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The Search for Unity: Notes for a History of Quantum Field Theory Author(s): Steven Weinberg 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 Field Theory for aHistory of Quantum Introduction field theory is the theory of matter and its interactions, which Quantum out and special relativity in the late of the fusion of grew quantum mechanics in the fol suffered frequent fluctuations 1920s. Its reputation among physicists came so close to at that field low times quantum theory dropping lowing years, be abandoned altogether. But now, partly as a result of a series of striking suc cesses over the last decade, quantum field theory has become the most widely framework for attacks on the fundamen accepted conceptual and mathematical a set of ultimate laws of nature were to If like tal problems of physics. something next in the be discovered few years (an eventuality by no means expected), these in the language of quantum field laws would probably have to be expressed theory. like a full his there does not exist anything To the best of my knowledge, tory of the past fifty years of quantum field theory. Existing histories of modern cover special and general relativity pretty thoroughly, and they take us physics through the early years of quantum mechanics, but their treatment of quantum the triumph of the statistical mechanics ends with interpretation generally a nature of not 1927. This around is only because of the fundamental pity, an but because its offers also quantum field theory, history interesting insight to the nature of scientific advance. It iswidely 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 body of doctrine, from which he must scientist's mind a language, amind-set, to break free in order advance further. There is great debate about the degree to are brought about by the individual scientific genius, which these revolutions to of dis able transcend the fixed ideas of his times, or by the accumulation there seems to crepancies between existing theory and experiment. However, be general agreement that the essential element of scientific progress is a deci sion to break with the past. Iwould not quarrel with this view, as applied to many of the major advances in the history of science. It certainly seems to apply to the great revolutions in in and of of this the century: quantum physics development special relativity of quantum field theory since 1930 pro mechanics. However, the development in which the essential element of progress has vides a curious counterexample, 17 WEINBERG STEVEN 18 been the realization, tum mechanics again and that a revolution again, were in the revolutions and relativity is unnecessary. sense the of If quan French Revo lution of 1789 or 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 that they could stay the same. enough I will try to tell this story here without To some extent, using mathematics. 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 is so which science nature how as behaves, so abstract, field quantum far removed from notions everyday of theory. cannot be a story of physics in one country. As we shall see, quantum and Britain, and was field theory had its birth in Europe, especially inGermany revived after World War II by a new generation in of theoretical physicists and center the United States. The has United the States been somewhat Japan of the intense activity of the last decade, but physicists from many countries in are and Asia have made essential And contributions. Europe although there a national in in discernible this styles they have played only minor role physics, or the social or the cultural history. It is not the national setting that has deter mined the direction of research in physics, but rather the logic of the subject itself, the need to understand nature as it really is. This article is not a history of quantum field theory, but only "notes for a This historians of history." A great deal of work needs to be done by professional I science in uncovering the story of the last half-century of theoretical physics. if this article were to spur someone to take on this overdue would be delighted task. Prehistory: Field Theory and Quantum Theory Before quantum field theory came field theory and quantum theory. The has been told again and again by able histo history of these earlier disciplines rians of science, so Iwill do no more here than remind the reader of the essential points. field theory was based on Newton's theory of was a not Newton himself did of fields?for him, gravitation gravitation. speak "ac in the universe, force which acts between every pair of material particles matter which they contain and propagates on all to solid of the cording quantity as the inverse square of the dis sides to immense distances, decreasing always of the eighteenth tances."1 It was the mathematical century who physicists The first successful to it convenient found classical replace this mutual action at a distance with a grav itational field, a numerical quantity (strictly speaking, a vector) which is defined at every point in space, which determines force acting on any the gravitational at from all the material which contributions that and receives point, particle particles parler?it one said at every other really made that the earth point. attracts this However, no difference was in Newton's the moon, or a mere mathematical fa?on theory of gravitation that the earth contributes field, and that it is this field at the location general gravitational which acts on the moon and holds it in orbit. de whether to the of the moon QUANTUM FIELD THEORY 19 an existence of their own with the devel really began to take on of the theory of electromagnetism. in the nineteenth Indeed, century opment It in 1849. into physics by Michael the word "field" was introduced Faraday to consider electromagnetic was still possible for Coulomb forces and Ampere or electric currents, but it as acting directly between pairs of electric charges became very much more natural to introduce an electric field and a magnetic of space, produced by all the charges and currents in the field as conditions Fields and universe, on turn in acting every charge current. and This interpretation that elec demonstrated after James Clerk Maxwell almost unavoidable waves travel at a finite which the of The force speed, speed light. tromagentic is not produced by the acts on electrons in the retina of my eye at this moment became electric currents on in atoms the sun at the same but moment, an by electro was produced by these currents about eight light wave, which magnetic now reached my eye. (As we shall see, an minutes and which has ago, only was to account in 1945 Richard and John Wheeler made attempt by Feynman a wave, this for work, of retardation forces electromagnetic in an action-at-a-distance but the idea did not catch on, and they both went frame on to more promising work). idea of a field as an indepen Maxwell himself did not yet adopt the modern dent inhabitant of our universe, with as much reality as the particles on which it acts. Instead (at least at first) he pictured electric and magnetic fields as distur bances in an medium?the underlying in only This one have would special frame, had aether?like tension in a rubber mem one obvious experimental implication?the waves would on the of observed electromagnetic depend speed of the speed observer with respect to the aether, just as the observed speed of elastic waves in a rubber membrane on the depends speed of the observer relative to the mem brane. Maxwell himself thought that his own field equations were in fact valid brane.2 at rest with respect to the aether.3 The notion of an aether that underlies the phenomena of electromagnetism persisted well into the to discover twentieth century, despite the repeated failure of experimentalists as it of the motion the earth aether of the effects revolves about the any through sun. the idea of a field as an But even with the problem of the aether unresolved, own in in its minds. Indeed, it became entity physicists' right grew stronger a to matter that itself is manifestation of electric and suppose popular ultimately a in the in theme of the theories electron fields, magnetic developed explored 1900-1905 by Joseph John Thomson, Wilhelm Wien,4 M. Abraham,5 Joseph in 1905 Larmor, Hendrik Antoon Lorentz,6 and Jules Henri Poincar?.7 Finally, the essential element needed to free electromagnetic from the need for an theory aether was supplied by the special theory of relativity of Albert Einstein.8 New rules were 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 that the observed specifically designed speed of a light wave would be just in that speed calculated Maxwell's the velocity of the observ theory, whatever er. Einstein's the effects of motion theory removed any hope of detecting through the aether, and although the aether lingered on in theorists' minds for a it eventually died away, leaving electromagnetic fields as things in them while, selves?the As tension it happens, in the membrane, it was the study but without the membrane. of electromagnetic phenomena that gave 20 STEVEN WEINBERG rise to the quantum nineteenth century, theory as well as to special relativity. By the end of the it was clear that the classical theories of electromagnetism were incapable of and statistical mechanics the energy of electro describing radiation at various wavelengths emitted by a heated opaque body. magnetic too much energy at very high The trouble was that classical ideas predicted so much energy in fact that the total energy per second emitted at frequencies, turn out to be infinite! In a paper read to the German all wavelengths would on December 15, 1900, a resolution of the problem was pro Physical Society for us to con posed by Max Karl Ernst Ludwig Planck.9 It will be worthwhile centrate on Planck's proposal for a moment, not only because it led to modern but also because an understanding of this idea is needed in quantum mechanics, order to understand what quantum field theory is about. Planck supposed that the electrons in a heated body are capable of oscillating back and forth at all possible frequencies, like a violin with a huge number of or of all Emission possible absorption of radiation at a given strings lengths. occurs frequency when the electron at oscillations that frequency give up ener field. The amount of energy gy to or receive energy from the electromagnetic an at radiated second therefore de any frequency per opaque body by being on amount at in of oscillations that the electron average energy pends particular frequency. It was in calculating this average energy that Planck made his revolutionary that the energy of any mode of oscillation is quan suggestion. He proposed an not to set with it oscillation tized?that that is is, any desired possible going as with allowed in but certain distinct values classical mechanics, energy, only that Planck assumed the difference between of the energy. More specifically, any two successive allowed values of the energy is always the same for a given mode of oscillation, and is equal to the frequency of the mode times a new of nature which has come to be called Planck's constant. It follows that the allowed states of the modes of oscillation of very high frequency are widely a great deal of energy to excite such amode separated in energy, so that it takes at all. But the rules of statistical mechanics tell us that the probability of finding a great deal of energy in any one mode of oscillation falls off rapidly with in constant creasing energy; hence the average energy in oscillations of very high frequency must fall off rapidly with the frequency, and the energy radiated by a heated must off with the also fall rapidly frequency of the radiation, thus avoiding body the catastrophe of an infinite total rate of radiation. to radiation Planck was not ready to apply the idea of energy quantization is has described Planck's view as follows: "Radiation itself. (George Gamow10 or store to can in the be returned which like butter, quar grocery only bought the butter as such can exist in any desired although ter-pound packages, amount.") It was Einstein11 who in 1905 proposed that radiation comes in bun to the later called photons, each with an energy proportional dles of energy, frequency. were together by Niels Bohr 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 an states with certain definite energies, but not generally equally spaced. When a a definite excited atom drops to a state of lower energy, it emits photon with In 1913 the ideas of Planck and Einstein brought FIELD QUANTUM THEORY 21 energy, equal to the difference of the energies of the initial and final atomic to a definite frequency, and it is states. Each definite photon energy corresponds that we see vividly displayed when we look at the bright lines these frequencies crossing the spectrum a fluorescent of lamp or a star. after, was early quantum theory, its brilliant ad hoc mathematical manipulation, justified by inspired guesswork, success in explaining the behavior of atoms and radiation. Quantum theory became the coherent scientific discipline known as quantum mechanics, through Max Born, Pascual Jordan, the work of Louis de Broglie, Werner Heisenberg, in 1925 and Erwin Adrian Maurice Paul Dirac, Pauli, Schroedinger Wolfgang 1926.13 Armed with this formalism, theorists were able to go back to the prob the allowed energy levels of material systems, and to repro lem of determining its origins in the duce the successful results first found by Bohr. But despite a coherent way still in mechanics dealt of thermal radiation, quantum theory not with in atoms?and radiation itself. only with material particles?electrons from Planck This to Bohr and for a decade The Birth ofQuantum Field Theory of 1925-1926 to fields The first application of the new quantum mechanics came one rather than particles in of the founding papers of quantum mechanics and Jordan14 turned their attention to the itself. In 1926, Born, Heisenberg, field in empty space, in the absence of any electric charges or electromagnetic currents. Their work can best be understood by an analogy with Planck's 1900 theory of thermal radiation. Planck, itwill be recalled, had treated the motion of the electrons in a heated an idealized picture, in which the electrons were replaced with body in terms of an unlimited number of modes of like a violin with a huge simple oscillation, further number of strings of all possible He had that the lengths. proposed allowed energies of any one mode of oscillation were separated by a definite quantity, equal to the frequency of the mode times Planck's constant. One of the products of the new quantum mechanics of 1925-1926 was a confirmation of it was proved that the energies of a simple oscillator, Planck's proposal: like a 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 this result, is that the energy required to produce any given displacement in the we pull a violin to the square of the displacement?as oscillator is proportional it becomes harder and string farther and farther from its equilibrium position, harder to produce any further displacement. But essentially the same is true of an electromagnetic field?the energy in one to mode of oscillation of the field is the of the field any square proportional strength?in a sense, to the its of square from "displacement" the normal state of field-free empty space. Thus, field the by applying to the electromagnetic 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 allowed values are separated by a basic unit of magnetic field is quantized?the the of the mode times Planck's constant. The physi energy, given by frequency cal interpretation of this result was immediate. The state of lowest energy is radiation-free empty space, and can be assigned an energy equal to zero. The STEVEN 22 WEINBERG times then have an energy equal to the frequency can and Planck's constant, be interpreted as the state of a single photon with that energy. The next state would have an energy twice as great, and therefore would be interpreted as containing two photons of the same energy. And so on. to the electromagnetic the application of quantum mechanics field had at Thus, idea of the photon on a firm mathematical foundation. last put Einstein's next state must lowest and Jordan had dealt only with the electromagnetic field Born, Heisenberg, it did not lead to any in empty space, so although their work was illuminating, The first "practical" use of quantum field predictions. important quantitative a 1927 paper of Paul Adrian Maurice Dirac.15 Dirac was was made in theory an old problem: how to calculate the rate at which atoms in grappling with radiation and drop into states of lower excited states would emit electromagnetic was so not correct in deriving an answer?the The much energy. difficulty formula had already been derived in an ad hoc sort of way by Born and Jordan16 this guessed-at formula and by Dirac17 himself. The problem was to understand as a mathematical crucial importance, one in which sists of an consequence of because the process "particles" excited atom, are quantum actually whereas after This mechanics. of spontaneous Before created. the event, the of of of radiation an is con the event, it consists was problem emission system atom in a state If quantum mechanics could not deal with of lower energy, plus one photon. not an it could be and of creation destruction, processes physical all-embracing theory. can best be understood theory of such processes quantum-mechanical to In the absence of any and fields oscillators. the between analogy by returning field is like an ensemble of com interaction with atoms, the electromagnetic energy is given to any mode of oscilla pletely isolated violin strings; whatever The tion, or equivalently, whatever the number of photons of a particular if an atom did not interact it will stay the same forever. Similarly, frequency, state itwas placed. But in whatever with radiation, itwould remain indefinitely atoms do interact with radiation, because electrons carry an electric charge. So the true analogy is with a set of violin strings that are weakly coupled together, as by the violin soundboard. knows what happens when one Every musician feed energy into the other modes of will gradually oscillator is set going?it this cannot happen oscillation until they are all excited. In quantum mechanics so instead the probability are the because gradu quantized, energies gradually was atom will be found originally stored in the ally increases that energy which that a photon will have been other words, field?in in the electromagnetic created. Dirac's firmed successful the universal treatment character of the of quantum spontaneous mechanics. emission However, of radiation con the world was and to be composed of two very different ingredients?particles still conceived in terms but were to of in both be described fields?which quantum mechanics, con were and like electrons protons very different ways. Material particles state of a system, one had to de ceived to be eternal; to describe the physical for finding each particle in any given region of space or scribe the probabilities a range of velocities. On the other hand, photons were supposed to be merely and of an underlying manifestation field, entity, the quantized electromagnetic could be freely created and destroyed. QUANTUM FIELD THEORY 23 It was not long before a way was found out of this distasteful dualism, toward a truly unified view of nature. The essential steps were taken in a 1928 and then in a pair of long papers in 1929 paper of Jordan and Eugene Wigner,18 1930 by Heisenberg and Pauli.19 (A somewhat different approach was also de veloped in 1929 by Enrico Fermi.20) They showed that material particles could as the quanta of various fields, in just the same way that the be understood is the field. There was supposed to be quantum of the electromagnetic photon one field for each type of elementary particle. Thus, the inhabitants of the uni verse were conceived to be a set of fields?an electron field, a proton field, an were reduced in status to mere epiphe field?and particles electromagnetic nomena. In its essentials, this point of view has survived to the present day, and forms the central dogma of quantum field theory: the essential reality is a set of all else is fields, subject to the rules of special relativity and quantum mechanics; a as derived of the quantum dynamics of these fields. consequence to matter had an immediate This field-theoretic implication: given approach just as pho enough energy, it ought to be possible to create material particles, tons are created when an atom loses energy. In 1932 Fermi21 used this aspect of quantum field theory to formulate a theory of the process of nuclear beta decay. in 1896 that a crystal containing uranium salts Ever since Becquerel discovered was known that nuclei were would fog a photographic subject to vari plate, it ous kinds of radioactive decay. In one of these modes of decay, known as beta the decay, nucleus emits an electron, and changes its own chemical properties. the 1920s it was believed that the nuclei are composed of protons Throughout so there was no great and electrons, paradox in supposing that every once in a while one of the electrons gets out. However, in 1931 Paul Ehrenfest and Julius a Robert Oppenheimer22 presented compelling though indirect argument that nuclei do not in fact contain electrons, and in 1932 Heisenberg23 in proposed 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 much suffered the same a beta place as decay? the Fermi's photon answer in the was radiative that decay the of electron an excited comes from atom?it is created in the act of decay, through an interaction of the field of the electron with the fields of the proton, and a hypothesized the neutron, the particle, neutrino. problem remained to be solved after 1930, in order for quantum field to take its modern form. In formulating the pre-field-theoretic theory theory of individual electrons, Dirac24 in 1928 had discovered that his equations had solu to electron states of tions corresponding negative energy, that is, with energy less than the zero energy of empty space. In order to explain why ordinary electrons do not fall down into these states, he was led in 1930 negative-energy to propose25 that almost all these states are already filled. The unfilled states, or "holes" in the sea of negative energy electrons would behave like particles of positive energy, just like ordinary electrons but with opposite electrical charge: plus instead of minus. Dirac thought at first that these "antiparticles" were the protons, but their true nature as a new 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 particles even without introducing the ideas of quantum field theory. Given One STEVEN 24 WEINBERG a electron can be lifted up into a positive enough energy, negative-energy to the creation of a positron (the hole in the negative energy state, corresponding energy sea) and an ordinary electron. And of course the reverse annihilation process can also occur. Dirac himself had always resisted the idea that quantum in field theory is needed to describe any sort of particle but photons. However, 1934 a pair of papers by Wendell and by Pauli and Furry and Oppenheimer27 Victor Weisskopf28 showed how quantum field theory naturally incorporates the idea of antimatter, without particles of negative introducing unobserved energy, and satisfactorily describes the creation and annihilation of particles and and particles and anti antiparticles. For most theorists, this settled the matter, now seen as are various of the quantum fields. particles coequal quanta It is important to understand that quantum field theory gave rise to a new view not only of particles but also of the forces among them. We can think of two charged particles interacting at a distance not by creating classical electro fields which act on one another, but by exchanging photons, which magnetic to one other kinds of force from the other. pass particle Similarly, continually can be produced by These other kinds of particle. exchanged parti exchanging cles are called virtual particles, and are not directly observable while they are because exchanged, being their as creation real (e.g., particles a free electron a an electron) would violate the law of conservation of turning into photon and that dictates the quantum-mechanical energy. However, uncertainty principle the energy of a system that survives for only a short time must be correspond so these virtual particles can be created in intermediate ingly highly uncertain, states of physical processes, but must be reabsorbed again very quickly. From this line of reasoning, one can infer that the force produced by the a range (the distance beyond which a it exchange of given type of particle has to the mass of the exchanged par falls off very rapidly) inversely proportional ticle. Thus, the photon, which has zero mass, gives rise to a force of infinite the range, familiar in an neutrons and a million predict a few inverse-square atomic nucleus force of Coulomb. was known between force The a to have range of a little protons less than so Hidekei Yukawa29 was able in 1936 to of a centimeter, millionth an new kind of particle, the meson, with amass the existence of entirely hundred times sumes that the energy for uncoupled simple values of the different that of the In electron. calculating these forces, one as a not just a sum of squares of fields, as density at point is but also contains products of the mechanical oscillators, fields (and their rates of change) at that point. These are the unknowns that have to be sought by our theo interactions multifield of quantum field theory, efforts. From the viewpoint retical and experimental all questions about the particles of which matter is composed and the forces that act among them are only are the fundamental what means quantum to an end?the fields, real and what problem is to determine are the interactions among them. The Problem of Infinities the early days of quantum field theory as if itwere a grand to triumph. This has been a somewhat distorted picture, from progress triumph the theory was thought to be subject to a grave for almost from the beginning I have described internal inconsistency. QUANTUM FIELD THEORY 25 a 1930 paper of Oppenheimer,30 who was problem first appeared in to calculate the effect on the energy of an atomic electron produced by its trying field. Just as the exchange of vir interaction with the quantum electromagnetic an two electrons produces tual photons between energy of interaction between so also in quantum field theory the emission of virtual photons and their them, same electron produces a self-energy, which might depend reabsorption by the as an on the atomic orbit occupied by the electron, and which might show up discov shift in atomic energy levels. Unfortunately, observable Oppenheimer of the electro the ered that the energy shift predicted quantum by theory field was infinite! magnetic The infinity here arises because when an electron in an atom turns briefly The into a photon the original tron involves and an a sum electron, these two in an infinite variety electron over all the ways that particles of ways. can The the momentum share the momentum self-energy can be of of the elec shared out, and can be, this sum turns out to because there is no limit to how large the momenta be infinite. It was not obvious that this would have to happen; after all, there are series in which one adds up an infinite number many examples of mathematical 1 + Vi + Va + Vs + . . .).How of terms and gets a finite result. (For example, of found that the the electron behaved more like ever, Oppenheimer self-energy and could hardly be interpreted as a finite the series 1 + 1 + 1 + 1 + ..., quantity. a bit a few years later, when WeisskopP1 a virtual electron, processes positron, and pho empty space, with the positron and photon then being over as a the original electron, leaving the new electron state. to This contribution the self-energy cancelled the real particle in the final worst part of the but the self-energy original infinity found by Oppenheimer, was left in the form of a sum over virtual momenta which behaved like the series . . ., and which 1 + Vi + Vi + lA + still could not be interpreted as a finite The problem was included the effects of ton are created out of annihilated along with ameliorated in which quantity. such as the "polarization of Similar infinities were found in other problems, and the the vacuum" by applied electric fields,32 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 it was shown in 1937 by Felix Bloch and Arnold Nord very low momenta; sieck34 that these infrared infinities all cancel in the total rates for collisions.) Of course, if one uses a theory to calculate an observable quantity, and finds that the answer is infinite, one concludes either that amathematical mistake has been the 1930s, the ac made, or that the original theory was no good. Throughout was that quantum field no theory was in fact cepted wisdom good, that itmight be useful here and there as a 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 toward a 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 compared with this energy unit. The analogy was with the other fundamental comes constants, Planck's constant and the speed of light. Quantum mechanics STEVEN 26 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 values as large as the speed of light. In the same way, Heisenberg supposed, some entirely new physical theory might be needed when energies exceed the fundamental unit, and some mechanism in this theory might wipe out the con tributions of virtual particles with such high energies, thus avoiding the prob lem of infinities. in idea drew the 1930s from the support (Heisenberg's observation that the showers of charged particles produced by high energy cos in quantum mic rays did not behave as expected This dis electrodynamics. of new particles, the crepancy was later realized to be due to the production and not to a failure of quantum field theory.) mesons, 2. John Archibald Wheeler36 in 1937 and Heisenberg in 194337 indepen a to which would have replaced dently proposed physics, positivistic approach a sort with different field of sometimes called an "S quantum theory theory, matrix theory," which would involve only directly measurable quantities. They do not actually allow us to follow what happens to reasoned that experiments electrons in atoms or in collision processes. Instead, it is only really possible to measure the energies and a few other properties of bound systems like atoms, for various collision processes. These quantities obey cer and the probabilities such as reality, conservation of probabilities, tain very general principles, smooth principles energy conservation dependence, that were supposed laws, to replace etc., it was and the assumptions these general of quantum field theory. 3. Dirac38 in 1942 suggested that quantum mechanics ought to be expanded as the initial or not to include states of which could appear negative probability, final state of any physical process, but which would have to be included among the states intermediate introduced into the in these sums over processes. the ways In that this manner, the minus intermediate signs states might share be out of the system, so that a finite answer would be obtained, just as lA + . . . is a finite quantity (the natural logarithm of 2)whereas lA + . . . is infinite. 4. As already mentioned, and John Wheeler39 Richard Feynman in 1945 of abandoning field theory altogether, considered the possibility replacing the interaction among particles with a direct action at a distance. field-mediated Some of these ideas have survived and are now part of the regular equipment In particular, the idea of a pure S-matrix of theoretical physics. theory has the momentum ? ? 1 Vi + Vi 1 + Vi + Vi + involve in the development of so-called "dispersion relations" which flourished are now a states of observable Also, only negative probability quantities.40 device, useful especially for dealing with the polarization handy mathematical none of these ideas of virtual photons.41 However, proved to be the key to the of infinities. problem than most theorists had The solution turned out to be far less revolutionary an had been found by Oppenhei the of electron that Recall energy expected. to receive an infinite contribution from the emis and Weisskopf mer, Waller, sion and reabsorption of "virtual" photons. An infinite self-energy of this sort in orbit in an atom, but also when appears not only when the electron ismoving it is at rest in empty space. But special relativity tells us that the energy of a mass by the famous formula E = mc2. Thus the particle at rest is related to its QUANTUM FIELD THEORY 27 electron mass found in tables of physical data could not be just the "bare" mass, the quantity appearing in our equations for the electron field, but would have to be identified with the bare mass plus the infinite "self mass," produced by the interaction of the electron with its own virtual photon cloud. This suggests that the bare mass might itself be infinite, with an infinity which just cancels the a mass to in finite total the self mass, leaving be identified with the mass infinity that is actually observed. Of course, it goes against the grain to*suppose that a field equations like the bare mass, which appears in our fundamental quantity we can never turn off the electron's virtual photon could be infinite; but after all, cloud to measure the bare mass, so no paradox arises. Similar remarks apply to other physical parameters, like the charge of the electron. Is it then possible that masses and bare charges cancel the infinities found in the infinities in the bare not quantum field theory, only when we calculate the total masses and charges, as well? This method, of eliminating but in all other calculations infinities by a redefinition of come to be called has into them parameters, physical absorbing renormalization. in 1936,42 and again idea was suggested by Weisskopf in the mid 1940s. However, it was far from obvious that the idea In order to eliminate infinities by absorbing them into redefini tions of physical parameters, the infinities must appear in only a special way, as The renormalization by Kramers43 would work. corrections to the observed values of these parameters. For instance, in order to absorb the infinite shift in atomic energy levels found by Oppenheimer into a 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 available methods in the 1930s and early 1940s were simply inadequate for the task of sorting out to see if they could all the infinities that might appear in all possible calculations was even more all be eliminated by renormalization. important, there Perhaps no reason to were no do so?there data which forced experimental compelling theorists to come to grips with these problems. And of course, physicists had other things on their minds from 1939 to 1945. Revival at Shelter Island on the foundations On June 1, 1947, a four-day conference of quantum a near at mechanics Shelter small island of the end Island, opened Long Island in New York State. The conference brought together young American physi cists from the new generation who had started their scientific work during the war at Los Alamos and the MIT Radiation as well as older Laboratory, physi cists who had been active in the 1930s. Among the younger participants was Willis Lamb, an experimentalist then working in the remarkable group of physi cists founded at Columbia University I. I. the re Rabi. Lamb announced by sults of a beautiful experiment, in which he and a student, R. C. Retherford, an effect of the had for the first time measured self-energy of the electron in the hydrogen atom.44 The existing theory of the hydrogen atom had been first advanced by Niels Bohr in 1913,12 then put on a 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 STEVEN 28 WEINBERG In the final version of this theory, in particular as formulated by Dirac, certain pairs of excited states of the atom were expected to have exactly equal to the two different ways that the spin energy. (These pairs of states correspond of the electron and the angular momentum associated with its revolution around the nucleus could combine to give a definite total angular momentum.) But this own with all effects of the of the electron its electro interaction theory ignored to field?the effects that had tried calculate when he magnetic Oppenheimer30 discovered the infinities. If such effects were real they would presumably shift the energies of these pairs of states, so that they would no longer be exactly Dirac.24 equal. and Retherford found. By using the new techniques of radiation that had come out of wartime work in radar, they that the energies of the first two excited states of hydrogen were and (the 2sy2 supposed to be equal according to the 1928 2pV2states), which is now Dirac differed by about 0.4 parts per million. This theory, actually known as the Lamb shift. at Shelter Island en in part by Lamb's results, the participants Stimulated of the underlying tered into an intense discussion theory. I was not at Shelter Island (having just entered high school) and I cannot trace the historical devel of quantum field theory that developed opment of the different reformulations most It be valuable for a historian of science to gather around that time. would at Shelter Island and succeeding confer the recollections of the participants This is what Lamb handling microwave were able to show ences, the read papers that were written at that time, and put together a coher ent account. I will sketch here only a few of the products of this period. The Lamb shift itself was first calculated by Hans Bethe,47 I believe on the to eliminate train ride back from Shelter Island. Using mass renormalization infinities, he obtained a in result reasonable agreement with the value an this was a nounced by Lamb. However, by Bethe himself, acknowledged were not that fully consistent with rough calculation, involving approximations the Special Theory of Relativity. of quantum field theory There were at least three general reformulations were out in the late 1940s that relativistic and that were worked thoroughly a treatment to of the infinities. allow systematic sufficiently simple and elegant well the Shelter been before One of these approaches had actually developed in and his colleagues Island Conference, Japan, but I by Sin-Itiro Tomonaga48 as States in the believe that their work had not yet become known in the United at summer of 1947. The two other approaches were contributed by participants and Richard Feynman.50 Shelter Island, Julian Schwinger49 one to associate a Feynman's work led to set of pictorial rules which allowed a definite numerical to each picture of how the momentum and energy quantity could flow through the intermediate states of any collision process: the probabil sum of these individual quan ity for the process is given by the square of the rules were very much more than a handy calculational tities. The Feynman an essential feature of quantum field theo algorithm, because they incorporated a and between symmetry antiparticles. Each line in Feynman particles ry?the a one at can of the and de end line created either represent particle diagram or an It is at this the other the other, way. equal antiparticle going stroyed treatment of particles and antiparticles that ensures that the quantities calcu QUANTUM FIELD THEORY 29 lated by Feynman diagrams are independent of the velocity of the observer, as at every stage of the calculation. of Relativity, required by the Special Theory states involving intermediate As had been shown long before by Weisskopf,31 a crucial role in down the of infinity, from antiparticles play cutting degree . . ., to more like disasters like 1 + 2 + 3 + something manageable 1 + + Vi Vi . . . .The Feynman rules ensured automatically the cancellations of the worst infinities, leaving over the more 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 umphs the the photons, of Dirac's electron, anomalous a number which moment magnetic 1928 theory24 was of its prediction characterizes the the One electron. of the magnetic strength of the of the tri moment electron's of inter action with magnetic field. How fields, and the strength of its own magnetic at Columbia in 1947 had revealed that the magnetic ever, experiments52 moment of the electron is actually a 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 renormalization of the charge of the electron, Schwinger was able to calculate a finite moment that was larger than the Dirac value by just 1.16 parts magnetic per thousand! Of course, both fects like the Lamb the experimental and the theoretical determinations of ef shift and the anomalous magnetic moment of the electron ex have been enormously improved since 1947.53 For instance, right now the moment value of the of the electron is than the Dirac perimental larger magnetic value by 1.15965241 parts per thousand, whereas the theory gives this anoma as 1.15965234 parts per thousand, with uncertainties lous magnetic moment of about 0.00000020 and 0.00000031 parts per thousand, cision of the agreement between theory and experiment The pre respectively. here can only be called spectacular. in 1949 showed that the formalisms of Schwinger Finally, Freeman Dyson54 and Tomonaga would yield the same graphical rules that had been found by also carried out an analysis of the infinities in general Feyn Feynman. Dyson man and sketched out a general proof that these infinities are always diagrams, of precisely the sort which could be removed by renormalization.55 As a gradu ate student in the mid-1950s, I learned the new approach to quantum field lucid papers. theory by reading Dyson's marvelously I should emphasize that the theory of Schwinger, Feynman, Tomonaga, and Dyson was not really a new physical theory. It was simply the old quantum field theory of Heisenberg, Pauli, Fermi, Oppenheimer, Furry, and Weisskopf, but cast in a form far more convenient for calculation, and equipped with a more realistic definition of masses like and charges. The physical parameters continued vitality of the old quantum field theory after fifteen years of attempts to find a substitute is truly impressive. This raises an 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 at but it least would have been infinite, formally to the order of of like the Lamb shift and guess possible magnitude quantities STEVEN 30 WEINBERG the correction to the magnetic moment of the electron. (An infinite series like 1 + Vi + Vi + !4 + . . . grows very slowly; after amillion terms, it is still less than 14.4.) Not only was this not done?most theorists seemed to have believed that these quantities were zero! Indeed, some evidence for what was later called in 1938, but to the best of my the Lamb shift had actually been discovered56 see whether to no of this checked the order of magnitude theorist knowledge, was or a more in less would be what quan expected reported energy splitting tum field theory. was quantum field theory not taken more seriously? One reason is the Why tremendous prestige of the Dirac theory24 of 1928, which had worked so well in for the fine structure of the hydrogen spectrum without including accounting more the of infinities dis Even effects. appearance important, self-energy credited quantum field theory altogether inmany physicists' minds. But I think that may not have been that the deepest reason is a psychological difficulty, is a huge apparent of There science.57 historians by sufficiently appreciated distance between the equations that theorists play with at their desks, and the practical of reality atomic and spectra collision a certain It takes processes. cour age to bridge this gap, and to realize that the products of thought and mathemat ics may actually have something to do with the real world. Of course, when a branch of science is well under way, there is continual give and take between and one gets used to the idea that the theory is about theory and experiment, the pressure of experimental real. Without data, the realization something comes harder. The great thing accomplished the discovery of the Lamb shift by our us was not so much to that it forced theories, as that it physical change us to take them seriously. forced Weak and Strong Interactions for quantum field theory was at a 1949, enthusiasm soon lead to an understanding it that would expected not the only dynamics of photons, electrons, and phenomena, For a few years after theorists high level. Many of all microscopic fidence?shares there it was However, positrons. began not in quantum a second depression, long field before theory which was there tumbled was to last another collapse on the physics for almost in con and bourse, twenty years. of the renormaliza Part of the problem arose from the limited applicability a of to renormalization be eliminated tion idea. In order for all infinities by to infinities masses for these and charges, it is necessary physical parameters like arise in only a limited number of ways, as corrections to masses, charges, etc., but not otherwise. Dyson's work54 showed that this would be the case for only a small class of quantum field theories, which are called renormalizable theories. and positrons The simplest theory of photons, (known as quantum electrons, in this sense, but most theories are not. electrodynamics) is renormalizable was one important class of physical phenomena which there Unfortunately, field theory. These were not be described by a renormalizable apparently could the weak interactions, which cause the radioactive beta decay of nuclei mentioned Fermi21 had in Field Theory. above in the section on the Birth of Quantum in 1932 which, with a few modifications, vented a theory of weak interactions FIELD QUANTUM approximation, culations all weak described adequately of that is, including rates. transition THEORY 31 interaction phenomena in the a only single simple Feynman as However, as soon this diagram was theory order of lowest in cal to the pushed next order of approximation, it exhibited infinities which could not be removed a of redefinition by 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 sum infinite quence ous of Feynman states consisting of intermediate To types. is the process each square diagram of the we sum these numbers a numerical quantities. a representing of definite associate of one each diagrams, quantity; in Now, se particular of particles the rate of vari of the electro quantum are very small; the quantities associated with complicated dynamics diagrams 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 factor is even smaller?at the typical energies of interactions, the corresponding it is 10"5 to 10~7. It is the rapid decrease of the elementary particle physics, contributions associated with complicated Feynman diagrams (together with the renormalizability of the theory) that makes it possible to carry calculations in quantum However, electrodynamics in addition to to such a fantastic and electromagnetic degree weak of accuracy. there interactions, is one in elementary other class of interaction known as the strong particle physics, interactions. It is the strong interactions that hold atomic nuclei together against the electrostatic repulsion of the protons they contain. For strong interactions, to the fine structure constant is the factor corresponding roughly of the order of one instead of 1/137, so are complicated Feynman diagrams just as important as ones. course are of is these interactions al (This simple why strong.) Thus, though attempts have been made time and time again to use quantum field theory in calculations of the nuclear force, it has never really worked in a con vincingly quantitative mesons and hyperons and then in accelerator kinds of strongly interacting particles called way. New were being discovered from 1947 on, first in cosmic rays laboratories, and quantum field theory was at first en thusiastically used to study their strong interactions, but again with little quan titative success. It was not that there was any in thinking of difficulty renormalizable quantum field theories that might account for the strong inter was just that actions?it having thought of such a theory, there was no way to use it to derive reliable and to test if it were true. quantitative predictions, The nonrenormalizability of the field theory of weak interactions and the uselessness of the field theory of strong interactions led in the early 1950s to a with quantum field theory. Some theorists turned disenchantment widespread to the study of symmetry and conservation laws, which can be ap principles detailed dynamical calculations. Others plied to physical phenomena without and Heisenberg, and worked to picked up the old S-matrix theory of Wheeler of interaction that would involve develop principles strong physics only observ able quantities. In both lines of work, quantum field was used heuristi theory but not as a basis for quantitative cally, as a guide to general principles, calculation. Now, as a result of work by many physicists over the last decade, quantum STEVEN 32 WEINBERG field theory has again become what it was in the late 1940s, the chief tool for a are quantum detailed understanding of elementary particle processes. There 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 in the new theories.58 lems of quantum field theory are surmounted 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 field theory of electromagnetic interactions. Just as electromagnetic interactions so are of the weak the among charged particles produced by photons, exchange interactions are produced by the exchange of particles called intermediate vector bosons and the strong interactions by the exchange of other particles called gluons. vector bosons, All these particles?photons, and gluons?have intermediate certain and have interactions symmetry prin spin, by powerful governed equal as gauge states that the known (A gauge symmetry principle ciples symmetries. fundamental equations do not change their form when the fields are subjected to certain whose transformations, theories these are so similar varies effect to quantum with and position electrodynamics, they time.) share Because its funda of being renormalizable. Indeed, the relation between weak and electromagnetic interactions is not merely one of analogy?the theory uni vector fies the two, and treats the fields of the photon and the intermediate of a single family of fields. bosons as members like the photon, but in The intermediate vector bosons are not massless, stead are believed to have perhaps 70 to 80 times the mass of a proton or neu the tron. This huge mass between is not due to any essential dissimilarity vector from the and boson but instead arises the intermediate fields, way photon field theory breaks down when the field that the symmetry of the underlying are solved. The family of intermediate vector bosons, of which the equations one heavy charged particle and its a to is is contain believed member, photon mental property, called antiparticle, the W+ and W~, and one even heavier neutral particle, called like nuclear the Z?. Exchange of the W produces the familiar weak interactions, a new kind of weak Z? would of the beta decay, whereas produce exchange in which the participating interaction, particles do not change their charge. in 1973, and are found to have just Such neutral current processes were discovered in All of the intermediate vector theories. these about the properties expected too heavy to have been produced with existing accelerator bosons are much them with colliding beams of facilities, but there are great hopes of producing too before and protons long. antiprotons the strong interactions may well have In contrast, the gluons which mediate zero mass. Such theories with massless gluons have a remarkable property as asymptotic freedom?at the very high energy or very short distances, now it is In decreases. of the interactions consequence, strength gradually gluon out detailed calculations of strong possible to use quantum field theory to carry it has been pos interaction processes at sufficiently high energy. In particular, a some in to account the observed features for of sible process such as high known energy electron-nucleon scattering. FIELD QUANTUM THEORY 33 Just as the gluon interactions become weak at high energy and short dis tances, they also become strong at low energy and long distances. For this rea son, it iswidely believed (though not yet proved) that particles which carry the quantity called color with which gluons interact (in the same sense that photons interact with electric charge) cannot be produced as separate free particles. The colored particles include the gluons themselves, which is presumably why as real never have been observed The colored gluons particles. particles are also believed to include the quarks discussed in this double issue o? Daedalus in the article by Sidney Drell. The observed strongly interacting particles such as neu trons, protons, and mesons are to be believed compound states, of consisting no net color. This a quarks, antiquarks, and gluons, but with picture represents over matter: the field the view of the funda of nearly complete triumph particle mental entities are the quark and gluon fields, which do not correspond to any particles that can be observed even in principle, whereas the observed strongly interacting particles are not elementary at all, but are mere consequences of an quantum field theory. are a unified gauge and hopes of theory of weak, electromagnetic, intermediate vector bosons, and gluons would strong interactions. The photon, then form part of a single family of fields. However, in order for this to be to to in there would have be other fields this possible, family, corresponding mass. one to the estimate, particles of extraordinarily expected high According mass of these new particles would be 1017 (a hundred thousand million million) proton masses. These masses are so high that it is no longer possible to ignore as done almost the gravitational fields of these particles, else in everywhere particle physics. strenuous to the present, efforts which continue Unfortunately, despite there still has not been found a satisfactory (e.g., renormalizable) quantum field of It is that ironic which theory gravitation. provided the first classi gravitation, cal field theory, has so far resisted incorporation into the general framework of quantum field theory. this history I have put great emphasis on the condition of renor Throughout that it should be possible to eliminate all infinities the malizability, requirement in a quantum field theory by a redefinition of a small number of physical param eters. Many physicists would disagree with this emphasis, and indeed, itmay or not, are be found that all quantum field theories, renormalizable eventually underlying There equally satisfactory. However, it has always seemed to me that the 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, to construct it is possible field theories of quantum we like, but in moment which the electron has any magnetic electromagnetism one a to of these moment of theories, only corresponding magnetic . . . times the Dirac value, is renormalizable. we 1.0011596523 as have Also, seen, it took a long time before it was found that there are any renormalizable theories at all of the weak interactions. We very much need a guiding principle to help us to pick the quantum field like renormalizability theory of the real world out of the infinite variety of conceivable quantum field theories. Thus, if to is some be with other I condition, renormalizability ultimately replaced STEVEN 34 WEINBERG be one that is equally or even more restrictive. After all, to describe the world as we find it, but to explain to the extent why it has to be the way it is. would hope that itwill we do not want merely greatest possible References I am I. I. Rabi, advice from Felix Bloch, Freeman Dyson, Francis Everitt, grateful for valuable and Victor Weisskopf. ^his often quoted summary of Newton's appears at the end of the Philoso theory of gravitation revised and annotated p. 546, in the translation of A. Motte, phiae Naturalis Principia Mathematica, by F. Cajori of California Press, 1966), p. 546. (Berkeley: University on 9th ed. (1875-1889); article "Ether" in The Encyclopedia Britannica, 2J. C. Maxwell, reprinted inW. D. Niven (New York: Dover, 1965), p. 763. (ed.), The Scientific Papers ofJames Clerk Maxwell see Maxwell's Also Treatise on Electricity and Magnetism, vol. 2 (New York: Dover, 1954), pp. 492 493. A useful short account of Maxwell's is given by C. W. F. Everitt, James Clerk life and work and Natural Maxwell?Physicist 3See Sees. 600-601, 668, Ann. Phys., 4W. Wien, 5 5M. Abraham,Nachr. Ges. (New York: Scribner, Philosopher and 769 of Maxwell, Treatise. (1901): 501. Wiss. (Gottingen), (1902): 20;Ann. 1974). 10(1903): Phys., 105;Phys. Zeit., 5 (1904): 576. 9 6H. A. Lorentz, 1 (1900): 498, 514; Versl. Kon. Akad. Wetenschap. (Amsterdam), Phys. Zeit., 6 (1904): 809; Encylop?die derMath. Wiss. V/2 (1904), pp. 63, (1900): 418; Proc. Roy. Acad. Amsterdam, 145. 21 (1905): 129. 7H. Poincar?, Rend. Cire. Mat. (Palermo), 8A. Einstein, Ann. Phys., 17 (1905): 891, 905; 18 (1905): 639. 9M. Planck, Verband. Deutsch. Phys. Ges., 2 (1900): 237; reprinted with an English translation by H. Kangro in Planck's Original Papers in Quantum Physics (New York: Wiley, 1972). 10G. Gamow, 1966), p. 23. Doubleday, Thirty Years That Shook Physics (Garden City, N.Y.: 17 (1905): 132. nA. Einstein, Ann. Phys., 26 (1913): 1, 476, 857. 12N. Bohr, Phil. Mag., Z. Phys., 33 (1925): 879; 13L. de Broglie, Compt. rend., 177 (1923): 507, 548, 630; W. Heisenberg, M. Born and P. Jordan, Z. Phys., 34 (1925): 858; P. A. M. Dirac, Proc. Roy. Soc., A109 (1925): 642; Ann. Phys., 79: 36 (1926): 336; E. Schr?dinger, M. Born, W. Heisenberg, and P. Jordan, Z. Phys., in in English translations of these papers are reprinted 361, 489; 80: 437; 81: 109 (all 1926). Many B. L. van der Waerden, Sources of Quantum Mechanics 1968). (New York: Dover, 36 (1926): 336, Sec. 3. and Jordan, Z. Phys., 14Born, Heisenberg, in an 15P. A. M. Dirac, article is reprinted Proc. Roy. Soc. (London), A114 (1947): 243. This invaluable collection of papers on quantum field theory, J. Schwinger (ed.), Quantum Electrodynamics (New York: Dover, 1958), referred to below as QED. 36 (1926): 336. 16Born and Jordan, Z. Phys., 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 56 (1929): 1; ibid, 59 (1930): and W. Pauli, Z. Phys., 168. 20E. Fermi, Rend. R. Ace. dei Lincei (6)9 (1929): 881. 88 (1934): 161. 21E. Fermi, Z. Phys., 22P. Ehrenfest and J. Oppenheimer, 37 (1931): 333. Phys. Rev., and F. Joliot, Compt. 23W. Heisenberg, Z. Phys., 77 (1932): 1; also see I. Curie-Joliot (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, rend., 194 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 (1934): 709. 28W. Pauli and V. Weisskopf, Helv. Phys. Acta, 29H. Yukawa, Soc. (Japan), 17 (3) (1935): 48. Proc. Phys.-Math. see I. Waller, Z. Phys. 59 (1930): 168; 35 (1930): 461. Also 30J. R. Oppenheimer, Phys. Rev., ibid., 62 (1930): 673. 31V. F. Weisskopf, Z. Phys., 89 (1934): 27; ibid., 90 (1934): 817. Also, Phys. Rev., 56 (1939): 72, in QED. reprinted in XVII Conseil Solvay de Physique 32P. A. M. Dirac, (Brussels, 1933), p. 203. 55 (1939): 959. 33S. M. Dancoff, Phys. Rev., in QED. 34F. Bloch and A. Nordsieck, Phys. Rev. 52 (1937): 54, reprinted QUANTUM FIELD THEORY 35 35W. Heisenberg, 36J. A. Wheeler, 37W. Heisenberg, 32 (1938): 20. Ann. Phys., 52 (1937): 1107. Phys. Rev., see C. Z. Phys., 120 (1943): 513, 673; Z. Naturforsch., 1 (1946): 608. Also Medd. ,23(1) M?ller, Kon. Dansk. Vid. Sels. Mat.-Fys. (1945); ibid., (19) (1946). (Heisenberg actually a fundamental an energy, introduced but the two are equivalent notions: any length rather than an energy, the energy of a photon with that wavelength). length determines 38P. A. M. Dirac, Proc. Roy. Soc. (London), A180 (1942): 1. and R. P. Feynman, Rev. Mod. Phys., 17 (1945): 157; ibid., 21 (1949): 425. 39J. A. Wheeler 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. 15 (6) (1936), esp. p. 34 and pp. 5-6; Medd., in QED. reprinted 43The only published report of Kramers' work that I can find is in the Rapports du VIH Conseil de Kramers 1950). However, Physique Salvay 1948 (ed. R. Stoops, Brussels, reported on his work at the Shelter Island Conference. 44W. E. Lamb, inQED. Jr. and R. C. Retherford, Phys. Rev., 72 (1947): 241, reprinted 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. Tomonaga, 1 (1946): 27; Z. Koba, T. Tati, and S. Prog. Theor. Phys. (Japan), and S. Tomonaga, ibid., 2 (1947): 101; S. Kanesawa ibid., 3 (1948): 276; Z. Koba and S. inQED. ibid., 3 (1948): 290. Some of these articles are reprinted 49J. Schwinger, Phys. Rev. 74 (1948): 1439; 75 (1949): 651; 76 (1949): 790; 82 (1951): (1953): 713; Proc. Nat. Acad. Sei. 37 (1951): 452. All but the first of these articles are Tomonaga, Tomonaga, 664, 914; 91 in reprinted QED. 50R. P. Feynman,/?^. Mod. Phys., 20(1948): 939, 1430; 76(1949): 749, 367;Phys. Rev. 74(1948): inQED. 769; 80 (1950): 440. Some of these articles are reprinted in QED. 51J. Schwinger, Phys. Rev., 73 (1948): 146, reprinted E. B. Nelson, and I. I. Rabi, Phys. Rev., 71 (1947): 914; D. E. R. S. Julian, S2J. E. Nafe, Nagel, and J. R. Zacharias, 72 (1947): 97; P. Kusch and H. M. Foley, Phys. Rev., 72 (1947): Phys. Rev., 1256. 53For early calculations of the Lamb shift, see J. B. French and V. F. Weisskopf, Phys. Rev., 75 ibid. (1949): 388; R. P. Feynman, (1949): 1240; N. M. Kroll and W. E. Lamb, Phys. Rev., 74 (1948): and S. Tomonaga, 1430; H. Fukuda, Y. Miyamoto, Prog. Theor. Phys. (Japan), 4 (1948): 47, 121. The article by Kroll and Lamb is reprinted in QED. 54F. J. Dyson, 75 (1949): 486, in QED. 1736, reprinted Phys. Rev., to complete Dyson's needed 55Elements later supplied by A. Salam, Phys. Rev., 82 proof were 118 (1960): 838. (1951): 217; 84 (1951): 426; and S. Weinberg, Phys. Rev. 56S. Pasternack, 1113. Also see W. V. Houston, 51 (1937): 446; P/ryj. ?w., Phys. Rev., 54(1938): R. C. Williams, results were reported by J. W. Drinkwater, O. Phys. Rev., 54 (1938): 558. Contrary and W. E. Williams, Proc. Roy. Soc, 174 (1940): 164. Richardson, 57I have recently discussed this point in a 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 subject on a technical to the see level, with references literature, original Rev. Mod. Phys., 46 (1974): 255. I have also accounts in e.g. S. Weinberg, given nonmathematical 231 (July 1974): 50; and in Bull. Am. Acad. Arts and Sciences, 24 (4) (1976): 13; Scientific American, in Plural, of (in Spanish) (1976); and in American Scientist, 65 (March 1977): 171. Aspects reprinted this subject are treated in two recent books: J. C. (Cam Gauge Theories of Weak Interactions Taylor, bridge, England: Cambridge University Achievements of Twentieth Century Physics Press, 1976); G. Feinberg, What Is theWorld Made Of? The Anchor (Garden City, N.Y.: 1977). Press/Doubleday,