The development of the space-time view of quantum electrodynamics

This is the html version of the file http://www.nobel.se/physics/laureates/1965/feynman-lecture.pdf.
G o o g l e automatically generates html versions of documents as we crawl the web.
To link to or bookmark this page, use the following url:

http://www.google.com/search?q=cache:nSLP55NKaOAJ:www.nobel.se/physics/laureates/1965/feynman-lecture.pdf+feynman+%22all+the+same+electron%22&hl=en&ie=UTF-8

Google is not affiliated with the authors of this page nor responsible for its content.

These search terms have been highlighted:

feynman

all

the

same

electron

Page 1

RI C H A R D P . FE Y N M A N

The development of the space-time view

of quantum electrodynamics

Nobel Lecture, December 11, 1965

We have a habit in writing articles published in scientific journals to make the

work as finished as possible, to cover all the tracks, to not worry about the

blind alleys or to describe how you had the wrong idea first, and so on. So

there isn’t any place to publish, in a dignified manner, what you actually did

in order to get to do the work, although, there has been in these days, some

interest in this kind of thing. Since winning the prize is a personal thing,

thought I could be excused in this particular situation, if I were to talk per-

sonally about my relationship to quantum electrodynamics, rather than to

discuss the subject itself in a refined and finished fashion. Furthermore, since

there are three people who have won the prize in physics, if they are all going

to be talking about quantum electrodynamics itself, one might become bored

with the subject. So, what I would like to tell you about today are the sequence

of events, really the sequence of ideas, which occurred, and by which I finally

came out the other end with an unsolved problem for which I ultimately

received a prize.

I realize that a truly scientific paper would be of greater value, but such a

paper I could publish in regular journals. So, I shall use this Nobel Lecture as

an opportunity to do something of less value, but which I cannot do elsewhere.

I ask your indulgence in another manner. I shall include details of anecdotes

which are of no value either scientifically, nor for understanding the develop-

ment of ideas. They are included only to make the lecture more entertaining.

I worked on this problem about eight years until the final publication in

1947.

The beginning of the thing was at the Massachusetts Institute of Tech-

nology, when I was an undergraduate student reading about the known phys-

ics, learning slowly about all these things that people were worrying about,

and realizing ultimately that the fundamental problem of the day was that

the quantum theory of electricity and magnetism was not completely satis-

factory. This I gathered from books like those of Heitler and Dirac. I was in-

spired by the remarks in these books; not by the parts in which everything

was proved and demonstrated carefully and calculated, because I couldn’t

Page 2

156

1 9 6 5

R I C H A R D P . F E Y N M A N

understand those very well. At the young age what I could understand were

the remarks about the fact that this doesn’t make any sense, and the last sen-

tence of the book of Dirac I can still remember, « It seems that some essentially

new physical ideas are here needed. » So, I had this as a challenge and an in-

spiration. I also had a personal feeling, that since they didn’t get a satisfactory

answer to the problem I wanted to solve, I don’t have to pay a lot of attention

to what they did do.

I did gather from my readings, however, that two things were the source

of the difficulties with the quantum electrodynamical theories. The first was

an infinite energy of interaction of the electron with itself. And this difficulty

existed even in the classical theory. The other difficulty came from some in-

finites which had to do with the infinite numbers of degrees of freedom in the

field. As I understood it at the time( as nearly as I can remember) this was simply

the difficulty that if you quantized the harmonic oscillators of the field (say in a

box) each oscillator has a ground state energy of (

and there is an infinite

number of modes in a box of every increasing frequency

w, and therefore

there is an infinite energy in the box. I now realize that that wasn’t a complete-

ly correct statement of the central problem; it can be removed simply by

changing the zero from which energy is measured. At any rate, I believed

that the difficulty arose somehow from a combination of the electron acting

on itself and the infinite number of degrees of freedom of the field.

Well, it seemed to me quite evident that the idea that a particle acts on itself,

that the electrical force acts on the same particle that generates it, is not a

necessary one-it is a sort of a silly one, as a matter of fact. And, so I suggested

to myself, that electrons cannot act on themselves, they can only act on other

electrons. That means there is no field at all. You see, if all charges contribute

to making a single common field, and if that common field acts back on all

the charges, then each charge must act back on itself. Well, that was where the

mistake was, there was no field. It was just that when you shook one charge,

another would shake later. There was a direct interaction between charges,

albeit with a delay. The law of force connecting the motion of one charge

with another would just involve a delay. Shake this one, that one shakes later.

The sun atom shakes; my eye electron shakes eight minutes later, because of a

direct interaction across.

Now, this has the attractive feature that it solves both problems at once.

First, I can say immediately, I don’t let the electron act on itself, I just let this

act on that, hence, no self-energy! Secondly, there is not an infinite number

of degrees of freedom in the field. There is no field at all; or if you insist on

Page 3

S P A C E - T I M E V I E W O F Q U A N T U M E L E C T R O D Y N A M I C S

157

thinking in terms of ideas like that of a field, this field is always completely

determined by the action of the particles which produce it. You shake this

particle, it shakes that one, but if you want to think in a field way, the field,

if it’s there, would be entirely determined by the matter which generates it,

and therefore, the field does not have any

independent

degrees of freedom and

the infinities from the degrees offreedom would then be removed. As a mat-

ter of fact, when we look out anywhere and see light, we can always « see »

some matter as the source of the light. We don’t just see light (except recently

some radio reception has been found with no apparent material source).

You see then that my general plan was to first solve the classical problem,

to get rid of the infinite self-energies in the classical theory, and to hope that

when I made a quantum theory of it, everything would just be fine.

That was the beginning, and the idea seemed so obvious to me and so ele-

gant that I fell deeply in love with it. And, like falling in love with a woman, it

is only possible if you do not know much about her, so you cannot see her

faults. The faults will become apparent later, but after the love is strong enough

to hold you to her. So, I was held to this theory, in spite of all difficulties, by

my youthful enthusiasm.

Then I went to graduate school and somewhere along the line I learned

what was wrong with the idea that an electron does not act on itself. When

you accelerate an electron it radiates energy and you have to do extra work

to account for that energy. The extra force against which this work is done is

called the force of radiation resistance. The origin of this extra force was iden-

tified in those days, following Lorentz, as the action of the electron itself The

first term of this action, of the electron on itself, gave a kind of inertia (not

quite relativistically satisfactory). But that inertia-like term was infinite for

a point-charge. Yet the next term in the sequence gave an energy loss rate,

which for a point-charge agrees exactly with the rate you get by calculating

how much energy is radiated. So, the force of radiation resistance, which is

absolutely necessary for the conservation of energy would disappear if I said

that a charge could not act on itself.

So, I learned in the interim when I went to graduate school the glaringly

obvious fault of my own theory. But, I was still in love with the original

theory, and was still thinking that with it lay the solution to the difficulties of

quantum electrodynamics. So, I continued to try on and off to save it some-

how. I must have some action develop on a given electron when I accelerate

it to account for radiation resistance. But, if I let electrons only act on other

electrons the only possible source for this action is another electron in the

Page 4

158

1 9 6 5

R I C H A R D P . F E Y N M A N

world. So, one day, when I was working for Professor Wheeler and could no

longer solve the problem that he had given me, I thought about this again and

calculated the following. Suppose I have two charges-I shake the first

charge, which I think of as a source and this makes the second one shake, but

the second one shaking produces an effect back on the source. And so, I cal-

culated how much that effect back on the first charge was, hoping it might

add up the force of radiation resistance. It didn’t come out right, of course,

but I went to Professor Wheeler and told him my ideas. He said, -yes, but

the answer you get for the problem with the two charges that you just men-

tioned will, unfortunately, depend upon the charge and the mass of the second

charge and will vary inversely as the square of the distance R, between the

charges, while the force ofradiation resistance depends on none of these things.

thought, surely, he had computed it himself, but now having become a pro-

fessor, I know that one can be wise enough to see immediately what some

graduate student takes several weeks to develop. He also pointed out some-

thing that also bothered me, that if we had a situation with many charges all

around the original source at roughly uniform density and if we added the

effect of all the surrounding charges the inverse R square would be compen-

sated by the R

in the volume element and we would get a result proportional

to the thickness of the layer, which would go to infinity. That is, one would

have an infinite total effect back at the source. And, finally he said to me, and

you forgot something else, when you accelerate the first charge, the second

acts later, and then the reaction back here at the source would be still later. In

other words, the action occurs at the wrong time. I suddenly realized what a

stupid fellow I am, for what I had described and calculated was just ordinary

reflected light, not radiation reaction.

But, as I was stupid, so was Professor Wheeler that much more clever. For

he then went on to give a lecture as though he had worked this all out before

and was completely prepared, but he had not, he worked it out as he went

along. First, he said, let us suppose that the return action by the charges in the

absorber reaches the source by advanced waves as well as by the ordinary re-

tarded waves of reflected light; so that the law ofinteraction acts backward in

time, as well as forward in time. I was enough of a physicist at that time not to

say, « Oh, no, how could that be? » For today all physicists know from study-

ing Einstein and Bohr, that sometimes an idea which looks completely para-

doxical at first, if analyzed to completion in all detail and in experimental

situations, may, in fact, not be paradoxical. So, it did not bother me any more

Page 5

S P A C E - T I M E V I E W O F Q U A N T U M E L E C T R O D Y N A M I C S

159

than it bothered Professor Wheeler to use advance waves for the back reaction

-a solution of Maxwell’s equations, which previously had not been physically

used.

Professor Wheeler used advanced waves to get the reaction back at the right

time and then he suggested this : If there were lots of electrons in the absorber,

there would be an index of refraction n, so, the retarded waves coming from

the source would have their wave lengths slightly modified in going through

the absorber. Now, if we shall assume that the advanced waves come back

from the absorber without an index-why? I don’t know, let’s assume they

come back without an index-then, there will be a gradual shifting in phase

between the return and the original signal so that we would only have to

figure that the contributions act as if they come from only a finite thickness,

that of the first wave zone. (More specifically, up to that depth where the

phase in the medium is shifted appreciably from what it would be in vacuum,

a thickness proportional to

). )

N ow, the less the number of electrons

in here, the less each contributes, but the thicker will be the layer that effec-

tively contributes because with less electrons, the index differs less from

The

higher the charges of these electrons, the more each contribute, but the thinner

the effective layer, because the index would be higher. And when we estimat-

ed it, (calculated without being careful to keep the correct numerical factor)

sure enough, it came out that the action back at the source was completely

independent of the properties of the charges that were in the surrounding ab-

sorber. Further, it was of just the right character to represent radiation resis-

tance, but we were unable to see if it was just exactly the right size. He sent

me home with orders to figure out exactly how much advanced and how

much retarded wave we need to get the thing to come out numerically right,

and after that, figure out what happens to the advanced effects that you would

expect if you put a test charge here close to the source? For if all charges gen-

erate advanced, as well as retarded effects, why would that test not be affected

by the advanced waves from the source?

I found that you get the right answer if you use half-advanced and half-

retarded as the field generated by each charge. That is, one is to use the solution

of Maxwell’s equation which is symmetrical in time and that the reason we

got no advanced effects at a point close to the source in spite of the fact that

the source was producing an advanced field is this. Suppose the source s sur-

rounded by a spherical absorbing wall ten light seconds away, and that the

test charge is one second to the right of the source. Then the source is as much

Page 6

160

1 9 6 5

R I C H A R D P . F E Y N M A N

as eleven seconds away from some parts of the wall and only nine seconds

away from other parts. The source acting at time t= o induces motions in the

wall at time +

Advanced effects from this can act on the test charge as

early as eleven seconds earlier, or at t= -

This is just at the time that the

direct advanced waves from the source should reach the test charge, and it

turns out the two effects are exactly equal and opposite and cancel out! At

the later time +

effects on the test charge from the source and from the walls

are again equal, but this time are of the same sign and add to convert the half-

retarded wave of the source to full retarded strength.

Thus, it became clear that there was the possibility that if we assume all

actions are via half-advanced and half-retarded solutions of Maxwell’s equa-

tions and assume that all sources are surrounded by material absorbing all the

the light which is emitted, then we could account for radiation resistance as

a direct action of the charges of the absorber acting back by advanced waves

on the source.

Many months were devoted to checking all these points. I worked to show

that everything is independent of the shape of the container, and so on, that

the laws are exactly right, and that the advanced effects really cancel in every

case. We always tried to increase the efficiency of our demonstrations, and to

see with more and more clarity why it works. I won’t bore you by going

through the details of this. Because of our using advanced waves, we also had

many apparent paradoxes, which we gradually reduced one by one, and saw

that there was in fact no logical difficulty with the theory. It was perfectly satis-

factory.

We also found that we could reformulate this thing in another way, and

that is by a principle of least action. Since my original plan was to describe

everything directly in terms of particle motions, it was my desire to represent

this new theory without saying anything about fields. It turned out that we

found a form for an action directly involving the motions of the charges only,

which upon variation would give the equations of motion of these charges.

The expression for this action

where

Xi,

is the four-vector position of the i

particle as a function of

Page 7

S P A C E - T I M E V I E W O F Q U A N T U M E L E C T R O D Y N A M I C S

161

some parameter

(a)

The first term is the integral of

proper time, the ordinary action of relativistic mechanics of free particles of

mass

(We sum in the usual way on the repeated index

m.) The second term

represents the electrical interaction of the charges. It is summed over each pair

of charges (the factor

is to count each pair once, the term

i=j

is omitted

to avoid self- action) .The interaction is a double integral over a delta function

of the square of space- time interval between two points on the paths. Thus,

interaction occurs only when this interval vanishes, that is, along light cones.

The fact that the interaction is exactly one- half advanced and half- retarded

meant that we could write such a principle of least action, whereas interaction

via

retarded waves alone cannot be written in such a way.

So, all of classical electrodynamics was contained in this very simple form.

It looked good, and therefore, it was undoubtedly true, at least to the beginner.

It automatically gave half- advanced and half-retarded effects and it was with-

out fields. By omitting the term in the sum when i = j, I omit self-interaction

and no longer have any infinite self-energy. This then was the hoped-for

solution to the problem of ridding classical electrodynamics of the infinities.

It turns out, of course, that you can reinstate fields if you wish to, but you

have to keep track of the field produced by each particle separately. This is

because to find the right field to act on a given particle, you must exclude the

field that it creates itself. A single universal field to which all contribute will

not do. This idea had been suggested earlier by Frenkel and so we called these

Frenkel fields. This theory which allowed only particles to act on each other

was equivalent to Frenkel’s fields using half- advanced and half-retarded solu-

tions.

There were several suggestions for interesting modifications of electro-

dynamics. We discussed lots of them, but I shall report on only one. It was to

replace this delta function in the interaction by another function,

which is not infinitely sharp. Instead of having the action occur only when the

interval between the two charges is exactly zero, we would replace the delta

function of I

by a narrow peaked thing. Let’s say that ƒ(Ζ) is large only near

Z= o width of order a

. Interactions will now occur when

- R

is of order

roughly where T is the time difference and

is the separation of the charges.

This might look like it disagrees with experience, but if a is some small dis-

tance, like

-13

cm, it says that the time delay Tin action is roughly

or approximately,-if

is much larger than a,

This means

that the deviation of time

from the ideal theoretical time

of Maxwell, gets

smaller and smaller, the further the pieces are apart. Therefore, all theories

Page 8

162 1 9 6 5 R I C H A R D P. F E Y N M A N

involving in analyzing generators, motors, etc., in fact, all of the tests of

electrodynamics that were available in Maxwell’s time, would be adequately

satisfied if a were

-13

cm. If R is of the order of a centimeter this deviation

in T is only

-26

parts. So, it was possible, also, to change the theory in a

simple manner and to still agree with all observations of classical electrody-

namics. You have no clue of precisely what function to put in for f, but it was

an interesting possibility to keep in mind when developing quantum electro-

dynamics.

It also occurred to us that if we did that (replace δ by ƒ) we could not rein-

state the term

=j in the sum because this would now represent in a relativis-

tically invariant fashion a finite action of a charge on itself. In fact, it was pos-

sible to prove that if we did do such a thing, the main effect of the self-action

(for not too rapid accelerations) would be to produce a modification of the

mass. In fact, there need be no mass m

, term, all the mechanical mass could

be electromagnetic self-action. So, if you would like, we could also have an-

other theory with a still simpler expression for the action A . In expression (1)

only the second term is kept, the sum extended over all

and j, and some func-

tion ƒ replaces δ. Such a simple form could represent all of classical electro-

dynamics, which aside from gravitation is essentially all of classical physics.

Although it may sound confusing, I am describing several different alterna-

tive theories at once. The important thing to note is that at this time we had

all these in mind as different possibilities. There were several possible solu-

tions of the difficulty of classical electrodynamics, any one of which might

serve as a good starting point to the solution of the difficulties of quantum

electrodynamics.

would also like to emphasize that by this time I was becoming used to a

physical point of view different from the more customary point of view. In

the customary view, things are discussed as a function of time in very great

detail. For example, you have the field at this moment, a differential equation

gives you the field at the next moment and so on; a method, which I shall call

the Hamilton method, the time differential method. We have, instead (in

(

)

say) a thing that describes the character of the path throughout all of space

and time. The behavior of nature is determined by saying her whole space-

time path has a certain character. For an action like (1)

the equations obtained

by variation

are no longer at all easy to get back into Hamiltonian

form. If you wish to use as variables only the coordinates of particles, then

you can talk about the property of the paths- but the path of one particle at a

given time is affected by the path of another at a different time. If you try to

Page 9

S P A C E - T I M E V I E W O F Q U A N T U M E L E C T R O D Y N A M I C S

163

describe, therefore, things differentially, telling what the present conditions

of the particles are, and how these present conditions will affect the future-

you see, it is impossible with particles alone, because something the particle

did in the past is going to affect the future.

Therefore, you need a lot of bookkeeping variables to keep track of what

the particle did in the past. These are called field variables. You will, also,

have to tell what the field is at this present moment, if you are to be able to see

later what is going to happen. From the overall space- time view of the least

action principle, the field disappears as nothing but bookkeeping variables in-

sisted on by the Hamiltonian method.

As a by-product of this same view, I received a telephone call one day at

the graduate college at Princeton from Professor Wheeler, in which he said,

« Feynman, I know why all electrons have the same charge and the same mass »

« Why? » « Because, they are all the same electron! » And, then he explained

on the telephone, « suppose that the world lines which we were ordinarily

considering before in time and space - instead of only going up in time were a

tremendous knot, and then, when we cut through the knot, by the plane

corresponding to a fixed time, we would see many, many world lines and

that would represent many electrons, except for one thing. If in one section

this is an ordinary electron world line, in the section in which it reversed itself

and is coming back from the future we have the wrong sign to the proper

time - to the proper four velocities - and that’s equivalent to changing the

sign of the charge, and, therefore, that part of a path would act like a positron. »

« But, Professor », I said, « there aren’t as many positrons as electrons. » « Well,

maybe they are hidden in the protons or something », he said. I did not take

the idea that all the electrons were the same one from him as seriously as I

took the observation that positrons could simply be represented as electrons

going from the future to the past in a back section of their world lines. That, I

stole !

To summarize, when I was done with this, as a physicist I had gained two

things. One, I knew many different ways of formulating classical electro-

dynamics, with many different mathematical forms. I got to know how to

express the subject every which way. Second, I had a point ofview-the over-

all space-time point of view - and a disrespect for the Hamiltonian method

of describing physics.

I would like to interrupt here to make a remark. The fact that electrodynam-

ics can be written in so many ways - the differential equations of Maxwell,

various minimum principles with fields, minimum principles without fields,

Page 10

164

1 9 6 5 R I C H A R D P. F E Y N M A N

all different kinds of ways,was something I knew, but I have never understood.

It always seems odd to me that the fundamental laws of physics, when dis-

covered, can appear in so many different forms that are not apparently iden-

tical at first, but, with a little mathematical fiddling you can show the relation-

ship. An example of that is the Schrödinger equation and the Heisenberg

formulation of quantum mechanics. I don’t know why this is - it remains a

mystery, but it was something I learned from experience. There is always an-

other way to say the same thing that doesn’t look at all like the way you said it

before. I don’t know what the reason for this is. I think it is somehow a repre-

sentation of the simplicity of nature. A thing like the inverse square law is just

right to be represented by the solution of Poisson’s equation, which, there-

fore, is a very different way to say the same thing that doesn’t look at all like

the way you said it before. I don’t know what it means, that nature chooses

these curious forms, but maybe that is a way of defining simplicity. Perhaps a

thing is simple if you can describe it fully in several different ways without im-

mediately knowing that you are describing the same thing.

was now convinced that since we had solved the problem of classical

electrodynamics (and completely in accordance with my program from M.

I.T., only direct interaction between particles, in a way that made fields un-

necessary) that everything was definitely going to be all right. I was convinced

that all I had to do was make a quantum theory analogous to the classical one

and everything would be solved.

So, the problem is only to make a quantum theory, which has as its classical

analog, this expression (1).

Now, there is no unique way to make a quantum

theory from classical mechanics, although all the textbooks make believe there

is. What they would tell you to do, was find the momentum variables and re-

place them by (h/i)(

δ/

δx), but I

couldn’t find a momentum variable, as there

wasn’t any.

The character of quantum mechanics of the day was to write things in the

famous Hamiltonian way - in the form of a differential equation, which de-

scribed how the wave function changes from instant to instant, and in terms of

an operator,

. If the classical physics could be reduced to a Hamiltonian

form, everything was all right. Now, least action does not imply a Hamilto-

nian form if the action is a function of anything more than positions and veloc-

ities at the same moment. If the action is of the form of the integral of a func-

tion, (usually called the Lagrangian) of the velocities and positions at the same

time

(2)

Page 11

S P A C E - T I M E V I E W O F Q U A N T U M E L E C T R O D Y N A M I C S

165

then you can start with the Lagrangian and then create a Hamiltonian and

work out the quantum mechanics, more or lessuniquely. But this thing

(

)

involves the key variables, positions, at two different times and therefore, it

was not obvious what to do to make the quantum-mechanical analogue.

tried - I would struggle in various ways. One of them was this; if I had

harmonic oscillators interacting with a delay in time, I could work out what

the normal modes were and guess that the quantum theory of the normal

modes was the same as for simple oscillators and kind of work my way back

in terms of the original variables. I succeeded in doing that, but I hoped then

to generalize to other than a harmonic oscillator, but I learned to my regret

something, which many people have learned. The harmonic oscillator is too

simple; very often you can work out what it should do in quantum theory

without getting much of a clue as to how to generalize your results to other

systems.

So that didn’t help me very much, but when I was struggling with this

problem, I went to a beer party in the Nassau Tavern in Princeton. There was

a gentleman, newly arrived from Europe (Herbert Jehle) who came and sat

next to me. Europeans are much more serious than we are in America because

they think that a good place to discuss intellectual matters is a beer party. So,

he sat by me and asked, « what are you doing » and so on, and I said, « I’m

drinking beer. » Then I realized that he wanted to know what work I was

doing and I told him I was struggling with this problem, and I simply turned

to him and said, ((listen, do you know any way of doing quantum mechanics,

starting withaction - where the action integral comes into the quantum me-

chanics? » « No », he said, « but Dirac has a paper in which the Lagrangian, at

least, comes into quantum mechanics. I will show it to you tomorrow. »

Next day we went to the Princeton Library, they have little rooms on the

side to discuss things, and he showed me this paper. What Dirac said was the

following : There is in quantum mechanics a very important quantity which

carries the wave function from one time to another, besides the differential

equation but equivalent to it, a kind of a kernal, which we might call

which carries the wave function ψ (x) known at time t, to the wave function

y (x’) at time, t +ε. Dirac points out that this function K was analogous to the

quantity in classical mechanics that you would calculate if you took the ex-

ponential of multiplied by the Lagrangian

imagining that these

two positions x,x’ corresponded t and t +ε. In other words,

is analogous to e

Page 12

166

1 9 6 5

R I C H A R D P

F E Y N M A N

Professor Jehle showed me this, I read it, he explained it to me, and I said,

« what does he mean, they are analogous; what does that mean, analogous?

What is the use of that? » He said, « you Americans ! You always want to find

a use for everything! » I said, that I thought that Dirac must mean that they

were equal. « No », he explained, « he doesn’t mean they are equal. » « Well »,

I said, « let’s see what happens if we make them equal. »

So I simply put them equal, taking the simplest example where the Lag-

rangian is

but soon found I had to put a constant of propor-

tionality A in, suitably adjusted. When I substituted

for to get

and just calculated things out by Taylor series expansion, out came the Schrö-

dinger equation. So, I turned to Professor Jehle, not really understanding, and

said, « well, you see Professor Dirac meant that they were proportional. » Pro-

fessor Jehle’s eyes were bugging out - he had taken out a little notebook and

was rapidly copying it down from the blackboard, and said, « no, no,this is an

important discovery. You Americans are always trying to find out how some-

thing can be used. That’s a good way to discover things! » So, I thought I was

finding out what Dirac meant, but, as a matter of fact, had made the discovery

that what Dirac thought was analogous, was, in fact, equal. I had then, at least,

the connection between the Lagrangian and quantum mechanics, but still

with wave functions and infinitesimal times.

It must have been a day or so later when I was lying in bed thinking about

these things, that I imagined what would happen if I wanted to calculate the

wave function at a finite interval later.

would put one of these factors

in here, and that would give me the

wave functions the next moment,

and then I could substitute that back

into (3) to get another factor of e

and give me the wave function the next

moment, t +

2ε,

and so on and so on. In that way I found myself thinking of a

large number of integrals, one after the other in sequence. In the integrand was

the product of the exponentials, which, of course, was the exponential of the

sum of terms like

εL. Now, L is the Lagrangian and ε is like the time interval

dt, so that if you took a sum of such terms, that’s exactly like an integral.

That’s like Riemann’s formula for the integral ∫

you just take the value

at each point and add them together. We are to take the limit as ε - 0, of

course. Therefore, the connection between the wave function of one instant

and the wave function of another instant a finite time later could be obtained

by an infinite number of integrals, (because ε goes to zero, of course) of ex-

Page 13

S P A C E

T I M E V I E W O F Q U A N T U M E L E C T R O D Y N A M I C S

167

ponential

where S is the action expression (2). At last, I had succeeded

in representing quantum mechanics directly in terms of the action S.

This led later on to the idea of the amplitude for a path; that for each pos-

sible way that the particle can go from one point to another in space-time,

there’s an amplitude. That amplitude is e to the

times the action for the

path. Amplitudes from various paths superpose by addition. This then is an-

other, a third way, of describing quantum mechanics, which looks quite dif-

ferent than that of Schrödinger or Heisenberg, but which is equivalent to

them.

Now immediately after making a few checks on this thing, what I wanted

to do, of course, was to substitute the action (

) for the other (2). The first

trouble was that I could not get the thing to work with the relativistic case of

spin one-half. However, although I could deal with the matter only non-

relativistically, I could deal with the light or the photon interactions perfectly

well by just putting the interaction terms of (I) into any action, replacing the

mass terms by the non-relativistic

When the action has a delay, as

it now had, and involved more than one time, I had to lose the idea of a wave

function. That is, I could no longer describe the program as; given the ampli-

tude for all positions at a certain time to compute the amplitude at another

time. However, that didn’t cause very much trouble. It just meant develop-

ing a new idea. Instead of wave functions we could talk about this; that if a

source of a certain kind emits a particle, and a detector is there to receive it,

we can give the amplitude that the source will emit and the detector receive.

We do this without specifying the exact instant that the source emits or the

exact instant that any detector receives, without trying to specify the state of

anything at any particular time in between, but by just finding the amplitude

for the complete experiment. And, then we could discuss how that amplitude

would change if you had a scattering sample in between, as you rotated and

changed angles, and so on, without really having any wave functions.

It was also possible to discover what the old concepts of energy and momen-

tum would mean with this generalized action. And, so I believed that I had a

quantum theory of classical electrodynamics - or rather of this new classical

electrodynamics described by action (

). I made a number of checks. If I took

the Frenkel field point of view, which you remember was more differential, I

could convert it directly to quantum mechanics in a more conventional way.

The only problem was how to specify in quantum mechanics the classical

boundary conditions to use only half-advanced and half-retarded solutions.

By some ingenuity in defining what that meant, I found that the quantum

Page 14

168

1 9 6 5 R I C H A R D P. F E Y N M A N

mechanics with Frenkel fields, plus a special boundary condition, gave me

back this action,

(1)

in the new form of quantum mechanics with a delay.

So, various things indicated that there wasn’t any doubt I had everything

straightened out.

It was also easy to guess how to modify the electrodynamics, if anybody

ever wanted to modify it. I just changed the delta to an ƒ, just as I would for

the classical case. So, it was very easy, a simple thing. To describe the old re-

tarded theory without explicit mention of fields I would have to write prob-

abilities, not just amplitudes. I would have to square my amplitudes and that

would involve double path integrals in which there are two S’s and so forth.

Yet, as I worked out many of these things and studied different forms and dif-

ferent boundary conditions. I got a kind of funny feeling that things weren’t

exactly right. I could not clearly identify the difficulty and in one of the short

periods during which I imagined I had laid it to rest, I published a thesis and

received my Ph.D.

During the war, I didn’t have time to work on these things very extensively,

but wandered about on buses and so forth, with little pieces of paper, and

struggled to work on it and discovered indeed that there was something

wrong, something terribly wrong. I found that if one generalized the action

from the nice Langrangian forms

(2)

to these forms

(1)

then the quantities

which I defined as energy, and so on, would be complex. The energy values of

stationary states wouldn’t be real and probabilities of events wouldn’t add

up to

100%.

That is, if you took the probability that this would happen and

that would happen -everything you could think of would happen, it would

not add up to one.

Another problem on which I struggled very hard, was to represent rela-

tivistic electrons with this new quantum mechanics. I wanted to do a unique

and different way-and not just by copying the operators of Dirac into some

kind of an expression and using some kind of Dirac algebra instead of ordinary

complex numbers. I was very much encouraged by the fact that in one space

dimension, I did find a way of giving an amplitude to every path by limiting

myself to paths, which only went back and forth at the speed of light. The

amplitude was simple (i

ε) to a power equal to the number ofvelocity reversals

where I have divided the time into steps ε and I am allowed to reverse velocity

only at such a time. This gives (as ε approaches zero) Dirac’s equation in two

dimensions - one dimension of space and one of time

Dirac’s wave function has four components in four dimensions, but in this

case, it has only two components and this rule for the amplitude of a path

Page 15

S P A C E - T I M E V I E W O F Q U A N T U M E L E C T R O D Y N A M I C S

169

automatically generates the need for two components. Because if this is the

formula for the amplitudes of path, it will not do you any good to know the

total amplitude of all paths, which come into a given point to find the am-

plitude to reach the next point. This is because for the next time, if it came in

from the right, there is no new factor if it goes out to the right, whereas, if it

came in from the left there was a new factor So, to continue this same infor-

mation forward to the next moment, it was not sufficient information to know

the total amplitude to arrive, but you had to know the amplitude to arrive

from the right and the amplitude to arrive to the left, independently. If you did,

however, you could then compute both of those again independently and thus

you had to carry two amplitudes to form a differential equation (first order in

time).

And, so I dreamed that if I were clever, I would find a formula for the am-

plitude of a path that was beautiful and simple for three dimensions of space

and one of time, which would be equivalent to the Dirac equation, and for

which the four components, matrices, and all those other mathematical funny

things would come out as a simple consequence - I have never succeeded in

that either. But, I did want to mention some of the unsuccessful things on

which I spent almost as much effort, as on the things that did work.

To summarize the situation a few years after the way, I would say, I had

much experience with quantum electrodynamics, at least in the knowledge

of many different ways of formulating it, in terms of path integrals of actions

and in other forms. One of the important by-products, for example, of much

experience in these simple forms, was that it was easy to see how to combine

together what was in those days called the longitudinal and transverse fields,

and in general, to see clearly the relativistic invariance of the theory. Because

of the need to do things differentially there had been, in the standard quantum

electrodynamics, a complete split of the field into two parts, one of which

is called the longitudinal part and the other mediated by the photons, or

transverse waves. The longitudinal part was described by a Coulomb potential

acting instantaneously in the Schrödinger equation, while the transverse part

had entirely different description in terms of quantization of the transverse

waves. This separation depended upon the relativistic tilt of your axes in space-

time. People moving at different velocities would separate the same field into

longitudinal and transverse fields in a different way. Furthermore, the entire

formulation ofquantum mechanics insisting, as it did, on the wave function at

a given time, was hard to analyze relativistically. Somebody else in a different

coordinate system would calculate the succession of events in terms of wave

Page 16

170

1 9 6 5

R I C H A R D P. F E Y N M A N

functions on differently cut slices of space-time, and with a different separa-

tion of longitudinal and transverse parts. The Hamiltonian theory did not

look relativistically invariant, although, of course, it was. One of the great

advantages of the overall point of view, was that you could see the relativistic

invariance right away - or as Schwinger would say - the covariance was mani-

fest. I had the advantage, therefore, of having a manifestedly covariant form

for quantum electrodynamics with suggestions for modifications and so on. I

had the disadvantage that if I took it too seriously - I mean, if I took it seriously

at all in this form, - I got into trouble with these complex energies and the

failure of adding probabilities to one and so on. I was unsuccessfully struggling

with that.

Then Lamb did his experiment, measuring the separation of the

and

levels of hydrogen, finding it to be about

1000

megacycles of frequency

difference. Professor Bethe, with whom I was then associated at Cornell, is a

man who has this characteristic : If there’s a good experimental number you’ve

got to figure it out from theory. So, he forced the quantum electrodynamics

of the day to give him an answer to the separation of these two levels. He

pointed out that the self-energy of an electron itself is infinite, so that the

calculated energy of a bound electron should also come out infinite. But, when

you calculated the separation of the two energy levels in terms of the corrected

mass instead of the old mass, it would turn out, he thought, that the theory

would give convergent finite answers. He made an estimate of the splitting

that way and found out that it was still divergent, but he guessed that was

probably due to the fact that he used an unrelativistic theory of the matter.

Assuming it would be convergent if relativistically treated, he estimated he

would get about a thousand megacycles for the Lamb-shift, and thus, made

the most important discovery in the history of the theory of quantum electro-

dynamics. He worked this out on the train from Ithaca, New York to Schen-

ectady and telephoned me excitedly from Schenectady to tell me the result,

which I don’t remember fully appreciating at the time.

Returning to Cornell, he gave a lecture on the subject, which I attended.

He explained that it gets very confusing to figure out exactly which infinite

term corresponds to what in trying to make the correction for the infinite

change in mass. If there were any modifications whatever, he said, even

though not physically correct, (that is not necessarily the way nature actually

works) but any modification whatever at high frequencies, which would

make this correction finite, then there would be no problem at all to figuring

out how to keep track of everything. You just calculate the finite mass correc-

Page 17

S P A C E - T I M E V I E W O F Q U A N T U M E L E C T R O D Y N A M I C S

171

tion to the electron mass

substitute the numerical values of + for

m in the results for any other problem and all these ambiguities would be

resolved. If, in addition, this method were relativistically invariant, then we

would be absolutely sure how to do it without destroying relativistically in-

variant.

After the lecture, I went up to him and told him, « I can do that for you, I’ll

bring it in for you tomorrow. » I guess I knew every way to modify quantum

electrodynamics known to man, at the time. So, I went in next day, and ex-

plained what would correspond to the modification of the delta-function to ƒ

and asked him to explain to me how you calculate the self-energy of an elec-

tron, for instance, so we can figure out if it’s finite.

I want you to see an interesting point. I did not take the advice of Professor

Jehle to find out how it was useful. I never used all that machinery which I

had cooked up to solve a single relativistic problem. I hadn’t even calculated

the self-energy of an electron up to that moment, and was studying the dif-

ficulties with the conservation of probability, and so on, without actually

doing anything, except discussing the general properties of the theory.

But now I went to Professor Bethe, who explained to me on the blackboard,

as we worked together, how to calculate the self-energy of an electron. Up to

that time when you did the integrals they had been logarithmically divergent.

told him how to make the relativistically invariant modifications that I

thought would make everything all right. We set up the integral which then

diverged at the sixth power of the frequency instead of logarithmically!

So, I went back to my room and worried about this thing and went around

in circles trying to figure out what was wrong because I was sure physically

everything had to come out finite, I couldn’t understand how it came out

infinite. I became more and more interested and finally realized I had to learn

how to make a calculation. So, ultimately, I taught myself how to calculate

the self-energy of an electron working my patient way through the terrible

confusion of those days of negative energy states and holes and longitudinal

contributions and so on. When I finally found out how to do it and did it with

the modifications I wanted to suggest, it turned out that it was nicely conver-

gent and finite, just as I had expected. Professor Bethe and I have never been

able to discover what we did wrong on that blackboard two months before,

but apparently we just went off somewhere and we have never been able to

figure out where. It turned out, that what I had proposed, if we had carried it

out without making a mistake would have been all right and would have

given a finite correction. Anyway, it forced me to go back over all this and to

Page 18

172

1 9 6 5

R I C H A R D P

F E Y N M A N

convince myself physically that nothing can go wrong. At any rate, the cor-

rection to mass was now finite, proportional to (mu

where a is the width

of that function ƒ which was substituted for δ. If you wanted an unmodified

electrodynamics, you would have to take a equal to zero, getting an infinite

mass correction. But, that wasn’t the point. Keeping a finite, I simply followed

the program outlined by Professor Bethe and showed how to calculate all the

various things, the scatterings of electrons from atoms without radiation, the

shifts of levels and so forth, calculating everything in terms of the experimen -

tal mass, and noting that the results as Bethe suggested, were not sensitive to a

in this form and even had a definite limit as a → 0.

The rest of my work was simply to improve the techniques then available

for calculations, making diagrams to help analyze perturbation theory

quicker. Most of this was first worked out by guessing - you see, I didn’t have

the relativistic theory of matter. For example, it seemed to me obvious that

the velocities in non-relativistic formulas have to be replaced by Dirac’s

matrix α or in the more relativistic forms by the operators

I just took my

guesses from the forms that I had worked out using path integrals for non-

relativistic matter, but relativistic light. It was easy to develop rules of what

to substitute to get the relativistic case. I was very surprised to discover that

it was not known at that time, that every one of the formulas that had been

worked out so patiently by separating longitudinal and transverse waves could

be obtained from the formula for the transverse waves alone, if instead of

summing over only the two perpendicular polarization directions you would

sum over all four possible directions of polarization. It was so obvious from

the action

(1)

that I thought it was general knowledge and would do it all the

time. I would get into arguments with people, because I didn’t realize they

didn’t know that; but, it turned out that all their patient work with the longi-

tudinal waves was always equivalent to just extending the sum on the two

transverse directions of polarization over all four directions. This was one of

the amusing advantages of the method. In addition, I included diagrams for

the various terms of the perturbation series, improved notations to be used,

worked out easy ways to evaluate integrals, which occurred in these problems,

and so on, and made a kind of handbook on how to do quantum electrody-

namics.

But one step of importance that was physically new was involved with the

negative energy sea of Dirac, which caused me so much logical difficulty. I got

so confused that I remembered Wheeler’s old idea about the positron being,

maybe, the electron going backward in time. Therefore, in the time depen-

Page 19

S P A C E - T I M E V I E W O F Q U A N T U M E L E C T R O D Y N A M I C S

173

dent perturbation theory that was usual for getting self-energy, I simply sup-

posed that for a while we could go backward in the time, and looked at what

terms

got by running the time variables backward. They were the same as

the terms that other people got when they did the problem a more complicat-

ed way, using holes in the sea, except, possibly, for some signs. These, I, at

first, determined empirically by inventing and trying some rules.

have tried to explain that all the improvements of relativistic theory were

at first more or less straightforward, semi-empirical shenanigans. Each time I

would discover something, however, I would go back and I would check it

so many ways, compare it to every problem that had been done previously

in electrodynamics (and later, in weak coupling meson theory) to see if it

would always agree, and so on, until I was absolutely convinced of the truth

of the various rules and regulations which I concocted to simplify all the work.

During this time, people had been developing meson theory, a subject I

had not studied in any detail. I became interested in the possible application

of my methods to perturbation calculations in meson theory. But, what was

meson theory? All I knew was that meson theory was something analogous

to electrodynamics, except that particles corresponding to the photon had a

mass. It was easy to guess the δ− function in (

), which was a solution of d’Alem-

bertian equals zero, was to be changed to the corresponding solution of d’A-

lembertian equals m

. Next, there were different kind of mesons-the one in

closest analogy to photons, coupled via

are called vector mesons- there

were also scalar mesons. Well, maybe that corresponds to putting unity in

place of the

I would here then speak of « pseudo vector coupling » and I

would guess what that probably was. I didn’t have the knowledge to under-

stand the way these were defined in the conventional papers because they

were expressed at that time in terms of creation and annihilation operators,

and so on, which, I had not successfully learned. I remember that when some-

one had started to teach me about creation and annihilation operators, that this

operator creates an electron, I said, « how do you create an electron? It dis-

agrees with the conservation of charge », and in that way, I blocked my mind

from learning a very practical scheme of calculation. Therefore, I had to find

as many opportunities as possible to test whether I guessed right as to what the

various theories were.

One day a dispute arose at a Physical Society meeting as to the correctness

of a calculation by Slotnick of the interaction of an electron with

neutron

using pseudo scalar theory with pseudo vector coupling and also, pseudo scalar

theory with pseudo scalar coupling. He had found that the answers were not

Page 20

I74 1 9 6 5

R I C H A R D P. F E Y N M A N

the same, in fact, by one theory, the result was divergent, although convergent

with the other. Some people believed that the two theories must give the same

answer for the problem. This was a welcome opportunity to test my guesses

as to whether I really did understand what these two couplings were. So, I

went home, and during the evening I worked out the electron neutron scat-

tering for the pseudo scalar and pseudo vector coupling, saw they were not

equal and subtracted them, and worked out the difference in detail. The

next day at the meeting, I saw Slotnick and said, « Slotnick, I worked it out

last night, I wanted to see if I got the same answers you do. I got a different

answer for each coupling - but, I would like to check in detail with you be-

cause I want to make sure of my methods. » And, he said, « what do you mean

you worked it out last night, it took me six months ! » And, when wecompared

the answers he looked at mine and he asked, « what is that Q in there, that

variable Q? » (I h d p

a ex ressions like (tan -

Q) /Q etc.). I said, « that’s the mo-

mentum transferred by the electron, the electron deflected by different angles. »

« Oh », he said, « no, I only have the limiting value as Q approaches zero; the

forward scattering. » Well, it was easy enough to just substitute Q equals zero

in my form and I then got the same answers as he did. But, it took him six

months to do the case of zero momentum transfer, whereas, during one eve-

ning I had done the finite and arbitrary momentum transfer. That was a thrill-

ing moment for me, like receiving the Nobel Prize, because that convinced

me, at last, I did have some kind of method and technique and understood

how to do something that other people did not know how to do. That was my

moment of triumph in which I realized I really had succeeded in working out

something worthwhile.

At this stage, I was urged to publish this because everybody said it looks like

an easy way to make calculations, and wanted to know how to do it. I had to

publish it, missing two things; one was proof of every statement in a mathemat-

ically conventional sense. Often, even in a physicist’s sense, I did not have a

demonstration of how to get all of these rules and equations from conventio-

nal electrodynamics. But, I did know from experience, from fooling around,

that everything was, in fact, equivalent to the regular electrodynamics and

had partial proofs of many pieces, although, I never really sat down, like

Euclid did for the geometers of Greece, and made sure that you could get it

all from a single simple set of axioms. As a result, the work was criticized, I

don’t know whether favorably or unfavorably, and the « method » was called

the aintuitive method». For those who do not realize it, however, I should

like to emphasize that there is a lot of work involved in using this

intuitive

Page 21

S P A C E - T I M E V I E W O F Q U A N T U M E L E C T R O D Y N A M I C S

I75

method» successfully. Because no simple clear proof of the formula or idea

presents itself, it is necessary to do an unusually great amount of checking and

rechecking for consistency and correctness in terms of what is known, by com-

paring to other analogous examples, limiting cases, etc. In the face of the lack

of direct mathematical demonstration, one must be careful and thorough to

make sure of the point, and one should make a perpetual attempt to demon-

strate as much of the formula as possible. Nevertheless, a very great deal more

truth can become known than can be proven.

It must be clearly understood that in all this work, I was representing the

conventional electrodynamics with retarded interaction, and not my half-

advanced and half-retarded theory corresponding to

(1).

I merely use

(1)

guess at forms. And, one of the forms I guessed at corresponded to changing δ

to a function ƒ of width a

, so that I could calculate finite results for all of the

problems. This brings me to the second thing that was missing when I publish-

ed the paper, an unresolved difficulty. With δ replaced by ƒ the calculations

would give results which were not « unitary », that is, for which the sum of the

probabilities of all alternatives was not unity. The deviation from unity was

very small, in practice, if a was very small. In the limit that I took a very tiny,

it might not make any difference. And, so the process of the renormalization

could be made, you could calculate everything in terms of the experimental

mass and then take the limit and the apparent difficulty that the unitary is

violated temporarily seems to disappear. I was unable to demonstrate that, as

a matter of fact, it does.

It is lucky that I did not wait to straighten out that point, for as far as

know,

nobody has yet been able to resolve this question. Experience with meson

theories with stronger couplings and with strongly coupled vector photons,

although not proving anything, convinces me that if the coupling were

stronger, or if you went to a higher order (137th order of perturbation theory

for electrodynamics), this difficulty would remain in the limit and there

would be real trouble. That is, I believe there is really no satisfactory quantum

electrodynamics, but I’m not sure. And, I believe, that one of the reasons for

the slowness of present-day progress in understanding the strong interactions

is that there isn’t any relativistic theoretical model, from which you can really

calculate everything. Although, it is usually said, that the difficulty lies in the

fact that strong interactions are too hard to calculate, I believe, it is really be-

cause strong interactions in field theory have no solution, have no sense-

they’re either infinite, or, if you try to modify them, the modification destroys

the unitarity. I don’t think we have a completely satisfactory relativistic quan-

Page 22

176

1 9 6 5

R I C H A R D P

F E Y N M A N

turn- mechanical model, even one that doesn’t agree with nature, but, at least,

agrees with the logic that the sum of probability of all alternatives has to be

100%.

Therefore, I think that the renormalization theory is simply a way to

sweep the difficulties of the divergences of electrodynamics under the rug.

am, of course, not sure of that.

This completes the story of the development of the space-time view of

quantum electrodynamics. I wonder if anything can be learned from it. I

doubt it. It is most striking that most of the ideas developed in the course of

this research were not ultimately used in the final result. For example, the

half-advanced and half-retarded potential was not finally used, the action

expression (

) was not used, the idea that charges do not act on themselves

was abandoned. The path-integral formulation of quantum mechanics was

useful for guessing at final expressions and at formulating the general theory

of electrodynamics in new ways - although, strictly it was not absolutely

necessary. The same goes for the idea of the positron being a backward

moving electron, it was very convenient, but not strictly necessary for the

theory because it is exactly equivalent to the negative energy sea point of

view.

We are struck by the very large number of different physical viewpoints and

widely different mathematical formulations that are all equivalent to one an-

other. The method used here, of reasoning in physical terms, therefore, appears

to be extremely inefficient. On looking back over the work, I can only feel a

kind of regret for the enormous amount of physical reasoning and mathe-

matically re-expression which ends by merely re-expressing what was pre-

viously known, although in a form which is much more efficient for the cal-

culation of specific problems. Would it not have been much easier to simply

work entirely in the mathematical framework to elaborate a more efficient

expression? This would certainly seem to be the case, but it must be remarked

that although the problem actually solved was only such a reformulation, the

problem originally tackled was the (possibly still unsolved) problem of avoid-

ante of the inifinities of the usual theory. Therefore, a new theory was sought,

not just a modification of the old. Although the quest was unsuccessful, we

should look at the question of the value of physical ideas in developing a new

theory.

Many different physical ideas can describe the same physical reality. Thus,

classical electrodynamics can be described by a field view, or an action at a

distance view, etc. Originally, Maxwell filled space with idler wheels, and

Faraday with fields lines, but somehow the Maxwell equations themselves are

Page 23

S P A C E - T I M E V I E W O F Q U A N T U M E L E C T R O D Y N A M I C S

I77

pristine and independent of the elaboration of words attempting a physical

description. The only true physical description is that describing the experi-

mental meaning of the quantities in the equation - or better, the way the

equations are to be used in describing experimental observations. This being

the case perhaps the best way to proceed is to try to guess equations, and dis-

regard physical models or descriptions. For example, McCullough guessed

the correct equations for light propagation in a crystal long before his col-

leagues using elastic models could make head or tail of the phenomena, or

again, Dirac obtained his equation for the description of the electron by an

almost purely mathematical proposition. A simple physical view by which all

the contents of this equation can be seen is still lacking.

Therefore, I think equation guessing might be the best method to proceed

to obtain the laws for the part of physics which is presently unknown. Yet,

when I was much younger, I tried this equation guessing and I have seen

many students try this, but it is very easy to go off in wildly incorrect and im-

possible directions.

think the problem is not to find the best or most efficient

method to proceed to a discovery, but to find any method at all. Physical

reasoning does help some people to generate suggestions as to how the un-

known may be related to the known. Theories of the known, which are de-

scribed by different physical ideas may be equivalent in all their predictions

and are hence scientifically indistinguishable. However, they are not psycho-

logically identical when trying to move from that base into the unknown. For

different views suggest different kinds of modifications which might be made

and hence are not equivalent in the hypotheses one generates from them in

ones attempt to understand what is not yet understood. I, therefore, think

that a good theoretical physicist today might find it useful to have a wide range

of physical viewpoints and mathematical expressions of the same theory (for

example, of quantum electrodynamics) available to him. This may be asking

too much of one man. Then new students should as a class have this. If every

individual student follows the same current fashion in expressing and think-

ing about electrodynamics or field theory, then the variety of hypotheses

being generated to understand strong interactions, say, is limited. Perhaps

rightly so, for possibly the chance is high that the truth lies in the fashionable

direction. But, on the off-chance that it is in another direction - a direction

obvious from an unfashionable view of field theory - who will find it? Only

someone who has sacrificed himself by teaching himself quantum electro-

dynamics from a peculiar and unusual point of view; one that he may have to

invent for himself.

say sacrificed himself because he most likely will get

Page 24

178

1 9 6 5

R I C H A R D P. F E Y N M A N

nothing from it, because the truth may lie in another direction, perhaps even

the fashionable one.

But, if my own experience is any guide, the sacrifice is really not great be-

cause if the peculiar viewpoint taken is truly experimentally equivalent to the

usual in the realm of the known there is always a range of applications and

problems in this realm for which the special viewpoint gives one a special

power and clarity of thought, which is valuable in itself. Furthermore, in the

search for new laws, you always have the psychological excitement of feeling

that possible nobody has yet thought of the crazy possibility you are looking

at right now.

So what happened to the old theory that I fell in love with as a youth?

Well, I would say it’s become an old lady, that has very little attractive left in

her and the young today will not have their hearts pound when they look at

her anymore. But, we can say the best we can for any old woman, that she

has been a very good mother and she has given birth to some very good chil-

dren. And, I thank the Swedish Academy of Sciences for complimenting one

of them. Thank you.