear. But if we don't have instruments that can tell which way the light goes, the interference effects come back! Very strange, indeed!
To understand this paradox, let me remind you of a most important principle: in order to correctly calculate the probability of an event, one must be very careful to de.fine the complete event clearly-in particular, what the initial conditions and the final conditions of the experiment are. You look at the equipment before and after the experiment, and look for changes. When we were calculating the probability that a photon gets from S to D with no detectors at A or B, the event was, simply, the detector at D makes a click. When a click at D was the only change in conditions, there was no way to tell which way the photon went, so there was interference.
When we put in detectors at A and B, we changed the problem. Now, it turns out, there are two complete eventstwo sets of final conditions-that are distinguishable: 1) the detectors at A and D go off, or 2) the detectors at B and D go off. When there are a number of possible final conditions in an experiment, we must calculate the probability of each as a separate, complete event.
To calculate the amplitude that the detectors at A and D go off, we multiply the arrows that represent the follow-

82

Chapter 3

ing steps: a photon goes from S to A, the photon goes from A to D, and the detector at D goes off. The square of the final arrow is the probability of this event-I %-the same as when the hole at B was closed, because both cases have exactly the same steps. The other complete event is the detectors at B and D go off. The probability of this event is calculated in a similar way, and is also the same as before-about l %.
If we want to know how often the detector at D goes off and we don't care whether it was A or B that went off in the process, the probability is the simple sum of the two events-2%. In principle, if there is something left in the system that we could have observed to tell which way the photon went, we have different "final states" (distinguishable final conditions), and we add the probabilities-not the amplitudes-for each final state. 1
I have pointed out these things because the more you see how strangely Nature behaves, the harder it is to make a model that explains how even the simplest phenomena actually work. So theoretical physics has given up on that.
We saw in the first lecture how an event can be divided into alternative ways and how the arrow for each way can be "added." In the second lecture, we saw how each way can be divided into successive steps, how the arrow for each step can be regarded as the transformation of a unit arrow,

1 The complete story on this situation is very interesting: if the detectors at A and B are not perfect, and detect photons only some of the time, then there are three distinguishable final conditions: l) the detectors at A and D go off; 2) the detectors at Band D go off, and 3) the detector at D goes off alone, with A and B unchanged (they are left in their initial state). The probabilities for the first two events are calculated in the way ex• plained above (except that there will be an extra step-a shrink for the probability that the detector at A [or B] goes off, since the detectors are not perfect). When D goes off alone, we can't separate the two cases, and Nature plays with us by bringing in interference-the same peculiar an• swer we would have had if there were no detectors (except that the final arrow is shrunk by the amplitude that the detectors do not go off). The final result is a mixture, the simple sum of all three cases (see Fig. 51). As the reliability of the detectors increases, we get less interference.

Electro~ns and Their Interactions 83
and how the arrows for each step can be "multiplied" by successive shrinks and turns. We are thus familiar with all the necessary rules for drawing and combining arrows (that represent bits and pieces of events) to obtain a final arrow, whose square is the probability of an observed physical event.
It is natural to wonder how far we can push this process of splitting events into simpler and simpler subevents. What are the smallest possible bits and pieces of events? Is there

Percentage of light

reaching D

(al

( b)

4%

4%

3%

3%

2%

2%

1%

1%

0%

0%

Distance SBD-SAD

(c)

(d)

4%

4%

3%

3%

2%

1%

1%

0%

0%

FIGURE 51. When there are no detectors at A or B, there zs interferencethe afflQunt of light varies from zero to 4% (a). When there are detectors at A and B that are I 00% reliable, there is no znteiference-the amount of light reaching Dis a constant2% (b). When the detectors at A and Bare not 100% reliable (i.e., when sometimes there zs nothing left in A or in B that can be detected), there are now three possible final conditions-A and D go off, B and D go off, and D goes off alone. The final curve is thus a mixture, made
up of contributions from each possible final condition. When the detectors at
A and Bare less reliable, there is more interference present. Thus the detectors m case (c) are less reliable than in case (d). Theprmciple regardingmteiference is: The probability of each of the different possible final conditions must be independently calculated by adding arrows and squaring the length of the ft nal arrow; after that, the several probabilities are added together m the normal fashion.

84

Chapter 3

a limited number of bits and pieces that can be compounded to form all the phenomena that involve light and electrons? Is there a limited number of "letters" in this language of quantum electrodynamics that can be combined to form "words" and "phrases" that describe nearly every phenomenon of Nature?
The answer is yes; the number is three. There are only three basic actions needed to produce all of the phenomena associated with light and electrons.
Before I tell you what these three basic actions are, I should properly introduce you to the actors. The actors are photons and electrons. The photons, particles of light, have been discussed at length in the first two lectures. Electrons were discovered in 1895 as particles: you could count them; you could put one of them on an oil drop and measure its electric charge. It gradually became apparent that the motion of these particles accounted for electricity in wires.
Shortly after electrons were discovered it was thought that atoms were like little solar systems, made up of a central, heavy part (called the nucleus) and electrons, which went around in "orbits," much like the planets do when they go around the sun. If you think that's the way atoms are, then you're back in 1910. In 1924 Louis De Broglie found that there was a wavelike character associated with
electrons, and soon afterwards, C. J. Davisson and L. H.
Germer of the Bell Laboratories bombarded a nickel crystal with electrons and showed that they, too, bounced off at crazy angles Gust like X-rays do), and that these angles could be calculated from De Broglie's formula for the wavelength of an electron.
When we look at photons on a large scale-much larger than the distance required for one stopwatch turn-the phenomena that we see are very well approximated by rules such as "light travels in straight lines," because there are enough paths around the path of minimum time to rein-

Electrons and Their Interactions 85
force each other, and enough other paths to cancel each other out. But when the space through which a photon moves becomes too small (such as the tiny holes in the screen), these rules fail-we discover that light doesn't have to go in straight lines, there are interferences created by two holes, and so on. The same situation exists with electrons: when seen on a large scale, they travel like particles, on definite paths. But on a small scale, such as inside an atom, the space is so small that there is no main path, no "orbit"; there are all sorts of ways the electron could go, each with an amplitude. The phenomenon of interference becomes very important, and we have to sum the arrows to predict where an electron is likely to be.
It's rather interesting to note that electrons looked like particles at first, and their wavish character was later discovered. On the other hand, apart from Newton making a mistake and thinking that light was "corpuscular," light looked like waves at first, and its characteristics as a particle were discovered later. In fact, both objects behave somewhat like waves, and somewhat like particles. In order to save ourselves from inventing new words such as "wavicles," we have chosen to call these objects "particles," but we all know that they obey these rules for drawing and combining arrows that I have been explaining. It appears that all the "particles" in Nature--quarks, gluons, neutrinos, and so forth (which will be discussed in the next lecture)-behave in this quantum mechanical way.
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. In a moment, I'll tell you those rules, or laws, out of which we can make

86

Chapter 3

the whole world (aside from the nuclei, and gravitation, as always!).
Now, the stage on which these actions take place is not just space, it is space and time. Until now, I have disregarded problems concerning time, such as exactly when a photon leaves the source and exactly when it arrives at the detector. Although space is really three-dimensional, I'm going to reduce it to one dimension on the graphs that I'm going to draw: I will show a particular object's location in space on the horizontal axis, and the time on the vertical axis.
The first event I am going to draw in space and timeor space-time, as I might inadvertently call it-is a baseball standing still (See Fig. 52). On Thursday morning, which

T3t----+-
T2
1j To..._--~----
Xo Space

FIGURE 52. The stage on which all ac-
tions m the universe talu place is spacetime Usually consisting offour dimensions (three for space and one for time), spacetime will be represented here in two dimensions--0ne for space, in the honzontal dimension, and one for time, in the vertical
Each time we look at the baseball (such as
at time T 1), it is in the same place This produces a "band of baseball" going straight up, as time goes on

I will label as T 0, the baseball occupies a certain space, which I will label as X0. A few moments later, at T 1, it occupies the same space, because it's standing still. A few moments later, at T 2, the baseball is still at X0. So the diagram of a baseball standing still is a vertical band, going straight up, with baseball all over it inside.
What happens ifwe have a baseball drifting in the weightlessness of outer space, going straight toward a wall? Well,

Electrons and Their Interactions

8 7

on Thursday morning (T0) it starts at Xo (see Fig. 53), but a little bit later, it's not in the same place-it has drifted over a little bit, to X1. As the baseball continues to drift, it creates a slanted "band of baseball" on the diagram of space-time. When the baseball hits the wall (which is standing still and is therefore a vertical band), it goes back the other way, exactly where it came from in space (Xo), but to a different point in time (T6).

FIGURE 53. A baseball dnftmg directly toward a wall at nght angles and then bounczng back to its ongmal location (shoum below the graph) is moving m one dimension and appears as a slanted "band of baseball " At times T1 and T2 , the baseball is getting closer to the wall, at T1 it hits the wall, and begins to go back

Time
T6

Ts

,;.

w

A

T3

L L

T2

lj
To L----~........,..____.___ Xo -X3
0 !WALL!

As for the time scale, it is most convenient to represent the time not in seconds, but in much smaller units. Since we will be dealing with photons and electrons, which move very rapidly, I am going to have a 45° angle represent something going the speed of light. For example, for a particle moving at the speed of light from X1T 1 to X2T 2, the horizontal distance between X1 and X2 is the same as the vertical distance between T 1 and T 2 (see Fig. 54). The factor by which time is stretched out (to make a 45° angle represent a particle going the speed of light) is called c, and you'll find e's flying around everywhere in Einstein's formulas-they are the result of the unfortunate choice of the second as the unit of time, rather than the time it takes light to go one meter.
Now, let's look at the first basic action in detail-a photon

88

Chapter 3

goes from place to place. I will draw this action as a wiggly line from A to B for no good reason. I should be more careful: I should say, a photon that is known to be at a given place at a given time has a certain amplitude to get to another place at another time. On my space-time graph (see Fig. 55), the photon at point A-at X1 and T 1-has an amplitude to appear at point B-X2 and T 2. The size of this amplitude I will call P(A to B).

TT2 (The time
,t tokes light to go 30cm)
J_T I

x,

x 2 Space

r-30 cm-,

FIGURE 54. The tzme scale I will use m these g;raphs wzll show particles going at the speed oflzght to be travelling at a 45-deg;ree angle through space-time. The amount of time zt takes light to go 30 centimeters-from X 1 to X2 or from X2 to X1-is about one-billionth of a second.

Time

A
Space

FIGURE 55. A photon (represented fry a wavy lint) has an amplitude to go

from a point A zn space-time to another point, B. This amplitude, which I

will call P(A to B), is cakulated from a formula that depends only on the

difference zn locatzon-(X X )-and the difference of the tzme-(T

-

-

2

1

2

T 1). In fact, it's a simple function that is the inverse of the difference of their

squares-an "interval," I, that can be wntten as (X2 -X1)2 - (T2 - T 1)2.

Electrons and Their Interactions

89

There is a formula for the size of this arrow, P(A to B). It is one of the great laws of Nature, and it's very simple. It depends on the difference in distance and the difference in time between the two points. These differences can be expressed mathematically2 as (X2 - Xi) and (T2 - T 1).
The major contribution to P(A to B) occurs at the conventional speed of light-when (X2 - X 1) is equal to (T2 - T 1)-where one would expect it all to occur, but there is also an amplitude for light to go faster (or slower) than the conventional speed of light. You found out that in the last lecture that light doesn't go only in straight lines; now, you find out that it doesn't go only at the speed of light!
It may surprise you that there is an amplitude for a photon to go at speeds faster or slower than the conventional speed, c. The amplitudes for these possibilities are very small compared to the contribution from speed c; in fact, they cancel out when light travels over long distances. However, when the distances are short-as in many of the

2 In these lectures, I am plotting a point's location in space in one dimension, along the x-axis. To locate a point in three-dimensional space, a "room" has to be set up, and the distance of the point from the floor and from each of two adjacent walls (all at right angles to each other) has to be measured. These three measurements can be labeled X1, Yi, and Z1. The actual distance from this point to a second point with measurements X2, Y2, ~ can be calculated usmg a "three-dimensional Pythagorean Theorem": the square of this actual distance is
(X2 - X1)2 + (Y2 - Y,)2 + (Z2 - Z,)2•
The excess of this over the time difference, squared-
(X2 - Xi)2 + (Y2 - Y,)2 + (Z2 - Z1)2 - (T2 - T 1) 2
-is sometimes called "the Interval," or /, and is the combination that Einstein's theory of relativity says that P(A to B) must depend on. Most of the contribution to the final arrow for P(A to B) is just where you would expect it-where the difference in distance is equal to the difference in time (that is, when/ 1s zero). But in addition; there is a contribution when I is not zero, that is inversely proportional to /: it points in the direction of 3 o'clock when / is more than zero (when light is going faster than c), and points toward 9 o'clock when / is less than zero. These later contributions cancel out in many circumstances (see Fig. 56).

90

Chapter 3

diagrams I will be drawing-these other possibilities become vitally important and must be considered.
So that's the first basic action, the first basic law of physics-a photon goes from point to point. That explains all about optics; that's the entire theory of light! Well, not quite: I left out polarization (as always), and the interaction of light with matter, which brings me to the second law.

I= 0 (speedC)

(slower than C)I< 0

I>O(faster than C)

FIGURE 56. When light goes at the speed C, the "interval," I, equals zero, and there is a large contribution in the 12 o'clock direction. When I is greater than zero, there is a small contribution in the three o'clock direction inversely proportional to I; when I is less than zero, there is a similar contribution in the nine o'clock direction. Thus light has an amplitude to go faster or slower than speed C, but these amplitudes cancel out over long distances.

The second action fundamental to quantum electrody-

namics is: An electron goes from point A to point B in

space-time. (For the moment we will imagine this electron

as a simplified, fake electron, with no polarization-what

the physicists call a "spin-zero" electron. In reality, electrons

have a type of polarization, which doesn't add anything to

the main ideas; it only complicates the formulas a little bit.)

The formula for the amplitude for this action, which I will

call E(A to B) also depends on (X -X and (T -T (in

)

)

2

1

2

1

the same combination as described in note 2) as well as on

a number I will call "n," a number that, once determined,

enables all our calculations to agree with experiment. (We

will see later how we determine n's value.) It is a rather

complicated formula, and I'm sorry that I don't know how

Electrons and Their Interactions

91

to explain it in simple terms. However, you might be interested to know that the formula for P(A to B)-a photon going from place to place in space-time-is the same as that for E(A to B)-an electron going from place to place-if n is set to zero.3
The third basic action is: an electron emits or absorbs a photon-it doesn't make any difference which. I will call this action a 'junction," or ..coupling." To distinguish electrons from photons in my diagrams, I will draw each electron going through space-time as a straight line. Every coupling, therefore, is a junction between two straight lines and a wavy line (see Fig. 58). There is no complicated formula for the amplitude of an electron to emit or absorb a photon; it doesn't depend on anything-it's just a number! This junction number I will call j-its value is about - 0.1: a shrink to about one-tenth, and half a turn.4
Well, that's all there is to these basic actions-except for some slight complications due to this polarization that we're

3 The formula for E(A to B) is complicated, but there is an interesting way to explain what it amounts to. E(A to B) can be represented as a giant sum of a lot of different ways an electron could go from point A to point B in space-time (see Fig. 57): the electron could take a "one-hop flight," going directly from A to B; it could take a "two-hop flight," stopping at an intermediate point C; it could take a "three-hop flight," stopping at points D and E, and so on. In such an analysis, the amplitude for each "hop"-from one point F to another point G-is P(F to G), the same as the amplitude for a photon to go from a point F to a point G. The amplitude for each "stop" is represented by n2, n being the same number I mentioned before which we used to make our calculations come out right.
The formula for E(A to B) is thus a series of terms: P(A to B) [the ''onehop" flight] + P(A to C)*n2*P(C to B) ["two-hop" flights, stopping at C]
+ P(A to D)*n2*P(D to E) * n2*P(E to B) ["three-hop" flights, stopping at
D and E] + ... for all possible intermediate points C, D, E, and so on. Note that when n increases, the nondirect paths make a greater con-
tribution to the final arrow. When n is zero (as for the photon), all terms with an n drop out (because they are also equal to zero), leaving only the first term, which is P(A to B). Thus E(A to B) and P(A to B) are closely related.
4 This number, the amplitude to emit or absorb a photon, is sometimes called the "charge" of a particle.

92
(a)

Chapter 3
( b)

Space

Space

FIGURE 57. An electron has an amplitude to go from point to point in space-time, which I will call "E(A to B).'' Although I will represent E(A to B) as a straight line between two points (a), we can think of it as the sum of many amplitudes (b)-among them, the amplitude for the electron to change direction at points C or C' on a "two-hop" path, and the amplitude to change direction at D and E on a "three-hop" path-in addition to the direct path from A to B. The number oftimes an electron can change direction is anywhere from zero to infinity, and the points at which the electron can change direction on its way from A to B in space-time are infinite. All are included in E(A toB).

Time

FIGURE 58. An ekctron, depicted by a

straight line, has a certain amplitude to

emit or absorb a photon, shown by a wavy

line. Since the amplitude to emit or absorb

is the same, I will call either case a "cou-

pling." The amplitude for a coupling is a

number that I will call j; it is about - 0.1

for the electron (this number is sometimes

called the "charge").

Space

always leaving out. Our nextjob is to put these three actions together to represent circumstances that are somewhat more complicated.
For our first example, let's calculate the probability that two electrons, at points l and 2 in space-time, end up at

Electrons and Their Interactions

93

points 3 and 4 (see Fig. 59). This event can happen in several ways. The first way is that the electron at I goes to 3--computed by putting I and 3 into the formula E(A to B), which I will write as E(l to 3)-and the electron at 2 goes to 4--computed by E(2 to 4). These are two "subevents" happening concomitantly, so the two arrows are multiplied to produce an arrow for this first way the event could happen. Therefore we write the formula for the "first-way arrow" as E(l to 3) * E(2 to 4).

4

4

2

2

Space

Space

FIGURE 59. To calculate the probability that electrons at points I and 2 in space-time end up at points 3 and 4, we calculate the ''first way" arrow for 1 going to 3 and 2 going to 4 with the formula for E(A to B); then we calculate the "second way'' arrow for 1 going to 4 and 2 going to 3 (a "crossover"). Finally, we add the ''first way" and "second way" arrows to arrive at a good approximation of the final arrow. (This is true for the fake, simplified "spin zero" electron. Had we included the polarization of the electron, we would have subtracted-rather than added-the two arrows.)

Another way this event could happen is that the electron at I goes to 4 and the electron at 2 goes to 3-again, two concomitant subevents. The "second-way arrow" is E( l to 4) * E(2 to 3), and we add it to the "first-way" arrow.5
This is a good approximation for the amplitude of this event. To make a more exact calculation that will agree more closely with the results of experiment, we must con-
s Had I included the effects of the polarization of the electron, the "second-way" arrow would have been "subtracted"-turned 180° and added. (More on this comes later in this lecture.)

94

Chapter 3

sider other ways this event could happen. For instance, for each of the two main ways the event can happen, one electron could go charging off to some new and wonderful place and emit a photon (see Fig. 60). Meanwhile, the other electron could go to some other place and absorb the pho-

Time 4

Space

Space

FIGURE 60. Two "other ways" the event in Fig. 59 could happen are: a photon is emitted at 5 and absorbed at 6 for each case. The final conditions of these alternatives are the same as for the other cases-two electrons went in, and two electrons came out-and these results are indistinguishab/,e from the other alternatives. There[ore the arrows for these "other ways" must be added to the arrows in Fig. 59 to arrive at a better approximation of the final arrow for the event.

ton. Calculating the amplitude for the first of these new ways involves multiplying the amplitudes for: an electron goes from l to the new and wonderful place, 5 (where it emits a photon), and then goes from 5 to 3; the other electron goes from 2 to the other place, 6 (where it absorbs the photon), and then goes from 6 to 4. We must remember to include the amplitude that the photon goes from 5 to 6. I'm going to write the amplitude for this way the event could happen in a high-class mathematical fashion, and you can follow along: E(l to 5)*}*E(5 to 3) * E(2 to 6)*}*E(6 to 4) * P(5 to 6)-a lot of shrinking and turning. (I'll let you figure out the notation for the other case, where the electron at I ends up at 4, and the electron at 2 ends up at 3.)6
6 The final conditions of the experiment for these more complicated ways are the same as for the simpler ways-electrons start at points 1 and

Electrons and Their Interactions 95
But wait: positions 5 and 6 could be anywhere in space and time-yes, anywhere-and the arrows for all of those positions have to be calculated and added together. You see it's getting to be a lot of work. Not that the rules are so difficult-it's like playing checkers: the rules are simple, but you use them over and over. So our difficulty in calculating comes from having to pile so many arrows together. That's why it takes four years of graduate work for the students to learn how to do this efficiently-and we're looking at an easy problem! (When the problems get too difficult, we just put them on the computer!)
I would like to point out something about photons being emitted and absorbed: if point 6 is later than point 5, we might say that the photon was emitted at 5 and absorbed at 6 (see Fig. 61). If 6 is earlier than 5, we might prefer to say the photon was emitted at 6 and absorbed at 5, but we could just as well say that the photon is going backwards in time! However, we don't have to worry about which way in space-time the photon went; it's all included in the formula for P(5 to 6), and we say a photon was "exchanged." Isn't it beautiful how simple Nature is!7
Now, in addition to the photon that is exchanged between 5 and 6, another photon could be exchanged-between two points, 7 and 8 (see Fig. 62). I'm too tired to write down all the basic actions whose arrows have to be multiplied, but-as you may have noticed-every straight line gets an E(A to B), every wavy line gets a P(A to B), and every coupling gets aj. Thus, there are six E(A to B)'s, two P(A to B)'s, and four j's-for every possible 5, 6, 7, and 8! That makes billions of tiny arrows that have to be multiplied and then added together!
2 and end up at points 3 and 4-so we cannot distinguish between these alternatives and the first two. Therefore we must add the arrows for these two ways to the two ways just previously considered.
7 Such an exchanged photon that never really appears in the initial or final conditions of the experiment is sometimes called a "virtual photon."