SCIENTIFIC

..AME RIC.AN" Established 1845

May 1963

Volume 208 Number 5

The Evolution of the Physicists Picture of Nature
An account of how ph)lsical theo/�Y has de(Jeloped in the past and how, in the light of this de(Jelopment, it can perhaps be expected to de(Jelop in the future

b,' P. A. M. Dirac

In this article I should like to discuss the development of general physical theory: how it developed in the past and how one may expect it to develop in the future. One can look on this con­ tinual development as a process of evo­ lution, a process that has been going on for several centuries.
The first main step in this process of evolution was brought about by Newton. Before Newton, people looked on the world as being essentially two-dimen­ sional-the two dimensions in which one can walk about-and the up-and-down dimension seemed to be something es­ sentially different. Newton showed how one can look on the up-and-down direc­ tion as being symmetrical with the other two directions, by bringing in gravita­ tional forces and showing how they take their place in physical theory. One can say that Newton enabled us to pass from a picture with two-dimensional sym­ metry to a picture with three-dimension­ al symmetry.
Einstein made another step in the same direction, showing how one can pass from a picture with three-dimen­ sional symmetry to a picture with four­ dimensional symmetry. Einstein brought in time and showed how it plays a role that is in many ways symmetrical with the three space dimensions. However, this symmetry is not quite perfect. With

Einstein's picture one is led to think of the world from a four-dimensional point of view, but the four dimensions are not completely symmetrical. There are some directions in the four-dimensional pic­ ture that are different from others: di­ rections that are called null directions, along which a ray of light can move; hence the four-dimensional picture is not completely symmetrical. Still, there is a great deal of symmetry among the four dimensions. The only lack of symmetry, so far as concerns the equations of phys­ ics, is in the appearance of a minus sign in the equations with respect to the time dimension as compared with the three
space dimensions [see top equation on page 50].
We have, then, the development from the three-dimensional picture of the world to the four-dimensional picture. The reader will probably not be happy with this situation, because the world still appears three-dimensional to his consciousness. How can one bring this appearance into the four-dimensional picture that Einstein requires the physi­ cist to have?
What appears to our consciousness is really a three-dimensional section of the four-dimensional picture. We must take a three-dimensional section to give us what appears to our consciousness at one time; at a later time we shall have a

different three-dimensional section. The task of the physicist consists largely of relating events in one of these sections to events in another section referring to a later time. Thus the picture with four­ dimensional symmetry does not give us the whole situation . This becomes par­ ticularly important when one takes into account the developments that have been brought about by quantum theory. Quantum theory has taught us that we have to take the process of observation into account, and observations usually require us to bring in the three-dimen­ sional sections of the four-dimensional picture of the universe.
The special theory of relativity, which Einstein introduced, requires us to put all the laws of physics into a form that displays four-dimensional symmetry. But when we use these laws to get results about observations, we have to bring in something additional to the four-dimen­ sional symmetry, namely the three-di­ mensional sections that describe our consciousness of the universe at a cer­ tain time.
Einstein made another most important contribution to the development of our physical picture: he put forward the general theory of relativity, which re­ quires us to suppose that the space of physics is curved. Before this physicists

45 © 1963 SCIENTIFIC AMERICAN, INC

had always worked with a flat space, the three-dimensional flat space of Newton which was then extended to the four­ dimensional flat space of special relativ­ ity. General relativity made a really im­ portant contribution to the evolution of our physical picture by requiring us to go over to curved space. The general re­ quirements of this theory mean that all the laws of physics can be formulated in curved four-dimensional space, and that they show symmetry among the four dimensions. But again, when we want to bring in observations, as we must if we look at things from the point of view of (Juantum theory, we have to refer to a section of this four-dimensional space. '<\lith the four-dimensional space curved, any section that we make in it also has to be curved, because in general we cannot give a meaning to a flat section in a curved space. This leads us to a picture in which we have to take curved three­ dimensional sections in the curved four­ dimensional space and discuss observa­ tions in these sections.
During the past fe.w years people have been trying to apply quantum ideas to

gravitation as well as to the other phenomena of physics, and this has led to a rather unexpected development, namely that when one looks at gravita­ tional theory from the point of view of the sections, one finds that there are some degrees of freedom that drop out of the theory. The gravitational field is a tensor field with 10 components. One finds that six of the components are ade­ quate for describing everything of physi­ cal importance and the other four can be dropped out of the equations. One can­ not, however, pick out the six important components from the complete set of 10 in any way that does not destroy the four-dimensional symmetry. Thus if one insists on preserving four-dimensional symmetry in the equations, one cannot adapt the theory of gravitation to a dis­ cussion of measurements in the way quantum theory requires without being forced to a more complicated description than is needed bv the physical situation. This result 'has led me to doubt how fundamental the four-dimensional re­ quirement in physics is. A few decades ago it seemed quite certain that one had

I S AAC NEWTON 0642-1727), with his law of gravitation, changed the physicist's picture of nature from one with two·dimensional symmetry to one with three·dimensional symmetry. This drawing of him was made in 1760 by James Ma cardel from a painting by Enoch Seeman.

to express the whole of physics in four­ dimensional form. But now it seems that four-dimensional symmetry is not of such overriding importance, since the descrip­ tion of nature sometimes gets simplified when one departs from it.
Now I should like to proceed to the developments that have been brought about by quantum theory. Quantum theory is the discussion of very small things, and it has formed the main sub­ ject of physics for the past 60 years. During this period phvsicists have been amassing quite a lot of experimental in­ formation and developing a theory to correspond to it, and this combination of theory and experiment has led to im­ pOitant developments in the physicist's picture of the world.
The quantum first made its appear­ ance when Planck discovered the need to suppose that the energy of electro­ magnetic waves can exist only in mul­ tiples of a certain unit, depending on the frequency of the waves, in order to ex­ plain the law of black-body radiation. Then Einstein discovered the same unit of energy occurring in the photoelectric effect. In this early work on quantum theory one simply had to accept the unit of energy without being able to incor­ pOl'ate it into a physical picture.
f' he first new picture that appeared
was Bohr's picture of the atom. It was a picture in which we had electrons mov­ ing about in certain well-defined orbits and occasionally making a jump from one orbit to another. We could not pic­ ture how the jump took place. We just had to accept it as a kind of discon­ tinuity. Bohr's picture of the atom worked only for special examples, essen­ tially when there was only one electron that was of importance for the problem under consideration. Thus the picture was an incomplete and primitive one.
The big advance in the lJuantum theory came in 1925, with the discovery of quantum mechanics. This advance was brought about independently by two men, Heisenberg first and Schrbdinger soon afterward, working from different points of view. Heisenberg worked keep­ ing close to the experimental evidence about spectra that was being amassed at that time, and he found out how the ex­ perimental information could be fitted into a scheme that is now known as matrix mechanics. All the experimental data of spectroscopy fitted beautifully into the scheme of matrix mechanics, and this led to quite a different picture of the atomic world. Schrbdinger worked from a more mathematical point of view, try­ ing to find a beautiful theory for describ-

46 © 1963 SCIENTIFIC AMERICAN, INC

ing atomic events, and was helped by De Broglie's ideas of waves associated with particles. He was able to extend De Broglie's ideas and to get a very beautiful equation, known as Schrodinger's wave equation, for describing atomic proc­ esses. Schrodinger got this equation by pure thought, looking for some beautiful generalization of De Broglie's ideas, and not by keeping close to the experimental development of the subject in the way Heisenberg did.
I might tell you the story I heard from Schrodinger of how, when he first got the idea for this equation, he immediate­ ly applied it to the behavior of the elec­ tron in the hydrogen atom, and then he got results that did not agree with ex­ periment. The disagreement arose be­ cause at that t ime it was not known that the electron has a spin. That, of course, was a great disappointment to Schro­ dinger, and it caused him to abandon the work for some months. Then he noticed that if he applied the theory in a more approximate way, not taking into ac­ count the refinements required by rela­ tivity, to this rough approximation his work was in agreement with observa­ tion. He published his first paper with only this rough approximation, and in that way Schrodinger's wave equation was presented to the world. Afterward, of course, when people found out how to take into account correctly the spin of the electron, the discrepancy between the results of applying Schrodinger's rel­ ativistic eeluation and the experiments was completelv cleared up.

I think there is a moral to this story, . namely that it is more important to have beauty in one's equations than to have them fit experiment. If Schrodinger had been more confident of h is work, he could have published it some months earlier, and he could have published a more accurate equation. That equation is now known as the Klein-Gordon equa­ tion, although it was really discovered by Schrodinger, and in fact was discovered by Schrodinger before he discovered his nonrelativistic treatment of the hydro­ gen atom. It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agree­ ment between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further developments of the theory.

ALBERT EINSTEIN 0879-1955), with his special theory of relativity, changed the physi· cist's picture from one with three·dimensional symmetry to one with four·dimensional sym· metry. This photo graph o f him and his wife and their daughter Margot was made in 1929.

That is how quantum mechanics was

discovered. It led to a drastic change

in the physicist's picture of the world,

perhaps the biggest that has yet taken

place. This change comes from our hav­

ing to give up the deterministic picture

we had always taken for granted. \Ve are

led to a theory that does not predict with

certainty what is going to happen in the

future but gives us information only

about the probability of occurrence of

various events. This giving up of deter­

minacv subje�t,

has and

been a ver'v controversial some people do not like it at

all. Einstein in particular never liked it.

Although Einstein was one of the great contributors to the development of quan­ tum mechanics, he still was always rath­ er hostile to the form that (luantum mechanics evolved into during his life­ time and that it still retains.
The hostility some people have to the giving up of the deterministic picture can be centered on a much discussed paper by Einstein, Podolsky and Rosen dealing with the difficulty one has in forming a consistent picture that still gives results according to the rules of quantum mechanics. The rules of quan­ tum mechanics are quite definite. People

47

© 1963 SCIENTIFIC AMERICAN, INC

NIELS BOHR 0885-1962) introduced the idea that the electron m oved a bout the nucleus in well·defined orbits. This photograph was m a de in 1922, nine years after the publication of his paper.

MAX PLANCK 0858-1947) introduced the idea that electro· ma gnetic ra diation consists o f quanta, or particles. This photo graph was made in 1913, 13 years a fter his original pa per was published.

know how to calculate results and how to compare the results of their calculations with experiment . Everyone is agreed on the formalism . It works so well that no­ body can afford to disagree with it . But still the picture that we are to set up behind this formalism is a subject of controversy .
I should like to suggest that one not worry too much about this controversy. I feel very strongly that the stage physics has reached at the present day is not the final stage. It is just one stage in the evo­ lution of our picture of nature, and we should expect this process of evolution to continue in the future, as biological evolution continues into the future. The present stage of physical theory is mere· ly a steppingstone toward the better stages we shall have in the future . One can be yuite sure that there will be better stages simply because of the difficulties that occur in the physics of today.
I should now like to dwell a bit on . the difficulties in the physics of the present day. The reader who is not an expert in the subject might get the idea that because of all these difficulties physical theory is in pretty poor shape and that the quantum theory is not much good . I should like to correct this impres­ sion by saying that quantum theory is an extremely good theory. It gives wonder­ ful agreement with observation over a wide range of phenomena. There is no doubt that it is a good theory, and the only reason physicists talk so much about

the difficulties in it is that it is precisely the difficulties that are interesting . The successes of the theory are all taken for granted . One does not get anywhere simply by going over the successes again and again, whereas by talking over the difficulties people can hope to make some progress .
The difficulties in quantum theory are of two kinds. I might call them Class One difficulties and Class Two difficulties . Class One difficulties are the difficulties I have already mentioned: How can one form a consistent picture behind the rules for the present quantum theory? These Class One difficulties do not really worry the physicist . If the physicist knows how to calculate results and com­ pare them with experiment, he is quite happy if the results agree with his ex­ periments, and that is all he needs. It is only the philosopher, wanting to have a satisfying description of nature, who is bothered by Class One difficulties .
There are, in addition to the Class One difficulties, the Class Two difficulties, which stem from the fact that the present laws of quantum theory are not always adequate to give any results. If one pushes the laws to extreme conditions­ to phenomena involving very high ener· gies or very small distances-one some· times gets results that are ambiguous or not really sensible at all . Then it is clear that one has reached the limits of appli­ cation of the theory and that some fur­ ther development is needed. The Class Two difficulties are important even for

the physicist, because they put a limita· tion on how far he can use the rules of quantum theory to get results compara­ ble with experiment.
I should like to say a little more about the Class One difficulties . I feel that one should not be bothered with them too much, because they are difficulties that refer to the present stage in the develop­ ment of our physical picture and are almost certain to change with future de­ velopment. There is one strong reason, I think, why one can be quite confident that these difficulties will change. There are some fundamental constants in na­ ture: the charge on the electron (desig.
nated e ) , Planck's constant divided by 27T (designated Ii) and the velocity of light (c ) . From these fundamental con­
stants one can construct a number that
has no dimensions: the number hc/e2.
That number is found by experiment to have the value 137, or something very close to 137. Now, there is no known reason why it should have this value rather than some other number . Various people have put forward ideas about it, but there is no accepted theory. Still, one can be fairly sure that someday physicists will solve the problem and explain why the number has this value. There will be a physics in the future that
works when lic/e2 has the value 137
and that will not work when it has any other value.
The physics of the future, of course, cannot have the three <luantities 11, e and
c all as fundamental (luantities. On Iv two

48 © 1963 SCIENTIFIC AMERICAN, INC

of them can be fundamental, and the third must be derived from those two. It is almost certain that c will be one of the two fundamental ones. The velocity of light, c, is so important in the four­ dimensional picture, and it plays such a fundamental role in the special theory of relativity, correlating our units of space and time, that it has to be fundamental. Then we are faced with the fact that of
the two quantities Ii and e, one will be
fundamental and one will be derived. If h is fundamental, e will have to be ex­ plained in some way in terms of the square root of h, and it seems most un­ likely that any fundamental theory can give e in terms of a square root, since square roots do not occur in basic equa­ tions. It is much more likely that e will
be the fundamental quantity and that h
will be explained in terms of e2. Then there will be no square root in the basic equations. I think one is on safe ground if one makes the guess that in the physi­ cal picture we shall have at some future stage e and c will be fundamental quan­
tities and Ii will be derived.
If h is a derived quantity instead of a fundamental one, our whole set of ideas
about uncertainty will be altered: Ii is
the fundamental quantity that occurs in the Heisenberg uncertainty relation con­ necting the amount of uncertainty in a position and in a momentum. This un­ certainty relation cannot play a funda­
mental role in a theory in which Ii itself
is not a fundamental quantity. I think one can make a safe guess that uncertain­ ty relations in their present form will not survive in the physics of the future.

and that perhaps it is quite impossible to get a satisfactory picture for this stage.
I have disposed of the Class One dif­ ficulties by saying that they are really not so important, that if one can make progress with them one can count one­ self lucky, and that if one cannot it is nothing to be genuinely disturbed about. The Class Two difficulties are the really serious ones. They arise primarily from the fact that when we apply our quan­ tum theory to fields in the way we have to if we are to make it agree with special relativity, interpreting it in terms of the three-dimensional sections I have men­ tioned, we have equations that at first look all right. But when one tries to solve them, one finds that they do not have any solutions. At this point we ought to say that we do not have a theory. But physi­ cists are very ingenious about it, and they have found a way to make prog­ ress in spite of this obstacle. They find that when they try to solve the equations, the trouble is that certain quantities that ought to be finite are actually in­ finite. One gets integrals that diverge instead of converging to something defi­ nite. Physicists have found that there is a

way to handle these infinities according to certain rules, which makes it possible to get definite results. This method is known as the renormalization method.
I shall merely explain the idea in words. We start out with a theory involving equations. In these equations there occur certain parameters: the charge of the electron, e, the mass of the electron, 111, and things of a similar nature. One then finds that these quantities, which appear in the original equations, are not equal to the measured values of the charge and the mass of the electron. The measured values differ from these by certain cor­ recting terms-6e, 6111 and so on-so
that the total charge is e + 6e and the total mass 111 + 6111. These changes
in charge and mass are brought about through the interaction of our elemen­ tary particle with other things. Then one says that e + 6e and 111 + 6111, being the observed things, are the important things. The original e and 111 are just mathematical parameters; they are un­ observable and therefore just tools one can discard when one has got far enough to bring in the things that one can com-

O f course there will not be a return to the determinism of classical physi­ cal theory. Evolution does not go back­ ward. It will have to go forward. There will have to be some new development that is quite unexpected, that we cannot make a guess about, which will take us still further from Classical ideas but which will alter completely the discus­ sion of uncertainty relations. And when this new development occurs, people will find it all rather futile to have had so much of a discussion on the role of ob­ servation in the theory, because they will have then a much better point of view from which to look at things. So I shall say that if we can find a way to describe the uncertainty relations and the in­ determinacy of present quantum me­ chanics that is satisfying to our philo­ sophical ideas, we can count ourselves lucky. But if we cannot find such a way, it is nothing to be really disturbed about. We simply have to take into ac­ count that we are at a transitional stage

LOUIS DE BROGLIE (1892- ) put forward the idea that particles a re a s s ociated with waves. This photo graph was m a de in 1929, five years a fter the appearance of his paper.

49 © 1963 SCIENTIFIC AMERICAN, INC

pare with observation. This would bc a
quite correct way to proceed if 6 e and 6m were small (or evcn if they
were not so small but finite) corrections. According to the actual theory, however,
6e and 6m are infinitely great. In spite
of that fact one can still use the formal­
ism and get results in terms of e + 6e and m + 6111, which one can interpret by saying that the original e and 111 have
to be minus infinity of a suitable amount
to compensate for the 6e and 6m that
are infinitely great. One can use the theory to get results that can be com­ pared with experiment, in particular for electrodynamics. The surprising thing is that in tbe case of electrodynamics one gets results that are in extremely good agreement with experiment. The agree­ ment applies to many significant fig­ ures-the kind of accuracy that previ­ ously one had only in astronomy. It is because of this good agreement that physicists do attach some value to the renormalization theory, in spite of its illogical character.
It seems to be quite impossible to put this theory on a mathematically sound basis. At one time physical theory was all built on mathematics that was inherently

sound. I do not say tInt phvs'cisls ,;j""ays use sound mathematics; they often use unsound steps in their calculations. But previously when they did so it was simply because of, one might say, lazi­ ness. They wanted to get results as quickly as possible without doing un­ necessary work. It was always possible for the pure mathematician to come along and make the theory sound by bringing in further steps, and perhaps by introducing quite a lot of cumbersome notation and other things that are desir­ able from a mathematical point of view in order to get everything expressed rigorously but do not contribute to the physical ideas. The earlier mathematics could always be made sound in that way, but in the renormalization theory we have a theory that has defied all the at­ tempts of the mathematician to make it sound. I am inclined to suspect that the renormalization theory is something that will not survive in the future, and that the remarkable agreement between its results and experiment should be looked on as a fluke.
This is perhaps not altogether surpris­ ing, because there have been similar flukes in the past. In fact, Bohr's elec-

FOUR-DIMENSIONAL SYMMETRY introdu('ed by the special theory of relativity is not quite perfect. Tbis equation is the expression for the invariant distance in four-dimensional .pa ce-time. The symbol s is the invariant distance; c, the speed of light; t, time; x, y and z, the three spatial dimensions. The d's are differentials. The lack of complete symmetry lies in the fact that the contribution from the time direction (c2dt2) does not have the same sign a s the contributions from the three spatial directions (- dx2, - dy2 and - dz2).
SCHRODINGER'S FIRST WAVE EQUATION did not fit experimental results because it did not take into account the spin of the electron, which was not known at the time. The equation is a generalization of De Broglie's equation for the motion of a free electron. The symhol e represents the charge on the electron; i, the square root of minus one; h, Planck's
constant; r, the distance froln the nucleus; t/J, S('hrodinger's wave function; nt, the m a s s of
the electron. The symbols resembling sixes turned backward are partial derivatives.
SCHRODINGER'S SECOND WAVE EQUATION is an approximation to the original equation, which does not take into a ccount the refinements that a re required by relativity.

tron-orbit theory was found to give very good agreement with observation as long as one confined oneself to one-electron problems. I think people will now say that this agreement was a fluke, because the basic ideas of Bohr's orbit theory have been superseded by something radically different. I believe the suc­ cesses of the renormalization theory will be on the same footing as the successes of the Bohr orbit theory applied to one­ electron problems.
T he renormalization theory bas re­ moved some of these Class Two dif­ ficulties, if one can accept the illogical character of discarding infinities, but it does not remove all of them. There are a good many problems left over concern­ ing particles other than those that come into electrodynamics: tbe new particles­ mesons of various kinds and neutrinos. Therc the theory is still in a primitive stage. It is fairly certain that there will have to be drastic changes in our funda­ mental ideas before these problems can be solved.
One of the problems is the one I have already mentioned about accounting for the number 137. Other problems are how to introduce the fundamental length to physics in some natural way, how to explain the ratios of the masses of the elementarv particles and how to explain their other properties. I believe separate ideas will be needed to solve these dis­ tinct problems and that they will be solved one at a time through successive stages in the future evolution of physics. At this point I find myself in disagree­ ment with most physicists. They are in­ clined to think one master idea will be discovered that will solve all these prob­ lems together. I think it is asking too mnch to hope that anyone will be able to solvc all these problems together. One should separate them one from another as much as possible and try to tackle them separately. And I believe the fu­ ture development of physics will consist of solving them one at a time, and that after any one of them has been solved there will still be a great mystery about how to attack further ones.
I might perhaps discuss some ideas I have had about how one can possibly attack some of these problems. None of these ideas has been worked ont very far, and I do not have much hope for any one of them. But I think they are worth mentioning briefly.
One of these ideas is to introduce something corresponding to the luminif­ erous ether, which was so popular among the physiCists of the 19th century. I said earlier that physics does not evolve back-

50 © 1963 SCIENTIFIC AMERICAN, INC

ward. When I talk about reintroducing the ether, I do not mean to go back to the picture of the ether that one had in the 19th century, but I do mean to intro­ duce a new picture of the ether that will conform to our present ideas of quantum theory. The objection to the old idea of the ether was that if you suppose it to be a fluid filling up the whole of space, in any place it has a definite velocity, which destroys the four-dimensional symmetry required by Einstein's special principle of relativity. Einstein's special relativity killed this idea of the ether.
But with our present quantum theory we no longer have to attach a definite velocity to any given physical thing, be­ cause the velocity is subject to uncer­ tainty relations. The smaller the mass of the thing we are interested in, the more important are the uncertainty relations. Now, the ether will certainly have very little mass, so that uncertainty relations for it will be extremely important. The velocity of the ether at some particular place should therefore not be pictured as definite, because it will be subject to un­ certainty relations and so may be any­ thing over a wide range of values. In that way one can get over the difficulties of reconciling the existence of an ether with the special theory of relativity.
There is one important change this will make in our picture of a vacuum. We would like to think of a vacuum as a region in which we have complete sym­ metry between the four dimensions of space-time as required by special relativ­ ity. If there is an ether subject to uncer­ tainty relations, it will not be possible to have this symmetry accurately. vVe can suppose that the velocity of the ether is equally likely to be anything within a wide range of values that would give the symmetry only approximately. We can­ not in any precise way proceed to the limit of allowing all values for the veloc­ ity between plus and minus the velocity of light, which we would have to do in order to make the symmetry accurate. Thus the vacuum becomes a state that is unattainable. I do not think that this is a phYSical objection to the theory. It would mean that the vacuum is a state we can approach very closely. There is no limit as to how closely we can approach it, but we can never attain it. I believe that would be quite satisfactory to the experimental physicist. It would, how­ ever, mean a departure from the notion of the vacuum that we have in the quantum theory, where we start off with the vacuum state having exactly the svmmetry required by special relativity.
That is one idea for the development of physics in the future that would

ERWIN S C HRODINGER (1887-1961) devised his wave e quation by extending De Bro g lie's
idea that waves are a s s ociated with particles to the electrons m oving around the nnclens. This photo graph was made in 1929, four years after he had published his second equation.

change our picture of the vacuum, but change it in a way that is not unaccept­ able to the experimental physicist. It has proved difficult to continue with the theory, because one would need to set up mathematically the uncertainty relations for the ether and so far some satisfactory theory along these lines has not been dis­ covered. If it could be developed satis­ factorily, it would give rise to a new kind of field in physical theory, which might help in explaining some of the elemen­ tary particles.
kother possible picture I should like
J. - to mention concerns the question of why all the electric charges that are ob­ served in nature should be multiples of
one elementary unit, e. 'Vhy does one
not have a continuous distribution of charge occurring in nature? The picture I propose goes back to the idea of Faradav lines of force and involves a development of this idea. The Faradav

lines of force are a wav of picturing elec­ tric fields. If we have an electric field in any region of space, then according to Faraday we can draw a set of lines that have the direction of the electric field. The closeness of the lines to one another gives a measure of the sh'ength of the field-they are close where the field is strong and less close where the field is weak. The Faraday lines of force give us a good picture of the electric field in classical theory.
vVhen we go over to quantum theory, we bring a kind of discreteness into our basic picture. We can suppose that the continuous distribution of Faraday lines of force that we have in the classical pic­ ture is replaced by just a few discrete lines of force with no lines of force be­ tween them.
Now, the lines of force in the Faraday picture end where there are charges. Therefore with these quantized Faraday Iines of force it would be reasonable to

51

© 1963 SCIENTIFIC AMERICAN, INC

suppose the charge associated with each line, which has to lie at the end if the line of force has an end, is always the same ( apart from its sign) , and is al­
ways just the electronic charge, - e or
+ e. This leads us to a picture of discrete Faraday lines of force, each associated
with a charge, - e or + e. There is a di­
rection attached to each line, so that the ends of a line that has two ends are not
the same, and there is a charge + e at one end and a charge - e at the other.
We may have lines of force extending to infinity, of course, and then there is no charge.
If we suppose that these discrete Faraday lines of force are something basic in physics and lie at the bottom of our picture of the electromagnetic field, we shall have an explanation of why
charges always occur in multiples of e .
This happens because i f w e have any particle with some lines of force ending on it, the number of these lines must be a whole number. In that way we get a picture that is qualitatively quite rea­ sonable.
We suppose these lines of force can

move about. Some of them, forming closed loops or simply extending from minus infinity to infinity, will correspond to electromagnetic waves. Others will have ends, and the ends of these lines will be the charges. We may have a line of force sometimes breaking. When that happens, we have two ends appearing, and there must be charges at the two ends. This process-the breaking of a line of force-would be the picture for the
creation of an electron ( e- ) and a posi­ tron ( e + ) . It would be quite a reason­
able picture, and if one could develop it ,
it would provide a theory in which e
appears as a basic quantity. I have not yet found any reasonable system of equa­ tions of motion for these lines of force, and so I just put forward the idea as a possible physical picture we might have in the future.
There is one very attractive feature in this picture. It will quite alter the discussion of renormalization. The re­ normalization we have in our present quantum electrodynamics comes from starting off with what people call a bare electron-an electron without a charge

WERNER HEISENBERG ( 1901- ) introduced matrix mechanics, which, like the Schro· dinger theory, accounted for the motions of the electron. This photograph was made in 1929.

o n it. A t a certain stage i n the theory one brings in the charge and puts it on the electron, thereby making the electron interact with the electromagnetic field. This brings a perturbation into the equa­ tions and causes a change in the mass of
the electron , the 6 111, which is to be
added to the previous mass of the elec­ tron. The procedure is rather roundabout because it starts off with the unphysical concept of the bare electron. Probably in the improved physical picture we shall have in the future the bare electron will not exist at all.
Now, that state of affairs is just what we have with the discrete lines of force. vVe can picture the lines of force as strings, and then the electron in the pic­ ture is the end of a string. The string it­ self is the Coulomb force around the electron. A bare electron means an elec­ tron without the Coulomb force around it. That is i nconceivable with this pic­ ture, just as it is inconceivable to think of the end of a piece of string without think­ ing of the string itself. This, I think, is the kind of way in which we should try to develop our physical picture-to bring in ideas that make inconceivable the things we do not want to have. Again we have a picture that looks reasonable, but I have not found the proper eyuations for de­ veloping it.
I might mention a third picture with which I have been dealing lately. It involves departing from the picture of the electron as a point and thinking of it as a kind of sphere with a finite size. Of course, it is really quite an old idea to picture the electron as a sphere, but previously one had the difficulty of dis­ cussing a sphere that is subject to ac­ celeration and to irregular motion. It will get distorted, and how is one to deal with the distortions? I propose that one should allow the electron to have, in general, an arbitrary shape and size. There will be some shapes and sizes in which it has less energy than in others, and it will tend to assume a spherical shape with a certain size in which the electron has the least energy.
This picture of the extended electron has been stimulated by the discovery of the mu meson, or muon, one of the new particles of physics. The muon has the surprising property of being almost iden­ tical with the electron except in one particular, namely, its mass is some 200 times greater than the mass of the elec­ tron. Apart from this disparity in mass the muon is remarkably similar to the electron, having, to an extremely high degree of accuracy, the same spin and the same magnetic moment in propor­ tion to its mass as the electron does. This

52 © 1963 SCIENTIFIC AMERICAN, INC

leads to the suggestion that the muon should be looked on as an excited elec­ tron. If the electron is a point, picturing how it can be excited becomes quite awkward. But if the electron is the most stable state for an object of finite size, the muon might just be the next most stable state in which the object under­ goes a kind of oscillation . That is an idea I have been working on recently . There are difficulties in the development of this idea, in particular the difficulty of bring­ ing in the correct spin.

I have mentioned three possible ways in which one might think of develop­ ing our physical picture. No doubt there will be others that other people will think of. One hopes that sooner or later someone will find an idea that really fits and leads to a big development . I am rather pessimistic about it and am in­ clined to think none of them will be good enough. The future evolution of basic physics-that is to say, a development that will really solve one of the funda­ mental problems, such as bringing in the fundamental length or calculating the ratio of the masses-may require some much more drastic change in our physi­ cal picture. This would mean that in our present attempts to think of a new physi­ cal picture we are setting our imagina­ tions to work in terms of inadequate physical concepts. If that is really the case, how can we hope to make progress in the future?
There is one other line along which one can still proceed by theoretical means. It seems to be one of the funda­ mental features of nature that funda­ mental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to un­ derstand it . You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed . We simply have to accept it . One could perhaps describe the situa­ tion by saying that God is a mathema­ tician of a very high order, and He used very advanced mathematics in construct­ ing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathe­ matics we can hope to understand the universe better.
This view provides us with another way in which we can hope to make ad­ vances in our theories . Just by studying mathematics we can bope to make a guess at the kind of mathematics that will come into the phvsics of the future .

LINES OF FORCE in an electromagnetic field, if they are a s s umed to be discrete in the quantum theory, suggest why electric charges always occur in multiples of the charge of the electron. In Dirac's view, when a line of force has two ends, there is a particle with charge - e, perhaps an electron, at one end and a particle with charge + e, perhaps a po sitron, at the other end. When a closed line of force is broken, an electron·po sitron pail' materializes.

A good many people are working on the mathematical basis of quantum theory, trying to understand the theory better and to make it more powerful and more beautiful. If someone can hit on the right lines along which to make this de­ velopment, it may lead to a future ad­ vance in which people will first discover the equations and then, after examining them, gradually learn how to apply them . To some extent that corresponds with the line of development that oc­ CUlTed with Schrodinger's discovery of his wave equation . Schrodinger discov­ ered the equation simply by .looking for an equation with mathematical beauty. When the equation was first discovered, people saw that it fitted in certain ways, but the general principles according to which one should apply it were worked out only some two or three years later. It may well be that the next advance in physics will come about along these lines : people first discovering the equa-

tions and then needing a few years of development in order to find the physical ideas behind the equations. My own be­ lief is that this is a more likely line of progress than trying to guess at physical pictures.
Of course, it may be that even this line of progress will fail, and then the only line left is the experimental one. Experi­ mental physiCists are continuing their work quite independently of theory, col­ lecting a vast storehouse of information. Sooner or later there will be a new Heisenberg who will be able to pick out the important features of this informa­ tion and see how to use them in a way similar to that in which Heisenberg used the experimental knowledge of spectra to build his matrix mechanics . It is in­ evitable that physics will develop ulti­ mately along these lines, but we mav
� have to wait quite a long t ime if peopl
do not get bright ideas for developing the theoret i cal side .

53

© 1963 SCIENTIFIC AMERICAN, INC