sign in sign up
<<
Home , Sidney Drell, Steven Weinberg

The Search for Unity: Notes for a History of Author(s): Source: Daedalus, Vol. 106, No. 4, Discoveries and Interpretations: Studies in Contemporary Scholarship, Volume II (Fall, 1977), pp. 17-35 Published by: The MIT Press on behalf of American Academy of Arts & Sciences Stable URL: http://www.jstor.org/stable/20024506 . Accessed: 29/08/2011 00:22

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

The MIT Press and American Academy of Arts & Sciences are collaborating with JSTOR to digitize, preserve and extend access to Daedalus.

http://www.jstor.org STEVEN WEINBERG

The Search for Unity: Notes for aHistory of Quantum Field Theory

Introduction

Quantum field theory is the theory of matter and its interactions, which grew out of the fusion of and special relativity in the late in 1920s. Its reputation among suffered frequent fluctuations the fol so came to lowing years, at times dropping low that quantum field theory close as a a of suc be abandoned altogether. But now, partly result of series striking cesses over the last decade, quantum field theory has become the most widely on fundamen accepted conceptual and mathematical framework for attacks the a nature were to tal problems of . If something like set of ultimate laws of be discovered in the next few years (an eventuality by no means expected), these to laws would probably have be expressed in the language of quantum field theory. a To the best of my knowledge, there does not exist anything like full his tory of the past fifty years of quantum field theory. Existing histories of modern cover us physics special and pretty thoroughly, and they take the of but their treatment of through early years quantum mechanics, quantum mechanics generally ends with the triumph of the statistical interpretation around 1927. This is a pity, not only because of the fundamental nature of an quantum field theory, but also because its history offers interesting insight to the nature of scientific advance. It is widely supposed that progress in science occurs in large or small revolu tions. In this view, the successes of previous revolutions tend to fasten upon the a a a must scientist's mind language, mind-set, body of doctrine, from which he to break free in order to advance further. There is great debate about the degree which these revolutions are brought about by the individual scientific genius, or able to transcend the fixed ideas of his times, by the accumulation of dis seems to crepancies between existing theory and experiment. However, there a be general agreement that the essential element of scientific progress is deci sion to break with the past. as I would not quarrel with this view, applied to many of the major advances seems in the . It certainly to apply to the great revolutions in physics in this century: the development of special relativity and of quantum mechanics. However, the development of quantum field theory since 1930 pro a vides curious counterexample, in which the essential element of progress has 17 18 STEVEN WEINBERG

been the and that a revolution is If realization, again again, unnecessary. quan tum mechanics and were revolutions in sense the relativity the of French Revo or lution of 1789 the Russian Revolution of 1917, then quantum field theory is more of the order of the Glorious Revolution of 1688: things changed only just so enough that they could stay the same. I to will try tell this story here without using mathematics. To some extent, Iwill let the history do the job of explication?I will go into some of the histori more cal developments fully than others, because they help to introduce ideas which are needed later. Even so, this is not an easy task?there is no branch of natural science which is so so far removed from notions of abstract, everyday how nature as field behaves, quantum theory. a This cannot be story of physics in one country. As we shall see, quantum was field theory had its birth in Europe, especially inGermany and Britain, and a new revived after World War II by generation of theoretical physicists in Japan and the . The United States has been somewhat the center of the intense activity of the last decade, but physicists from many countries in are Europe and Asia have made essential contributions. And although there a discernible national styles in physics, they have played only minor role in this It not or or history. is the national the social the cultural setting that has deter mined the direction of research in physics, but rather the logic of the subject as itself, the need to understand nature it really is. a This article is not history of quantum field theory, but only "notes for a history." A great deal of work needs to be done by professional historians of science in uncovering the story of the last half-century of theoretical physics. I were on would be delighted if this article to spur someone to take this overdue task.

Prehistory: Field Theory and Quantum Theory

Before quantum field theory came field theory and quantum theory. The history of these earlier disciplines has been told again and again by able histo rians of science, so Iwill do no more here than remind the reader of the essential

points. The first successful classical field theory was based on Newton's theory of not was a gravitation. Newton himself did speak of fields?for him, gravitation force which acts between every pair of material particles in the universe, "ac matter on cording to the quantity of solid which they contain and propagates all as sides to immense distances, decreasing always the inverse square of the dis was tances."1 It the mathematical physicists of the eighteenth century who found it convenient to this mutual action at a distance with a replace grav a a itational field, numerical quantity (strictly speaking, vector) which is defined on at every point in space, which determines the gravitational force acting any particle at that point, and which receives contributions from all the material at other this was a mere mathematical de particles every point. However, fa?on no parler?it really made difference in Newton's theory of gravitation whether one said that the earth attracts the moon, or that the earth contributes to the at moon general gravitational field, and that it is this field the location of the which acts on the moon and holds it in orbit. QUANTUM FIELD THEORY 19

on an own Fields really began to take existence of their with the devel opment in the nineteenth century of the theory of . Indeed, the word "field" was introduced into physics by Michael Faraday in 1849. It to was still possible for Coulomb and Ampere consider electromagnetic forces as or acting directly between pairs of electric charges electric currents, but it an a became very much more natural to introduce electric field and magnetic field as conditions of space, produced by all the charges and currents in the and in turn on and current. This universe, acting every charge interpretation became almost unavoidable after James Clerk Maxwell demonstrated that elec waves at a tromagentic travel finite speed, the speed of light. The force which on acts electrons in the retina of my eye at this moment is not produced by the currents in atoms on the sun at the same but an electro electric moment, by a was currents magnetic wave, light wave, which produced by these about eight now we an minutes ago, and which has only reached my eye. (As shall see, was to attempt made in 1945 by and John Wheeler account for this retardation of forces in an action-at-a-distance frame electromagnetic on work, but the idea did not catch on, and they both went to more promising work). a as an Maxwell himself did not yet adopt the modern idea of field indepen our as on dent inhabitant of universe, with much reality as the particles which it at as acts. Instead (at least first) he pictured electric and magnetic fields distur bances in an medium?the aether?like tension in a rubber mem underlying brane.2 This would have had one obvious experimental implication?the waves on observed speed of electromagnetic would depend the speed of the as observer with respect to the aether, just the observed speed of elastic waves in a on rubber membrane depends the speed of the observer relative to the mem own were brane. Maxwell himself thought that his field equations in fact valid in one at rest with to the aether.3 The notion of an only special frame, respect aether that underlies the phenomena of electromagnetism persisted well into the twentieth century, despite the repeated failure of experimentalists to discover as any effects of the motion through the aether of the earth it revolves about the sun. a as an But even with the problem of the aether unresolved, the idea of field own entity in its right grew stronger in physicists' minds. Indeed, it became a popular to suppose that matter itself is ultimately manifestation of electric and a magnetic fields, theme explored in the theories of the electron developed in 1900-1905 by Joseph John Thomson, ,4 M. Abraham,5 Joseph Larmor, Hendrik Antoon Lorentz,6 and Jules Henri Poincar?.7 Finally, in 1905 an the essential element needed to free electromagnetic theory from the need for was aether supplied by the special theory of relativity of .8 New were rules given for the way that the observed space and time coordinates of an event change with changes in the velocity of the observer. These new rules were so a wave specifically designed that the observed speed of light would be just that speed calculated inMaxwell's theory, whatever the velocity of the observ er. Einstein's theory removed any hope of detecting the effects of motion on through the aether, and although the aether lingered in theorists' minds for a as while, it eventually died away, leaving electromagnetic fields things in them selves?the tension in the membrane, but without the membrane. was As it happens, it the study of electromagnetic phenomena that gave 20 STEVEN WEINBERG

to as rise the quantum theory well as to special relativity. By the end of the it was nineteenth century, clear that the classical theories of electromagnetism were and statistical mechanics incapable of describing the energy of electro at a magnetic radiation various wavelengths emitted by heated opaque body. was The trouble that classical ideas predicted too much energy at very high so frequencies, much energy in fact that the total energy per second emitted at all wavelengths would turn out to be infinite! In a paper read to the German on a Physical Society December 15, 1900, resolution of the problem was pro us con posed by Max Karl Ernst Ludwig Planck.9 It will be worthwhile for to centrate on Planck's proposal for a moment, not only because it led to modern an quantum mechanics, but also because understanding of this idea is needed in order to understand what quantum field theory is about. a are Planck supposed that the electrons in heated body capable of oscillating at a a back and forth all possible frequencies, like violin with huge number of or a strings of all possible lengths. Emission absorption of radiation at given occurs when the electron oscillations at that ener frequency frequency give up or gy to receive energy from the electromagnetic field. The amount of energy an being radiated per second by opaque body at any frequency therefore de on amount at pends the average of energy in electron oscillations that particular frequency. was It in calculating this average energy that Planck made his revolutionary suggestion. He proposed that the energy of any mode of oscillation is quan an tized?that is, that it is not possible to set oscillation going with any desired energy, as in classical mechanics, but only with certain distinct allowed values of the energy. More specifically, Planck assumed that the difference between same a any two successive allowed values of the energy is always the for given mode of and is to the of the mode times a new oscillation, equal frequency constant of nature which has come to be called Planck's constant. It follows that are the allowed states of the modes of oscillation of very high frequency widely a a separated in energy, so that it takes great deal of energy to excite such mode us at all. But the rules of statistical mechanics tell that the probability of finding a great deal of energy in any one mode of oscillation falls off rapidly with in creasing energy; hence the average energy in oscillations of very high frequency a must fall off rapidly with the frequency, and the energy radiated by heated body must also fall off rapidly with the frequency of the radiation, thus avoiding an rate the catastrophe of infinite total of radiation. to Planck was not ready to apply the idea of energy radiation as is itself. (George Gamow10 has described Planck's view follows: "Radiation or to store like butter, which can be bought returned the grocery only in quar as can in ter-pound packages, although the butter such exist any desired comes amount.") It was Einstein11 who in 1905 proposed that radiation in bun an to dles of energy, later called , each with energy proportional the

frequency. were In 1913 the ideas of Planck and Einstein brought together by in his theory of atomic spectra.12 Like the hypothetical modes of oscilla tion in Planck's work, atoms in Bohr's theory are supposed to exist in distinct not an states with certain definite energies, but generally equally spaced. When a a a excited atom drops to state of lower energy, it emits with definite QUANTUM FIELD THEORY 21

to of the initial and final atomic energy, equal the difference of the energies to a states. Each definite photon energy corresponds definite frequency, and it is we see we at lines these frequencies that vividly displayed when look the bright the of a fluorescent or a star. spectrum lamp to a was This early quantum theory, from Planck Bohr and for decade after, inspired guesswork, ad hoc mathematical manipulation, justified by its brilliant success in explaining the behavior of atoms and radiation. Quantum theory as became the coherent scientific discipline known quantum mechanics, through the work of , , , Pascual Jordan, , Paul Adrian Maurice Dirac, and Erwin Schroedinger in 1925 1926.13 Armed with this formalism, theorists were able to go back to the prob to lem of determining the allowed energy levels of material systems, and repro duce the successful results first found by Bohr. But despite its origins in the a theory of thermal radiation, quantum mechanics still dealt in coherent way not only with material particles?electrons in atoms?and with radiation itself.

The Birth ofQuantum Field Theory new to The first application of the quantum mechanics of 1925-1926 fields came one rather than particles in of the founding papers of quantum mechanics to itself. In 1926, Born, Heisenberg, and Jordan14 turned their attention the or electromagnetic field in empty space, in the absence of any electric charges can an currents. Their work best be understood by analogy with Planck's 1900 theory of thermal radiation. Planck, itwill be recalled, had treated the motion of the electrons in a heated an were body in terms of idealized picture, in which the electrons replaced with an a a unlimited number of modes of simple oscillation, like violin with huge number of strings of all possible lengths. He had further proposed that the one were a allowed energies of any mode of oscillation separated by definite quantity, equal to the frequency of the mode times Planck's constant. One of was a the products of the new quantum mechanics of 1925-1926 confirmation of was a a Planck's proposal: it proved that the energies of simple oscillator, like violin string, were indeed quantized, in just the way that Planck had guessed. The essential feature of the dynamics of simple oscillations, used in obtaining to this result, is that the energy required produce any given displacement in the we a oscillator is proportional to the square of the displacement?as pull violin string farther and farther from its equilibrium position, it becomes harder and harder to produce any further displacement. same an But essentially the is true of electromagnetic field?the energy in one any mode of oscillation of the field is proportional to the square of the field a to the of its from the normal state strength?in sense, square "displacement" to of field-free empty space. Thus, by applying the electromagnetic field the same mathematical methods that they had used for material oscillators, Born et al. were able to show that the energy of each mode of oscillation of an electro are a magnetic field is quantized?the allowed values separated by basic unit of constant. energy, given by the frequency of the mode times Planck's The physi cal interpretation of this result was immediate. The state of lowest energy is radiation-free and can be an to zero. The empty space, assigned energy equal 22 STEVEN WEINBERG

an next lowest state must then have energy equal to the frequency times can as a Planck's constant, and be interpreted the state of single photon with that energy. The next state would have an energy twice as great, and therefore as would be interpreted containing two photons of the same energy. And so on. Thus, the application of quantum mechanics to the electromagnetic field had at a last put Einstein's idea of the photon on firm mathematical foundation. Born, Heisenberg, and Jordan had dealt only with the electromagnetic field was in empty space, so although their work illuminating, it did not lead to any use important quantitative predictions. The first "practical" of quantum field was a was theory made in 1927 paper of Paul Adrian Maurice Dirac.15 Dirac an to rate at atoms grappling with old problem: how calculate the which in states excited states would emit electromagnetic radiation and drop into of lower was so an energy. The difficulty not much in deriving answer?the correct an formula had already been derived in ad hoc sort of way by Born and Jordan16 was and by Dirac17 himself. The problem to understand this guessed-at formula as a mathematical of mechanics. This was of consequence quantum problem crucial importance, because the process of spontaneous emission of radiation is one in which are created. Before the the con "particles" actually event, system sists of an excited atom, whereas after the event, it consists of an atom in a state one of lower energy, plus photon. If quantum mechanics could not deal with not an processes of creation and destruction, it could be all-embracing physical

theory. can The quantum-mechanical theory of such processes best be understood to In by returning the analogy between fields and oscillators. the absence of any an com interaction with atoms, the electromagnetic field is like ensemble of to mode of oscilla pletely isolated violin strings; whatever energy is given any or whatever the number of of a tion, equivalently, photons particular same if an atom not interact frequency, it will stay the forever. Similarly, did was with radiation, itwould remain indefinitely in whatever state it placed. But atoms electrons an electric So do interact with radiation, because carry charge. a are the true analogy is with set of violin strings that weakly coupled together, one as by the violin soundboard. Every musician knows what happens when oscillator is set going?it will gradually feed energy into the other modes of are cannot oscillation until they all excited. In quantum mechanics this happen are so the gradually because the energies quantized, instead probability gradu was atom will be found ally increases that energy which originally stored in the a have been in the electromagnetic field?in other words, that photon will created. treatment of radiation con Dirac's successful of the spontaneous emission the world was firmed the universal character of quantum mechanics. However, still conceived to be composed of two very different ingredients?particles and fields?which were both to be described in terms of quantum mechanics, but in were con very different ways. Material particles like electrons and protons state a one to ceived to be eternal; to describe the physical of system, had de or scribe the probabilities for finding each particle in any given region of space were to a range of velocities. On the other hand, photons supposed be merely an and manifestation of underlying entity, the quantized electromagnetic field, could be freely created and destroyed. QUANTUM FIELD THEORY 23

a was out It was not long before way found of this distasteful dualism, were a toward a truly unified view of nature. The essential steps taken in 1928 a paper of Jordan and ,18 and then in pair of long papers in 1929 was 1930 by Heisenberg and Pauli.19 (A somewhat different approach also de veloped in 1929 by .20) They showed that material particles could be understood as the quanta of various fields, in just the same way that the was photon is the quantum of the electromagnetic field. There supposed to be one field for each type of . Thus, the inhabitants of the uni verse were conceived to be a set of fields?an electron field, a proton field, an were status to mere electromagnetic field?and particles reduced in epiphe to nomena. In its essentials, this point of view has survived the present day, and a forms the central dogma of quantum field theory: the essential reality is set of fields, subject to the rules of special relativity and quantum mechanics; all else is a derived as consequence of the quantum dynamics of these fields. to matter an This field-theoretic approach had immediate implication: given as enough energy, it ought to be possible to create material particles, just pho tons are created when an atom loses energy. In 1932 Fermi21 used this aspect of a quantum field theory to formulate theory of the process of nuclear . a Ever since Becquerel discovered in 1896 that crystal containing uranium salts a was would fog photographic plate, it known that nuclei were subject to vari ous kinds of radioactive decay. In one of these modes of decay, known as beta the nucleus emits an and its own chemical decay, electron, changes properties. was are Throughout the 1920s it believed that the nuclei composed of protons so was no once a and electrons, there great paradox in supposing that every in while one of the electrons gets out. However, in 1931 Paul Ehrenfest and Julius a Robert Oppenheimer22 presented compelling though indirect argument that not nuclei do in fact contain electrons, and in 1932 Heisenberg23 proposed in stead that nuclei consist of protons and the newly discovered neutral particles, the neutrons. The mystery was, where did the electron come from when a nucleus suffered a answer was comes from beta decay? Fermi's that the electron much the same as the in the radiative of an excited atom?it place photon decay an is created in the act of decay, through interaction of the field of the electron a with the fields of the proton, the neutron, and hypothesized particle, the neutrino.

One problem remained to be solved after 1930, in order for quantum field to theory take its modern form. In formulating the pre-field-theoretic theory of individual electrons, Dirac24 in 1928 had discovered that his equations had solu tions to states corresponding electron of negative energy, that is, with energy less than the zero of energy empty space. In order to explain why ordinary electrons not do fall down into these negative-energy states, he was led in 1930 to states are propose25 that almost all these already filled. The unfilled states, or in sea "holes" the of negative energy electrons would behave like particles of like positive energy, just ordinary electrons but with opposite electrical charge: at plus instead of minus. Dirac thought first that these "" were the true as a new protons, but their nature kind of particle was revealed with the discovery26 of the positron in cosmic rays in 1932. Dirac's theory of antimatter allowed for a kind of creation and annihilation of even particles without introducing the ideas of quantum field theory. Given 24 STEVEN WEINBERG

a can a enough energy, negative-energy electron be lifted up into positive to a energy state, corresponding the creation of positron (the hole in the negative course energy sea) and an ordinary electron. And of the reverse annihilation process can also occur. Dirac himself had always resisted the idea that quantum field theory is needed to describe any sort of particle but photons. However, in 1934 a pair of papers by Wendell Furry and Oppenheimer27 and by Pauli and Victor Weisskopf28 showed how quantum field theory naturally incorporates the idea of antimatter, without introducing unobserved particles of negative energy, and satisfactorily describes the creation and annihilation of particles and antiparticles. For most theorists, this settled the matter, and particles and anti now as particles are seen coequal quanta of the various quantum fields. to a new It is important to understand that quantum field theory gave rise can view not only of particles but also of the forces among them. We think of a not two charged particles interacting at distance by creating classical electro act on one magnetic fields which another, but by exchanging photons, which to continually pass from one particle the other. Similarly, other kinds of force can be produced by exchanging other kinds of particle. These exchanged parti are are not are cles called virtual particles, and directly observable while they because their creation as real a free electron being exchanged, particles (e.g., a an turning into photon and electron) would violate the law of conservation of energy. However, the quantum-mechanical uncertainty principle dictates that the energy of a system that survives for only a short time must be correspond so can in ingly highly uncertain, these virtual particles be created intermediate must states of physical processes, but be reabsorbed again very quickly. one can From this line of reasoning, infer that the force produced by the a a which it exchange of given type of particle has range (the distance beyond to mass falls off very rapidly) inversely proportional the of the exchanged par a ticle. Thus, the photon, which has zero mass, gives rise to force of infinite the familiar force of Coulomb. The force between range, inverse-square protons an was to a a and neutrons in atomic nucleus known have range of little less than a million millionth of a centimeter, so Hidekei Yukawa29 was able in 1936 to an new a mass predict the existence of entirely kind of particle, the meson, with a that the electron. In these one as few hundred times of calculating forces, a not a sum as sumes that the energy density at point is just of squares of fields, of the for uncoupled simple mechanical oscillators, but also contains products at values of the different fields (and their rates of change) that point. These our multifield interactions are the unknowns that have to be sought by theo retical and experimental efforts. From the viewpoint of quantum field theory, matter that all questions about the particles of which is composed and the forces act them are means to an end?the real is to determine among only problem what are the fundamental quantum fields, and what are the interactions among them.

The Problem of Infinities as were a I have described the early days of quantum field theory if it grand to a progress from triumph triumph. This has been somewhat distorted picture, was to to a for almost from the beginning the theory thought be subject grave internal inconsistency. QUANTUM FIELD THEORY 25

a who was The problem first appeared in 1930 paper of Oppenheimer,30 to on an its trying calculate the effect the energy of atomic electron produced by as vir interaction with the quantum electromagnetic field. Just the exchange of two an tual photons between electrons produces energy of interaction between them, so also in quantum field theory the emission of virtual photons and their same a reabsorption by the electron produces self-energy, which might depend on as an the atomic orbit occupied by the electron, and which might show up observable shift in atomic energy levels. Unfortunately, Oppenheimer discov ered that the energy shift predicted by the quantum theory of the electro was magnetic field infinite! an an atom turns The infinity here arises because when electron in briefly into a and an these two can share the momentum of photon electron, particles an the original electron in infinite variety of ways. The self-energy of the elec a sum over momentum can tron involves all the ways that the be shared out, and can sum out to because there is no limit to how large the momenta be, this turns was to are be infinite. It not obvious that this would have happen; after all, there one an many examples of mathematical series in which adds up infinite number a + + + + . . of terms and gets finite result. (For example, 1 Vi Va Vs .).How more ever, Oppenheimer found that the self-energy of the electron behaved like + + a the series 1 + 1 + 1 1 ..., and could hardly be interpreted as finite quantity. a a The problem was ameliorated bit few years later, when WeisskopP1 a included the effects of processes in which virtual electron, positron, and pho are ton created out of empty space, with the positron and photon then being new over as a annihilated along with the original electron, leaving the electron real particle in the final state. This contribution to the self-energy cancelled the worst part of the original infinity found by Oppenheimer, but the self-energy was left in the form of a sum over virtual momenta which behaved like the series + + + + . . a 1 Vi Vi lA ., and which still could not be interpreted as finite quantity. Similar infinities were found in other problems, such as the "polarization of the vacuum" by applied electric fields,32 and the scattering of electrons by the electric fields of atoms.33 (One of the few bright spots in this otherwise dis couraging picture came in the treatment of infinities associated with photons of very low momenta; it was shown in 1937 by and Arnold Nord sieck34 that these infrared infinities all cancel in the total rates for collisions.) Of if one uses a to calculate an observable and finds that course, theory quantity, the answer is infinite, one concludes either that amathematical mistake has been or was no ac made, that the original theory good. Throughout the 1930s, the was was no cepted wisdom that quantum field theory in fact good, that itmight as a be useful here and there stopgap, but that something radically new would have to be added in order for it to make sense. The problem of infinities in fact has provided the single greatest impetus a toward radical revision of quantum field theory. Some of the ideas which were tried out in the 1930s and 1940s are listed below: a 1. In 1938 Heisenberg35 proposed that there is fundamental unit of energy, and that quantum field theory works only at scales of energy that are small was compared with this energy unit. The analogy with the other fundamental constants, Planck's constant and the speed of light. Quantum mechanics comes 26 STEVEN WEINBERG

as into play when ratios of energies and frequencies approaches values as small Planck's constant, whereas special relativity is needed when velocities approach as as same values large the speed of light. In the way, Heisenberg supposed, some new entirely physical theory might be needed when energies exceed the some fundamental unit, and mechanism in this theory might wipe out the con tributions of virtual particles with such high energies, thus avoiding the prob lem of infinities. (Heisenberg's idea drew support in the 1930s from the cos observation that the showers of charged particles produced by high energy as mic rays did not behave expected in . This dis crepancy was later realized to be due to the production of new particles, the a mesons, and not to failure of quantum field theory.) 2. John Archibald Wheeler36 in 1937 and Heisenberg in 194337 indepen a dently proposed positivistic approach to physics, which would have replaced a an quantum field theory with different sort of theory, sometimes called "S matrix theory," which would involve only directly measurable quantities. They us reasoned that experiments do not actually allow to follow what happens to or electrons in atoms in collision processes. Instead, it is only really possible to a measure the energies and few other properties of bound systems like atoms, cer and the probabilities for various collision processes. These quantities obey as tain very general principles, such reality, conservation of probabilities, smooth conservation and it was these energy dependence, laws, etc., general were to principles that supposed replace the assumptions of quantum field theory. to 3. Dirac38 in 1942 suggested that quantum mechanics ought be expanded to not as or include states of negative probability, which could appear the initial final state of any physical process, but which would have to be included among the intermediate states in these In this minus be processes. manner, signs might sums over states out introduced into the the ways that the intermediate share the momentum of the system, so that a finite answer would be obtained, just as ? + ? + . . . a 1 Vi Vi lA is finite quantity (the natural logarithm of 2) whereas 1 + Vi + Vi + lA + . . . is infinite. 4. As already mentioned, Richard Feynman and John Wheeler39 in 1945 considered the possibility of abandoning field theory altogether, replacing the a at a field-mediated interaction among particles with direct action distance. now Some of these ideas have survived and are part of the regular equipment a of theoretical physics. In particular, the idea of pure S-matrix theory has flourished in the development of so-called "dispersion relations" which involve states are now a only observable quantities.40 Also, of negative probability handy mathematical device, useful especially for dealing with the polarization none to to of virtual photons.41 However, of these ideas proved be the key the problem of infinities. The solution turned out to be far less revolutionary than most theorists had an expected. Recall that the energy of electron had been found by Oppenhei an mer, Waller, and Weisskopf to receive infinite contribution from the emis sort sion and reabsorption of "virtual" photons. An infinite self-energy of this an appears not only when the electron ismoving in orbit in atom, but also when us a it is at rest in empty space. But special relativity tells that the energy of mass = particle at rest is related to its by the famous formula E mc2. Thus the QUANTUM FIELD THEORY 27

mass not electron found in tables of physical data could be just the "bare" mass, our to the quantity appearing in equations for the electron field, but would have be identified with the bare mass plus the infinite "self mass," produced by the own interaction of the electron with its virtual photon cloud. This suggests that an the bare mass might itself be infinite, with infinity which just cancels the a mass mass infinity in the self mass, leaving finite total to be identified with the a that is actually observed. Of course, it goes against the grain to*suppose that our quantity like the bare mass, which appears in fundamental field equations could be infinite; but after all, we can never turn off the electron's virtual photon cloud to measure the bare so no arises. Similar remarks to mass, paradox apply other physical parameters, like the charge of the electron. Is it then possible that masses the infinities in the bare and bare charges cancel the infinities found in masses quantum field theory, not only when we calculate the total and charges, as but in all other calculations well? This method, of eliminating infinities by a come absorbing them into redefinition of physical parameters, has to be called . was The renormalization idea suggested by Weisskopf in 1936,42 and again by Kramers43 in the mid 1940s. However, it was far from obvious that the idea would work. In order to eliminate infinities by absorbing them into redefini must a as tions of physical parameters, the infinities appear in only special way, to in to corrections the observed values of these parameters. For instance, order a absorb the infinite shift in atomic energy levels found by Oppenheimer into redefinition of the electron mass, the infinite part of the self-energy would have to be the same for all atomic energy levels. The mathematical methods available were in the 1930s and early 1940s simply inadequate for the task of sorting out all the infinities that might appear in all possible calculations to see if they could was all be eliminated by renormalization. Perhaps even more important, there no compelling reason to do so?there were no experimental data which forced to theorists to come grips with these problems. And of course, physicists had on to other things their minds from 1939 1945.

Revival at Shelter Island

a On June 1, 1947, four-day conference on the foundations of quantum a near mechanics opened at Shelter Island, small island the end of Long Island in State. The conference brought together young American physi new cists from the generation who had started their scientific work during the war at as as Los Alamos and the MIT Radiation Laboratory, well older physi cists who had been active in the 1930s. Among the younger participants was an , experimentalist then working in the remarkable of physi cists founded at by I. I. Rabi. Lamb announced the re a sults of beautiful experiment, in which he and a student, R. C. Retherford, an had for the first time measured effect of the self-energy of the electron in the atom.44 hydrogen The existing theory of the hydrogen atom had been first advanced by Niels on a Bohr in 1913,12 then put sound mathematical foundation by the quantum mechanics of 1925-1926,13 and finally corrected to include effects of relativity and the spin of the electron by Heisenberg and Jordan,45 C. G. Darwin,46 and 28 STEVEN WEINBERG

as Dirac.24 In the final version of this theory, in particular formulated by Dirac, states atom were to certain pairs of excited of the expected have exactly equal energy. (These pairs of states correspond to the two different ways that the spin of the electron and the angular momentum associated with its revolution around a the nucleus could combine to give definite total angular momentum.) But this own theory ignored all effects of the interaction of the electron with its electro magnetic field?the effects that Oppenheimer30 had tried to calculate when he discovered the infinities. If such effects were real they would presumably shift so no the energies of these pairs of states, that they would longer be exactly equal. new This is what Lamb and Retherford found. By using the techniques of come handling microwave radiation that had out of wartime work in radar, they were able to show that the energies of the first two excited states of hydrogen were to to (the 2sy2and 2pV2states), which supposed be equal according the 1928 Dirac theory, actually differed by about 0.4 parts per million. This is now known as the . Stimulated in part by Lamb's results, the participants at Shelter Island en was tered into an intense discussion of the underlying theory. I not at Shelter I cannot trace Island (having just entered high school) and the historical devel opment of the different reformulations of quantum field theory that developed a around that time. It would be most valuable for historian of science to gather the recollections of the participants at Shelter Island and succeeding confer read the that were written at that and a coher ences, papers time, put together a ent account. I will sketch here only few of the products of this period. on The Lamb shift itself was first calculated by ,47 I believe the mass to train ride back from Shelter Island. Using renormalization eliminate a an infinities, he obtained result in reasonable agreement with the value as was a nounced by Lamb. However, acknowledged by Bethe himself, this were rough calculation, involving approximations that not fully consistent with the Special Theory of Relativity. There were at least three general reformulations of quantum field theory were worked out in the late 1940s that were thoroughly relativistic and that a treatment sufficiently simple and elegant to allow systematic of the infinities. One of these approaches had actually been developed well before the Shelter Island Conference, by Sin-Itiro Tomonaga48 and his colleagues in Japan, but I believe that their work had not yet become known in the United States in the were at summer of 1947. The two other approaches contributed by participants Shelter Island, Julian Schwinger49 and Richard Feynman.50 a one to Feynman's work led to set of pictorial rules which allowed associate a to momentum definite numerical quantity each picture of how the and energy states could flow through the intermediate of any collision process: the probabil sum ity for the process is given by the square of the of these individual quan more a tities. The Feynman rules were very much than handy calculational an algorithm, because they incorporated essential feature of quantum field theo a ry?the symmetry between particles and antiparticles. Each line in Feynman a at one diagram can represent either particle created end of the line and de or an It is stroyed at the other, going the other way. this equal treatment of and that ensures that the calcu particles antiparticles quantities QUANTUM FIELD THEORY 29

are as lated by Feynman diagrams independent of the velocity of the observer, at required by the Special Theory of Relativity, every stage of the calculation. states As had been shown long before by Weisskopf,31 intermediate involving a antiparticles play crucial role in cutting down the degree of infinity, from + + + . . more disasters like 1 2 3 ., to something manageable like 1 + Vi + Vi . . . .The rules ensured the cancellations Feynman automatically over more of the worst infinities, leaving the manageable infinities, which could be eliminated by renormalization. Before the end of 1947 Schwinger51 had used his approach to carry out what I believe was the first calculation of another effect of the electron's cloud of virtual the anomalous moment of the electron. One of the tri photons, magnetic was umphs of Dirac's 1928 theory24 its prediction of the magnetic moment of the a number which characterizes the of the electron's inter electron, strength own action with magnetic fields, and the strength of its magnetic field. How at ever, experiments52 Columbia in 1947 had revealed that the magnetic a moment of the electron is actually little larger than the Dirac value, by 1.15 to 1.21 parts per thousand. By absorbing the infinite effects of virtual photons into a was renormalization of the charge of the electron, Schwinger able to calculate a was finite magnetic moment that larger than the Dirac value by just 1.16 parts per thousand! Of course, both the experimental and the theoretical determinations of ef fects like the Lamb shift and the anomalous magnetic moment of the electron now ex have been enormously improved since 1947.53 For instance, right the moment perimental value of the magnetic of the electron is larger than the Dirac anoma value by 1.15965241 parts per thousand, whereas the theory gives this lous magnetic moment as 1.15965234 parts per thousand, with uncertainties of about 0.00000020 and 0.00000031 parts per thousand, respectively. The pre can cision of the agreement between theory and experiment here only be called spectacular. Finally, Freeman Dyson54 in 1949 showed that the formalisms of Schwinger same and Tomonaga would yield the graphical rules that had been found by an Feynman. Dyson also carried out analysis of the infinities in general Feyn man out a diagrams, and sketched general proof that these infinities are always sort a of precisely the which could be removed by renormalization.55 As gradu ate new student in the mid-1950s, I learned the approach to quantum field theory by reading Dyson's marvelously lucid papers. I should emphasize that the theory of Schwinger, Tomonaga, Feynman, and was not a new was Dyson really physical theory. It simply the old quantum field of theory Heisenberg, Pauli, Fermi, Oppenheimer, Furry, and Weisskopf, but cast in a form more a far convenient for calculation, and equipped with more realistic definition of masses physical parameters like and charges. The continued vitality of the old quantum field theory after fifteen years of attempts to a find substitute is truly impressive. an This raises interesting historical question. All the effects that were calcu lated in the great days of 1947-1949 could have been estimated if not actually calculated at any time after 1934. True, without the renormalization idea, the answers would have been formally infinite, but at least it would have been to possible guess the order of magnitude of quantities like the Lamb shift and 30 STEVEN WEINBERG

the correction to the magnetic moment of the electron. (An infinite series like + . . . a 1 + Vi Vi + !4 + grows very slowly; after million terms, it is still less than 14.4.) Not only was this not done?most theorists seemed to have believed was that these quantities were zero! Indeed, some evidence for what later called the Lamb shift had actually been discovered56 in 1938, but to the best of my see knowledge, no theorist checked to whether the order of magnitude of this was or a reported energy splitting more less what would be expected in quan tum field theory. was not more reason Why quantum field theory taken seriously? One is the so tremendous prestige of the Dirac theory24 of 1928, which had worked well in structure accounting for the fine of the hydrogen spectrum without including more self-energy effects. Even important, the appearance of infinities dis I credited quantum field theory altogether inmany physicists' minds. But think a not that the deepest reason is psychological difficulty, that may have been a sufficiently appreciated by historians of science.57 There is huge apparent at distance between the equations that theorists play with their desks, and the of atomic and collision It takes a certain cour practical reality spectra processes. age to bridge this gap, and to realize that the products of thought and mathemat a ics may actually have something to do with the real world. Of course, when branch of science is well under way, there is continual give and take between one to about theory and experiment, and gets used the idea that the theory is the something real. Without the pressure of experimental data, realization comes harder. The great thing accomplished by the discovery of the Lamb shift us our as was not so much that it forced to change physical theories, that it forced us to take them seriously.

Weak and Strong Interactions was a For a few years after 1949, enthusiasm for quantum field theory at soon to an high level. Many theorists expected that it would lead understanding not and of all microscopic phenomena, only the dynamics of photons, electrons, it was not before there was another in con positrons. However, long collapse on and fidence?shares in quantum field theory tumbled the physics bourse, there a second which was to last for almost began depression, twenty years. arose of the renormaliza Part of the problem from the limited applicability tion idea. In order for all infinities to be eliminated by a renormalization of masses it is for these infinities to physical parameters like and charges, necessary arise in a limited number of as corrections to only ways, masses, charges, etc., be case for a but not otherwise. Dyson's work54 showed that this would the only are small class of quantum field theories, which called renormalizable theories. as The simplest theory of photons, electrons, and positrons (known quantum but most theories are not. electrodynamics) is renormalizable in this sense, was one which Unfortunately, there important class of physical phenomena not a field These were apparently could be described by renormalizable theory. the weak interactions, which cause the radioactive beta decay of nuclei mentioned above in the section on the Birth of Quantum Field Theory. Fermi21 had in a vented a theory of weak interactions in 1932 which, with few modifications, QUANTUM FIELD THEORY 31

adequately described all phenomena in the lowest order of a approximation, that is, including only single simple in cal culations of transition rates. as soon as this was to the However, theory pushed next order of approximation, it exhibited infinities which could not be removed a by redefinition of physical quantities. The other major problem had to do with the limited validity of the approxi mation techniques used in 1947-1949. Any physical process is represented by an infinite sum of each one a se Feynman diagrams, representing particular states quence of intermediate consisting of definite numbers of particles of vari ous To each we associate a numerical the rate of the types. diagram quantity; is the of the sum of these in electro process square quantities. Now, quantum are dynamics the quantities associated with complicated diagrams very small; for each additional photon line there is one additional factor of a small number known as thefine-structure constant, roughly 1/137. In the Fermi theory of weak even interactions, the corresponding factor is smaller?at the typical energies of to elementary , it is 10"5 10~7. It is the rapid decrease of the contributions with associated complicated Feynman diagrams (together with the renormalizability of the theory) that makes it possible to carry calculations in to such a fantastic of quantum electrodynamics degree accuracy. in addition to and weak one However, electromagnetic interactions, there is as other class of interaction in elementary particle physics, known the strong interactions. It is the strong interactions that hold atomic nuclei together against the electrostatic repulsion of the protons they contain. For strong interactions, the factor to structure constant corresponding the fine is roughly of the order of one instead so are as of 1/137, complicated Feynman diagrams just as important ones. of course is are simple (This why these interactions strong.) Thus, al use though attempts have been made time and time again to quantum field in never theory calculations of the nuclear force, it has really worked in a con New vincingly quantitative way. kinds of strongly interacting particles called mesons and were being discovered from 1947 on, first in cosmic rays and then in accelerator and field was at first en laboratories, quantum theory to thusiastically used study their strong interactions, but again with little quan titative success. It was not that was there any difficulty in thinking of renormalizable quantum field theories that might account for the strong inter was actions?it just that having thought of such a theory, there was no way to use it to derive reliable quantitative predictions, and to test if it were true. The nonrenormalizability of the field theory of weak interactions and the uselessness of the field of a theory strong interactions led in the early 1950s to widespread disenchantment with quantum field theory. Some theorists turned to the study of symmetry principles and conservation laws, which can be ap to plied physical phenomena without detailed dynamical calculations. Others the old S-matrix picked up theory of Wheeler and Heisenberg, and worked to of develop principles physics that would involve only observ able In both was quantities. lines of work, quantum field theory used heuristi as a to not as a cally, guide general principles, but basis for quantitative calculation. as a Now, result of work by many physicists over the last decade, quantum 32 STEVEN WEINBERG

was a field theory has again become what it in the late 1940s, the chief tool for detailed understanding of elementary particle processes. There are quantum field theories of the weak and strong interactions, called gauge theories, which are not subject to the old problems of nonrenormalizability and incalculability, and which are to some extent even true! We are still in the midst of this revival, and I will not try to outline its history, but will only summarize how the old prob lems of quantum field theory are surmounted in the new theories.58 The essence of the new theories is that the weak and the strong interactions are described in a way that is almost identical to the successful older quantum as field theory of electromagnetic interactions. Just electromagnetic interactions are so among charged particles produced by the exchange of photons, the weak interactions are produced by the exchange of particles called intermediate vector and the strong interactions by the exchange of other particles called . All these particles?photons, intermediate vector bosons, and gluons?have equal spin, and have interactions governed by certain powerful symmetry prin as ciples known gauge symmetries. (A gauge symmetry principle states that the are to fundamental equations do not change their form when the fields subjected certain whose effect varies with and Because transformations, position time.) theories are so similar to share its funda these quantum electrodynamics, they mental property, of being renormalizable. Indeed, the relation between weak not one and electromagnetic interactions is merely of analogy?the theory uni fies the two, and treats the fields of the photon and the intermediate vector a bosons as members of single family of fields. The intermediate vector bosons are not massless, like the photon, but in are to mass a or neu stead believed to have perhaps 70 80 times the of proton mass to tron. This huge is not due any essential dissimilarity between the photon and the intermediate vector fields, but instead arises from the way that the symmetry of the underlying field theory breaks down when the field are vector equations solved. The family of intermediate bosons, of which the one photon is a member, is believed to contain heavy charged particle and its called the W+ and and one even heavier neutral called antiparticle, W~, particle, the Z?. Exchange of the W produces the familiar weak interactions, like nuclear a new beta decay, whereas exchange of the Z? would produce kind of weak not interaction, in which the participating particles do change their charge. to Such processes were discovered in 1973, and are found have just vector about the properties expected in these theories. All of the intermediate to bosons are much too heavy have been produced with existing accelerator facilities, but there are great hopes of producing them with colliding beams of too protons and antiprotons before long. have In contrast, the gluons which mediate the strong interactions may well zero mass. Such theories with massless have a remarkable gluons property or known as asymptotic freedom?at very high energy very short distances, the it is now strength of the interactions gradually decreases. In consequence, out possible to use quantum field theory to carry detailed calculations of strong interaction processes at sufficiently high energy. In particular, it has been pos as sible to account for some of the features observed in a process such high

energy electron-nucleon scattering. QUANTUM FIELD THEORY 33

as Just the gluon interactions become weak at high energy and short dis at rea tances, they also become strong low energy and long distances. For this son, it is widely believed (though not yet proved) that particles which carry the same sense quantity called color with which gluons interact (in the that photons as interact with ) cannot be produced separate free particles. The colored particles include the gluons themselves, which is presumably why as gluons have never been observed real particles. The colored particles are also believed to include the quarks discussed in this double issue o? Daedalus in the as neu article by . The observed strongly interacting particles such and mesons are believed to be of trons, protons, compound states, consisting no a quarks, antiquarks, and gluons, but with net color. This picture represents over nearly complete triumph of the field the particle view of matter: the funda are mental entities the quark and gluon fields, which do not correspond to any can even particles that be observed in principle, whereas the observed strongly are not at but are mere of an interacting particles elementary all, consequences underlying quantum field theory. are a There hopes of unified of weak, electromagnetic, and strong interactions. The photon, intermediate vector bosons, and gluons would a then form part of single family of fields. However, in order for this to be to possible, there would have be other fields in this family, corresponding to mass. one particles of extraordinarily high According to estimate, the expected mass of these new particles would be 1017 (a hundred thousand million million) masses. so no proton These masses are high that it is longer possible to ignore as the gravitational fields of these particles, done almost everywhere else in particle physics. Unfortunately, despite strenuous efforts which continue to the present, not a there still has been found satisfactory (e.g., renormalizable) quantum field theory of gravitation. It is ironic that gravitation, which provided the first classi cal so field theory, has far resisted incorporation into the general framework of quantum field theory. Throughout this history I have put great emphasis on the condition of renor malizability, the requirement that it should be possible to eliminate all infinities a a a in quantum field theory by redefinition of small number of physical param eters. Many physicists would disagree with this emphasis, and indeed, it may eventually be found that all quantum field theories, renormalizable or not, are it has seemed to me that the equally satisfactory. However, always requirement of renormalizability has just the kind of restrictiveness that we need in a funda mental physical theory. There are very few renormalizable quantum field theo ries. For instance, it is possible to construct quantum field theories of in we electromagnetism which the electron has any magnetic moment like, but one to a only of these theories, corresponding magnetic moment of . 1.0011596523 . . times the Dirac value, is renormalizable. Also, as we have a seen, it took long time before it was found that there are any renormalizable theories at all of a the weak interactions. We very much need guiding principle like to us to renormalizability help pick the quantum field theory of the real out world of the infinite variety of conceivable quantum field theories. Thus, if renormalizability is ultimately to be replaced with some other condition, I 34 STEVEN WEINBERG

one would hope that it will be that is equally or even more restrictive. After all, we not want as do merely to describe the world we find it, but to explain to the greatest possible extent why it has to be the way it is.

References am I grateful for valuable advice from Felix Bloch, , Francis Everitt, I. I. Rabi, and . at ^his often quoted summary of Newton's theory of gravitation appears the end of the Philoso phiae Naturalis Principia Mathematica, p. 546, in the translation of A. Motte, revised and annotated by F. Cajori (Berkeley: University of California Press, 1966), p. 546. C. on in 9th 2J. Maxwell, article "Ether" The Encyclopedia Britannica, ed. (1875-1889); reprinted inW. D. Niven (ed.), The Scientific Papers ofJames Clerk Maxwell (New York: Dover, 1965), p. 763. Also see on 2 Maxwell's Treatise Electricity and Magnetism, vol. (New York: Dover, 1954), pp. 492 493. A useful short account of Maxwell's life and work is given by C. W. F. Everitt, James Clerk Maxwell? and Natural Philosopher (New York: Scribner, 1974). 3See Sees. 600-601, 668, and 769 of Maxwell, Treatise. 4W. Wien, Ann. Phys., 5 (1901): 501. 5M. Abraham,Nachr. Ges. Wiss. (Gottingen), (1902): 20;Ann. Phys., 10(1903): 105;Phys. Zeit., 5 (1904): 576. 6H. A. Lorentz, Phys. Zeit., 1 (1900): 498, 514; Versl. Kon. Akad. Wetenschap. (Amsterdam), 9 (1900): 418; Proc. Roy. Acad. Amsterdam, 6 (1904): 809; Encylop?die derMath. Wiss. V/2 (1904), pp. 63, 145. 7H. Poincar?, Rend. Cire. Mat. (Palermo), 21 (1905): 129. 8A. Einstein, Ann. Phys., 17 (1905): 891, 905; 18 (1905): 639. an 9M. Planck, Verband. Deutsch. Phys. Ges., 2 (1900): 237; reprinted with English translation by in H. Kangro in Planck's Original Papers Quantum Physics (New York: Wiley, 1972). 10G. Gamow, Thirty Years That Shook Physics (Garden City, N.Y.: Doubleday, 1966), p. 23. 17 nA. Einstein, Ann. Phys., (1905): 132. 26 12N. Bohr, Phil. Mag., (1913): 1, 476, 857. W. 13L. de Broglie, Compt. rend., 177 (1923): 507, 548, 630; Heisenberg, Z. Phys., 33 (1925): 879; M. Born and P. Jordan, Z. Phys., 34 (1925): 858; P. A. M. Dirac, Proc. Roy. Soc., A109 (1925): 642; 36 Ann. 79: M. Born, W. Heisenberg, and P. Jordan, Z. Phys., (1926): 336; E. Schr?dinger, Phys., are 361, 489; 80: 437; 81: 109 (all 1926). Many of these papers reprinted in English translations in van B. L. der Waerden, Sources of Quantum Mechanics (New York: Dover, 1968). 14Born, Heisenberg, and Jordan, Z. Phys., 36 (1926): 336, Sec. 3. an 15P. A. M. Dirac, Proc. Roy. Soc. (London), A114 (1947): 243. This article is reprinted in on invaluable collection of papers quantum field theory, J. Schwinger (ed.), Quantum Electrodynamics (New York: Dover, 1958), referred to below as QED. 16Born and Jordan, Z. Phys., 36 (1926): 336. 17P. A. M. Dirac, Proc. Roy. Soc. (London), A112 (1926): 661, Sec. 5. 18P. Jordan and E. Wigner, Z. Phys., 47 (1928): 631, reprinted inQED. 19W. Heisenberg and W. Pauli, Z. Phys., 56 (1929): 1; ibid, 59 (1930): 168. 20E. Fermi, Rend. R. Ace. dei Lincei (6)9 (1929): 881. 21E. Fermi, Z. Phys., 88 (1934): 161. 22P. Ehrenfest and J. Oppenheimer, Phys. Rev., 37 (1931): 333. see 194 23W. Heisenberg, Z. Phys., 77 (1932): 1; also I. Curie-Joliot and F. Joliot, Compt. rend., (1932): 273. 24P. A. M. Dirac, Proc. Roy. Soc. (London), A117 (1928): 610. 25P. A. M. Dirac, Proc. Roy. Soc. (London), A126 (1930): 360; Proc. Camb. Phil. Soc, 26(1930): 361. 26C. D. Anderson, Science, 76 (1932): 238; Phys. Rev., 43 (1933): 491. 27W. H. Furry and J. R. Oppenheimer, Phys. Rev., 45 (1934): 245. 7 709. 28W. Pauli and V. Weisskopf, Helv. Phys. Acta, (1934): 29H. Yukawa, Proc. Phys.-Math. Soc. (Japan), 17 (3) (1935): 48. 461. see Z. 59 30J. R. Oppenheimer, Phys. Rev., 35 (1930): Also I. Waller, Phys. (1930): 168; ibid., 62 (1930): 673. 31V. F. Weisskopf, Z. Phys., 89 (1934): 27; ibid., 90 (1934): 817. Also, Phys. Rev., 56 (1939): 72, reprinted in QED. 203. 32P. A. M. Dirac, in XVII Conseil Solvay de Physique (Brussels, 1933), p. 33S. M. Dancoff, Phys. Rev., 55 (1939): 959. in 34F. Bloch and A. Nordsieck, Phys. Rev. 52 (1937): 54, reprinted QED. QUANTUM FIELD THEORY 35

35W. Heisenberg, Ann. Phys., 32 (1938): 20. 36J. A. Wheeler, Phys. Rev., 52 (1937): 1107. 37W. see Heisenberg, Z. Phys., 120 (1943): 513, 673; Z. Naturforsch., 1 (1946): 608. Also C. M?ller, Kon. Dansk. Vid. Sels. Mat.-Fys. Medd. ,23(1) (1945); ibid., (19) (1946). (Heisenberg actually a an are introduced fundamental length rather than energy, but the two equivalent notions: any an a length determines energy, the energy of photon with that wavelength). 38P. A. M. Dirac, Proc. Roy. Soc. (London), A180 (1942): 1. 39J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys., 17 (1945): 157; ibid., 21 (1949): 425. 40See e.g. G. F. Chew, The S-Matrix Theory of Strong Interactions (New York: Benjamin, 1961). 41See e.g. S. N. Gupta, Proc. Phys. Soc, 63 (1950): 681; 64 (1951): 850. 42V. F. Weisskopf, Kon. Dan. Vid. Sel. Mat.-Fys. Medd., 15 (6) (1936), esp. p. 34 and pp. 5-6; reprinted in QED. 43The of can only published report Kramers' work that I find is in the Rapports du VIH Conseil de 1948 Physique Salvay (ed. R. Stoops, Brussels, 1950). However, Kramers reported on his work at the Shelter Island Conference. 44W. E. R. Lamb, Jr. and C. Retherford, Phys. Rev., 72 (1947): 241, reprinted in QED. 45W. Heisenberg and P. Jordan, Z. Phys., 37 (1926): 263. 46C. G. Darwin, Proc. Roy. Soc. (London), A116 (1927): 227. 47H. Bethe, Phys. Rev. 72 (1947): 339, reprinted in QED. 48S. Theor. 1 Tomonaga, Prog. Phys. (Japan), (1946): 27; Z. Koba, T. Tati, and S. Tomonaga, 2 S. Kanesawa S. ibid., (1947): 101; and Tomonaga, ibid., 3 (1948): 276; Z. Koba and S. Tomonaga, ibid., 3 (1948): 290. Some of these articles are reprinted in QED. 49J. Schwinger, Phys. Rev. 74 (1948): 1439; 75 (1949): 651; 76 (1949): 790; 82 (1951): 664, 914; 91 Nat. 37 are (1953): 713; Proc. Acad. Sei. (1951): 452. All but the first of these articles reprinted in QED. 50R. P. Feynman,/?^. Mod. Phys., 20(1948): 367;Phys. Rev. 74(1948): 939, 1430; 76(1949): 749, 80 are 769; (1950): 440. Some of these articles reprinted in QED. 51J. Schwinger, Phys. Rev., 73 (1948): 146, reprinted in QED. E. E. B. and I. 71 S2J. Nafe, Nelson, I. Rabi, Phys. Rev., (1947): 914; D. E. Nagel, R. S. Julian, and R. 72 J. Zacharias, Phys. Rev., (1947): 97; P. Kusch and H. M. Foley, Phys. Rev., 72 (1947): 1256. 53For calculations of the Lamb see B. early shift, J. French and V. F. Weisskopf, Phys. Rev., 75 N. M. Kroll and W. (1949): 1240; E. Lamb, ibid. (1949): 388; R. P. Feynman, Phys. Rev., 74 (1948): H. Y. and 1430; Fukuda, Miyamoto, S. Tomonaga, Prog. Theor. Phys. (Japan), 4 (1948): 47, 121. The article by Kroll and Lamb is reprinted in QED. 54F. J. Dyson, Phys. Rev., 75 (1949): 486, 1736, reprinted in QED. 55Elements needed to were complete Dyson's proof later supplied by A. Salam, Phys. Rev., 82 (1951): 217; 84 (1951): 426; and S. Weinberg, Phys. Rev. 118 (1960): 838. 56S. Pasternack, Phys. Rev., 54(1938): 1113. Also see W. V. Houston, P/ryj. ?w., 51 (1937): 446; R. C. 54 558. were Williams, Phys. Rev., (1938): Contrary results reported by J. W. Drinkwater, O. Richardson, and W. E. Williams, Proc. Roy. Soc, 174 (1940): 164. 57I a have recently discussed this point in quite different context in The First Three Minutes?A Modern View of the Origin of the Universe (New York: Basic Books, 1977), Ch. 6. 58For a review of this on a to subject technical level, with references the original literature, see S. Rev. Mod. 46 e.g. Weinberg, Phys., (1974): 255. I have also given nonmathematical accounts in Scientific American, 231 (July 1974): 50; and in Bull. Am. Acad. Arts and Sciences, 24 (4) (1976): 13; in and in reprinted (in Spanish) Plural, (1976); American Scientist, 65 (March 1977): 171. Aspects of this are treated in two recent subject books: J. C. Taylor, Gauge Theories of Weak Interactions (Cam G. bridge, England: Cambridge University Press, 1976); Feinberg, What Is theWorld Made Of? The Achievements Twentieth N.Y.: of Century Physics (Garden City, Anchor Press/Doubleday, 1977).