Conceptual foundations of the unified theory of Weak andelectroraiagnetic interactions*'

Steven Weinberg

Lyman Laboratory of Physics, Harvard Universityand Harvard —Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138 U.S.A.

Our job in physics is to see things simply, to under-stand a great many complicated phenomena in a unifiedway, in terms of a few simple principles. At times,our efforts are illuminated by a brilliant experiment,such as the 1973 discovery of neutral current neutrinoreactions. But even in the dark times between experi-mental breakthroughs, there always continues a steadyevolution of theoretical ideas, leading almost imper-ceptibly to changes in previous beliefs. In this talk, Iwant to discuss the development of two lines of thoughtin theoretical physics. One of them is the slow growthin our understanding of symmetry, and in pa, rticular,broken or hidden symmetry. The other is the oldstruggle to come to terms with the infinities in quantumfield theories. To a remarkable degree, our presentdetailed theories of elementary particle interactionscan be understood deductively, as consequences of sym-metry principles and of a principle of renormalizabilitywhich is invoked to deal with the infinities. I will alsobriefly describe how the convergence of these lines ofthought led to my own work on the unification of weakand electromagnetic interactions. For the most part,my talk will center on my own gradual education inthese matters, because that is one subject on which Ican speak with some confidence. With rather less con-fidence, I will also try to look ahead, and suggest whatrole these lines of thought may play in the physics ofthe future.

Symmetry principles made their appearance in twen-tieth century physics in 1905 with Einstein s identifica-tion of the invariance group of space and time. Withthis as a precedent, symmetries took on a characterin physicists' minds as a pro~i principles of universalvalidity, expressions of the simplicity of nature at itsdeepest level. So it was painfully difficult in the 1930sto realize that there are internal symmetries, such asisospin conservation, ' having nothing to do with spaceand time, symmetries which are far from self-evident,and that only govern what are now called the strong in-teractions. The 1950s saw the discovery of anotherinternal symmetry —the conservation of strange-ness' —which ip not obeyed by the weak interactions,and even one of the supposedly sacred symmetries ofspace-time —parity —was also found to be violated byweak interactions. ' Instead of moving toward unity,physicists. were learning that different interactions are

This lecture was delivered December 8, 1979, on the occa-sion of the presentation of the 1979 Nobel Prizes in Physics.

~This manuscript is referenced according to the author 's ownstyle, and not according to RMP's customary British refer-ence system. We trust that this will not inconvenience thereader.

apparently governed by quite different symmetries.Matters became yet more confusing with the recognitionin the early 1960s of a symmetry group —the "eight-fold way" —which is not even an exact symmetry of thestrong interactions.

These are all "global" symmetries, for which thesymmetry transformations do not depend on position inspace and time. It had been recognized' in the 1920sthat quantum electrodynamics has another symmetryof a far more powerful kind, a "local" symmetry undertransformations in which the electron field suffers aphase change that can vary freely from point to pointin space-time, and the electromagnetic vector potentialundergoes a corresponding gauge transformation. To-day this would be called a U(l) gauge symmetry, be-cause a simple phase change can be thought of as multi-plication by a 1&& 1 unitary matrix. The extension tomore complicated groups was made by Yang and Mills'in 1954 in a seminal paper in which they showed how toconstruct an SU(2) gauge theory of strong interactions.(The name "SU(2)" means that the group of symmetrytransformations consists of 2&2 unitary matrices thatare "special, " in that they have determinant unity. )But here again it seemed that the symmetry, if real atall, would have to be approximate, because at leaston a naive level gauge invariance requires that vectorbosons like the photon would have to be massless, andit seemed obvious that the strong interactions are notmediated by massless particles. The old question re-mained: if symmetry principles are an expression ofthe simplicity of nature at its deepest level, then howcan there be such a thing as an approximate symmetry?Is nature only approximately simple?

Sometime in 1960 or early 1961, I learned of an ideawhich had originated earlier in solid state physics andhad been brought into particle physics by those likeHeisenberg, Nambu, and Goldstone, who had worked inboth areas. It was the idea of "broken symmetry, "that the Hamiltonian and commutation relations of aquantum theory could possess an exact symmetry, andthat the physical states might nevertheless not provideneat representations of the symmetry. In particular,a symmetry of the Hamiltonian might turn out to be nota symmetry of the vacuum.

As theorists sometimes do, I.fell in love with thisidea. But as often happens with love affairs, at first Iwas rather confused about its implications. I thought(as it turned out, wrongly) that the approximate sym-metries —parity, isospin, strangeness, the eightfoldway —might really be exact a priori symmetry princi-ples, and that the observed violations of these sym-metries might somehow be brought about by spontaneoussymmetry breaking. It was therefore rather disturbing

Reviews of Modern Physics, Vol. 52, No. 3, July 1989 -Copyright 1980 The Nobel Foundation

Steven Weinberg: Unified theory of weak and electromagnetic interactions

for me to hear of a result of Goldstone, ' that in at leastone simple case the spontaneous breakdown of a con-tinuous symmetry like isospin would necessarily entailthe existence of a massless spin zero particle —whatwould today be called a "Goldstone boson. " It seemedobvious that there could not exist any new type of mass-less particle of this sort which would not already havebeen discovered.

I had long discussions of this problem with Goldstoneat Madison in the summer of 1961, and then with Salamwhile I was his guest at Imperial College in 1961-62.The three of us soon were able to show that Goldstonebosons must in fact occur whenever a symmetry likeisospin or strangeness is spontaneously broken, andthat their masses then remain zero to all orders ofperturbation theory. I remember being so discouragedby these zero masses that when we wrote our jointpaper on the subject, ' I added an epigraph to the paperto underscore the futility of supposing that anythingcould be explained in terms of a noninvariant vacuumstate: it was Lear's retort to Cordelia, "Nothing willcome of nothing: speak again. " Of course, The Physi-cal Aevieu protected the purity of the physics litera-ture, and removed the quote. ' Considering the future ofthe noninvariant vacuum in theoretical physics, it wasjust as well.

There was actually an exception to this proof, pointedout soon afterwards by Higgs, Kibble, and others. 'They showed-that if the broken symmetry is a local,gauge symmetry, like electromagnetic gauge invariance,then although the Goldstone bosons exist formally, andare in some sense real, they can be eliminated by agauge transformation, so that they do not appear asphysical particles. The missing Goldstone bosonsappear instead as helicity zero states of the vectorparticles, which thereby acquire a mass.

I think that at the time physicists who heard aboutthis exception generally regarded it as a technicality.This may have been because of a new development intheoretical physics, which suddenly seemed to changethe role of Goldstone bosons from that of unw'anted in-truders to that of welcome friends.

In 1964 Adler and Weisberger' independently derivedsum rules which gave the ratio g„/gv of axial-vectorto vector coupling constants in beta decay in terms ofpion-nucleon cross sections. One way of looking attheir calculation (perhaps the most common way at thetime) was as an analog to the old dipole sum rule inatomic physics: a complete set of hadronic states isinserted in the commutation relations of the axial vec-tor currents. This is the approach memorialized inthe name of "current algebra. "" But there was anotherway of looking at the Adler-%eisberger sum rule. Onecould suppose that the strong interactions have an ap-proximate symmetry, based on the group SU(2) x SU(2),and that this symmetry is spontaneously broken, givingrise among other things to the nucleon masses. Thepion is then identified as (approximately) a Goldstoneboson, with small but nonzero mass, an idea that goesback to Nambu. '2 Although the SU(2) x SU(2) symmetryis spontaneously broken, it still has a great deal ofpredictive power, but its predictions take the form ofapproximate formulas, which give the matrix elements

for low energy pionic reactions. In this approach, theAdler-Weisberger sum rule is obtained by using thepredicted pion nucleon scattering lengths in conjunctionwith a well-known sum rule, ' which years earlier hadbeen derived from the dispersion relations for pion-nucleon scattering.

In these calculations one is really using not only thefact that the strong interactions have a spontaneouslybroken approximate SU(2) && SU(2) symmetry, but alsothat the currents of this symmetry group are, up to anoverall constant, to be identified with the vector andaxial vector currents of beta decay. (With this assump-tion g„/gv gets into the picture through the Goldberger-Treiman relation, "which gives g„/gv in terms of thepion decay constant and the pion nucleon coupling. ) Here,in this relation between the currents of the symmetriesof the strong interactions and the physical currents ofbeta decay, there was a tantalizing hint of a deep con-nection between the weak interactions and the strong in-teractions. But this connection was not really under-stood for almost a decade.

I spent the years 1965-67 happily developing the im-plications of spontaneous symmetry breaking for thestrong interactions. " It was this work that led io my1967 paper on weak and electromagnetic unification.But before I come to that I have to go back in historyand pick up one other line of thought, having to do withthe problem of infinities in quantum field theory.

I believe that it was Oppenheimer and %aller in. 1930who independently first noted that quantum field theorywhen pushed beyond the lowest approximation yieldsultraviolet divergent results for radiative self-ener-gies. Professor %aller told me last night that when hedescribed this result to Pauli, Pauli did not believe it.It must have seemed that these infinities would be adisaster for the quantum field theory that had just beendeveloped by Heisenberg and Pauli in 1929-30. Andindeed, these infinities did lead to a sense of dis-couragement about quantum field theory, and many at-tempts were made in the 1930s and early 1940s to findalternatives. The problem was solved (at least forquantum electrodynamics) after the war, by Feynman,Schwinger, and Tomonaga, "and Dyson. " It was found thatall infinities disappear if one identifies the observed finitevalues of the electron mass and charge, not with theparameters m and e appearing in the Lagrangian, butwith the electron mass and charge thai are caleuEatedfrom rn and e, when one takes into account the fact thatthe electron and photon are always surrounded withclouds of virtual photons and electron-positron pairs. "Suddenly all sorts of calculations became possible, andgave results in spectacular agreement with experi-ment.

But even after this success, opinions differed as tothe significance of the ultraviolet divergences in quan-tum field theory. Many thought —and some still dothink —that what had been done was just io sweep thereal problems under the rug. And it soon became clearthat there was only a limited class of so-called "re-normalizable" theories in which the infinities could beeliminated by absorbing them into a redefinition, or a"renormalizaiion, " of a finite number of physical pa-rameters. (Roughly speaking, in renormalizable theor-

Rev; Mod. Phys. , Vol. 52, No. 3, July 1980

Steven Weinberg: Unified theory of weak and e)ectromagnetic interactions 517

ies no coupling constants can have the dimensions ofnegative powers of mass. But every time we add a fieldor a space-time derivative to an interaction, me reducethe dimensionality of the associated coupling constant.So only a few simple types of interaction can be re-normalizable. ) In particular, the existing Fermi theoryof weak interactions clearly was noi renormalizable.(The Fermi coupling constant has the dimensions of[mass] .) The sense of discouragement about quantumfie1d theory persisted into ihe 1950s and 1960s.

I learned about renormalization theory as a graduatestudent, mostly by reading Dyson's papers. " From thebeginning it seemed to me to be a wonderful thing thatvery few quantum field theories are renormalizable.Limitations of this sort are, after all, what we mostavant; not mathematical methods which can make senseout of an infinite variety of physically irrelevant theor-ies, but methods which carry constraints, becausethese constraints may point the way toward the one truetheory. In particular, I was very impressed by thefact that-quantum electrodynamics could in a sense bederived from symmetry principles and the constraintof renormalizabiliiy; the only Lorentz invariant andgauge invaria, nt renormahzable Lagrangian for photonsand electrons is precisely the original Dirac Langran-gian of QED. Of course, that is not the way Diraccame to his theory. He had the benefit of the informa-tion gleaned in centuries of experimentation on electro-magnetism, and in order to fix the final form of histheory he relied on ideas of simplicity (specifically,on what is sometimes called minimal electromagneticcoupling). But we have to look ahead, to try to maketheories of phenomena which have noi been so wellstudied experimentally, and we may not be able to trustpurely formal ideas of simplicity. I thought that re-normalizability might be the key criterion, which alsoin a more general context would impose a precise kindof simplicity on our theories and help us to pick out theone true physical theory out of the infinite variety ofconceivable quantum field theories. As I will explainlater, I would say this a bit differently today, bui I ammore convinced than ever that the use of renormaliza-bility as a constraint on our theories of the observedinteractions is a good strategy. Filled with enthusiasmfor renormalization theory, I wrote my Ph. D. thesisunder Sam Treiman in 1957 on the use of a limitedversion of renormalizability to set constraints on theweak interactions, and a little later I worked out arather tough little theorem which completed the proofby Dyson" and Salam" that ultraviolet divergencesreally do cancel out to all orders in nominally re-normalizable theories. But none of this seemed to helpwith the important problem, of how to make a re-normalizable theory of weak interactions.

Now, back to 1967. I had been considering the im-plications of the broken SU(2)x SU(2) symmetry of thestrong interactions, and I thought of trying out the ideathat perhaps tlie SU(2) x SU(2) symmetry was a "local, '*

not merely a "global, " symmetry. That is, the stronginteractions might be described by something like aYang-Mills theory, but in addition to the vector p mes-ons of the Yang-Mills theory, there would also be axialvector A1 mesons. To give the p meson a mass, it was

necessary to insert a common p and A. l mass term inthe Lagrangian, and the spontaneous breakdown of theSU(2)x SU(2) symmetry would then split the p and%1by something like the Higgs mechanism, but since thetheory would not be gauge invariant the pions would re-main as physical Goldstone bosons. This theory gavean intriguing result, that the 4 I/P mass ratio should bev 2, and in trying to understand this result without re-lying on perturbation theory, I discovered certain sumrules, the "spectral function sum rules, ""which turnedout to have a variety of other uses. But the SU(2) x SU(2)theory was not gauge invariant, and hence it could notbe reriormalizable, 24 so I was not too enthusiastic aboutit." Of course, if I did not insert the P-Al mass termin the Lagrangian, then the theory would be gauge in-variant and renormalizable, and the A. l mould be mas-sive. But then there would be no pions and the p mes-ons would be massless, in obvious contradiction (tosay the least) with observation.

At some point in the fall of 1967, I thirik while drivingio my office at MIT, ii occurred to me that I had beenapplying the right ideas io the wrong problem. It is notthe p meson that is massless: it is the photon. And iispartner is not the @1, but the massive intermediatebosons, which since the time of Yukawa had been sus-pected to be the mediators of the weak interactions.The weak and electromagnetic interactions could thenbe described" in a unified may in terms of an exact butspontaneously broken gauge symmetry tOf c. ourse,not necessarily SU(2) x SU(2).] And this theory wouldbe renormalizable like quantum electrodynamics be-cause it is gauge invariant like quantum electrodynam-ics.

It was not difficult to develop a concrete model whichembodied these ideas. I had little confidence then in myunderstanding of strong interactions, so I decided toconcentrate on leptons. There are two left-handedelectron-type leptons, the v, l. and eL, , and one right-handed electron-type lepton, the e&, so I started withthe group U(2) x U(1): all unitary 2x 2 matrices actingon the left-handed e-type leptons, together with allunitary 1& 1 matrices acting on the right-handed e-typelepton. Breaking up U(2) into unimodular transforma-tions and phase iransformations, one could say that thegroup was SU(2) x U(l) x U(1). But then one of the U(1)'scould be identified with ordinary lepton number, andsince lepton number appears to be conserved and thereis no massless vector particle coupled to it, I decidedto exclude it from the group. This left the four-pa-rameter group SU(2)x U(1). The spontaneous breakdownof SU(2) x U(1) to the U(1) of ordinary electromagneticgauge invariance would give masses to three of the fourvector gauge bosons: the charged bosons ~', and aneutral boson that I called the Z . The fourth bosonwould automatically remain massless, and could beidentified as the photon. Knowing the strength of theordinary charged current weak interactions like betadecay which are mediated by ~', the mass of the +"was then determined as about 40 GeV/sin8, where 8 isthe y-Zo mixing angle.

To go further, one had to make some hypothesis aboutthe mechanism for the breakdown of SU(2) x U(1). Theonly kind of field in a renormalizable SU(2) x U(1) theory

Rev. Mod. Phys. , Vol. 52, No. 3, July 1980

518 Steven NJI'einberg: Unified theory of weak and etectromagnetic interactions

whose vacuum expectation values could give the elec-tron a mass is a spin zero SU(2) doublet (g', @ ), so forsimp1icity I assumed that these were the only scalarfields in the theory. The mass of the Z' was then deter-mined as about 80 GeV/sin26. This fixed the strengthof the neutral current weak interactions. Indeed, justas in QED, once one decides on the menu of fields inthe theory, all details of the theory are completely de-termined by. symmetry principles and renormalizability,with just a few free parameters: the lepton charge andmasses, the Fermi coupling constant of beta decay, themixing angle 6, and the mass of the scalar particle.(lt was of crucial importance to impose the constraintof renormalizability; otherwise weak interactions wouldreceive contributions from SU(2) x U(l)-invariant four-fermion couplings as well as from vector boson ex-change, and the theory would lose most of iis pre-dictive power. ) The naturalness of the whole theory iswell demonstrated by the fact that much the same theorywas independently developed" by Salam in 1968.

The next question now was renormalizability. TheFeynman rules for Yang-Mills theories with unbrokengauge symmetries had been worked out" by deWitt,Faddeev, and Popov and others, and it was known thatsuch theories are renormalizable. But in 196V I didnot know how to prove that this renormalizability wasnot spoiled by the spontaneous symmetry breaking. Iworked on the problem on and off for several years,partly in collaboration with students, "but I made littleprogress. With hindsight, my main difficulty was thatin quantizing the vector fields I adopted a gauge nowknown as the unitarity gauge: this gauge has severalwonderful advantages, it exhibits the true particle spec-trum of the theory, but it has the disadvantage of mak-ing renormalizability totally obscure.

Finally, in 1971 't Hooft" showed in a beautiful paperhow the problem could be solved. He invented a gauge,like the "Feynman gauge" in QED, in which the Feyn-man rules manifestly lead to only a finite number oftypes of ultraviolet divergence. It was also necessaryto show that these infinities satisfied essentially thesame constraints as the Lagrangian itself, so that theycould be absorbed into a redefinition of the parametersof the theory. (This was plausible, but not easy toprove, because a gauge invariant theory can be quan-tized only after one has picked a specific gauge, so itis not obvious that the ultraviolet divergences satisfythe same gauge invariance constraints as the Lagran-gian itself. ) The proof was subsequently completed"by Lee and Zinn- Justin and by 't Hooft and Veltman.More recently, Becchi, Rouei, and Stora" have in-vented an ingenious method for carrying out this sortof proof, by using a global supersymmetry of gaugetheories which is preserved even when we choose aspecific gauge.

I have to admit that when I first saw 't Hooft's paperin 1971, I was not convinced that he had found the wayto prove renormalizability. The trouble was not with't Hooft, but with me: I was simply not familiar enoughwi. th the path integral formalism on which 't Hooft'swork was based, and I wanted to see a derivation of theFeynman rules in 't Hooft's gauge from canonical quan-tization. That was soon supplied (for a limited class of

gauge theories) by a paper of Ben Lee,"and after Lee' spaper I was ready to regard the renormalizability of theunified theory as essentially proved.

By this time, many theoretical physicists were be-coming convinced of the general approach that Salamand I had adopted: that is, the weak and electromagneticinteractions are governed by some group of exact localgauge symmetries; this group is spontaneously brokento U(l), giving mass to all the vector bosons except thephoton; and the theory is renormalizable. What wasnot so clear was that our specific simple model was theone chosen by nature. That, of course, was amatterforexperiment to decide.

It was obvious even back in 196V that the best way totest the theory would be by searching for neutral cur-rent weak interactions, mediated by the neutral inter-mediate vector boson, the Z . Of course, the pos-sibility of neutral currents was nothing new. There hadbeen speculations" about possible neutral currents asfar back as 193V by Gamow and Teller, Kemmer, andWentzel, and again in 1958 by Bludman and Leite-Lopes. Attempts at a unified weak and electromagnetictheory had been made" by Glashow and Salam and Wardin the early 1960s, and these had neutral currents withmany of the features that Salam and I encountered indeveloping the 1967-68 theory. But since one of thepredictions of our theory was a value for the mass ofthe Z', it made a definite prediction of the strength ofthe neutral currents. More importarit, now we had acomprehensive quantum field theory of the weak andelectromagnetic interactions that was physically andmathematically satisfactory in the same sense as quan-tum electrodynamics —a theory that treated photons andintermediate vector bosons on the same footing, thatwas based on an exact symmetry principle, and that al-lowed one to carry calculations to any desired degreeof accuracy. To test this theory, it had now becomeurgent to settle the question of the existence of the neu-tral currents.

Late in 1971, I carried out a study of the experimentalpossibilities. 37 The results were striking. Previousexperiments had set upper bounds on the rates of neu-tral current processes which were rather low, andmany people had received the impression that neutral cur-rents were pretty well ruled out, butI found that in fact the196V-68 theory predicted quite low rates, low enoughin fact to have escaped clear detection up to that time.For instance, experiments" a few years earlier hadfound an upper bound of 0.12+0.06 on the ratio of aneutral current process, the elastic scattering of muonneutrinos by protons, to the corresponding chargedcurrent process, in which a muon is produced. I founda predicted ratio of 0.15 to 0.35, depending on the valueof the Z'-y mixing angle 6. So there was every reasonto look a little harder.

As everyone knows, neutral currents were finallydiscovered" in 19V3. There followed years of carefulexperimental study on the detailed properties of theneutral currents. It would take me too far from mysubject to survey these experiments, ' so I will just saythat they have confirmed the 1967-68 theory with steadi-ly improving precision for neutrino-nucleon and neutrino-electron neutral current reactions, and since the re-

Rev. Mod. Phys. , Vol. 52, No. 3, July 1980

Steven N/einberg: Unified theory of weak and electromagnetic interactions

markable SLA.C-Yale experiment" last year, for theelectron-nucleon neutral currents as well.

This is all very nice. But I must say that I would nothave been too disturbed if it had turned out that thecorrect theory was based on some other spontaneouslybroken gauge group, with very different neutral cur-rents. One possibility was a clever SU(2) theory pro-posed in 1972 by Georgi and Glashow, ' which h@s noneutral currents at all. The important thing to me wasthe idea of an exact spontaneously broken gauge symmetry, which connects the weak and electromagnetic'interactions, and allows these interactions to be re-normalizable. Of this I was convinced, if only becauseit fitted my conception of the way that nature ought tobe.

There were two other relevant theoretical develop-ments in the early 1970s, before the discovery of neu-tral currents, that I must mention here. One is theimportant work of Glashow, Iliopoulos, and Maiani onthe charmed quark. ' Their work provided a, solutionto what otherwise would have been a serious problem,that of neutral strangeness changing currents. I leavethis topic for Professor Glashow's talk. The othertheoretical development has to do specifically with thestrong interactions, but it will take us back to one ofthe themes of my talk, the theme of symmetry.

In 1973, Politzer and Gross and Wilczek discovered"a remarkable property of Yang-Mills theories whichthey called "asymptotic freedom" —the effective couplingconstant" decreases to zero as the characteristic en-ergy of a process goes to infinity. It seemed that thismight explain the experimental fact that the nucleon be-haves in high energy deep inelastic electron scatteringas if it consists of essentially free quarks. " But therewas a problem. In order to give masses to the vectorbosons in a gauge theory of strong interactions onewould want to include strongly interacting scalar fields,and these would generally destroy asymptotic freedom.Another difficulty, one that particularly bothered me,was that in a unified theory of weak and electromagneticinteractions the fundamental weak coupling is of thesame order as the electronic charge, e, so the effectsof virtual intermediate vector bosons would introducemuch too large violations of parity and strangenessconservation, of order I/13V, into the strong inter-actions of the scalars with each other and with thequarks. 4' At some point in the spring of 1973 it occur-red to me (and independently to Gross and Wilczek) thatone could do away with strongly interacting scalar fieldsaltogether, allowing the strong interaction gauge sym-metry to remain unbroken so that the vector bosons,or "gluons, " are massless, and relying on the increaseof the strong forces with increasing distance to explainwhy quarks as well as the massless gluons are not seenin the laboratory. " Assuming no strongly interactingscalars, three~" colors" of quarks (as indicated by ear-lier work of several authors" ), and an SU(3) gaugegroup, one then had a specific theory of strong inter-actions, the theory now generally known as quantumchromodynamics.

Experiments since then have increasingly confirmedQCD a,s the correct theory of strong interactions. Whatconcerns me here, though, is its impact on our under-

standing of symmetry principles. Once again, the con-straints of gauge invariance and renormalizabilityproved enormously powerful. These constraints forcethe Lagrangian to be so simple, that the strong inter-actions in QCD must conserve strangeness, chargeconjugation, and (apart from problems50 having to dowith instantons) parity. One does not have to assumethese symmetries as a PrioH. principles; there issimply no way Chat the Lagrangian can be complicatedenough to violate them. With one additional assump-tion, that the u and d quarks have relatively smallmasses, the strong interactions must also satisfy theapproximate SU(2) x SU(2) symmetry of current alge-bra, which when spontaneously broken leaves us withigospin. If the + qua, rk mass is also not too large, thenone gets the whole eightfold way as an approximatesymmetry of the strong interactions. And the breakingof this SU(3) && SU(3) symmetry by quark masses has justthe (3, 3) + (3, 3) form required to account for the pion-pion scattering lengths" and the Gell-Mann Okubo massformulas. Furthermore, with weak and electromagneticinteractions also described by a gauge theory, the weakcurrents a,re necessarily just the currents associatedwith these strong interaction symmetries. In otherwords, pretty much the whole pattern of approximatesymmetries of strong, weak, and electromagneticinteractions that puzzled us so much in the 1950s and1960s now stands explained as a. simple consequen. ceof strong, weak, and electromagnetic gauge invariance,plus renormalizability. Internal symmetry is now atthe point where space-time symmetry was in Einstein'sday. All the approximate internal symmetries are ex-plained dynamically. Qn a fundamental level, there areno approximate or par'tial symmetries; there are onlyexact symmetries which govern all interactions.

I now want to look ahead a bit, and comment on thepossible future development of the ideas of symmetryand renormalizability.

We are still confronted with the question whether thescalar particles that are responsible for the spontaneousbreakdown of the electroweak gauge symmetry SU(2)&& U(l) are really elementary. If they are, then spinzero semiweakly decaying "Higgs -bosons" should befound at energies comparable with those needed to pro-duce the intermediate vector bosons. On the other hand,it may be that the scalars are composites. " The Higgsbosons would then be indistinct broad states at veryhigh mass, analogous to the possible s-wave enhance-ment in m —m scattering. There would probably alsoexist lighter, more slowly decaying, scalar particlesof a rather different type, known as pseudo-Goldstonebosons. " And there would have to exist a new class of"extra strong" interactions" to provide the bindingforce, extra strong in the sense that asymptotic free-dom sets in not at a few hundred MeV, as in QCD, butat a few hundred GeV. This "extra strong" force wouldbe felt by new families of fermions, and would givethese fermions masses of the order of several hundredGeV. We shall see.

Of the four (now three) types of interactions, onlygravity has resisted incorporation into a renormalizablequantum field theory. This may just mean that we arenot being clever enough in our mathematical treatment

Rev. Mod. Phys. , Vol. 62, Na. 3, July 1980

520 Steven Weinberg: Unified theory of weak and electromagnetic interactions

of general relativity. But there is another possibilitythat seems to me quite plausible. The constant ofgravity defines a unit of energy known as the I'lanckenergy, about 10" GeV. This is the energy at whichgravitation becomes effectively a strong interaction,so that at this energy one can no longer ignore its ultra-violet divergences. It may be that there is a wholeworld of new physics with unsuspected degrees of free-dom at these enormous energies, and that generalrelativity does not provide an adequate framework forunderstanding the physics of these superhigh, energy de-grees of freedom. When we explore gravitation or otherordinary phenomena, with particle masses and ener-gies no greater than a TeV or so, we may be lea, mingonly about an "effective" field theory; that is, one inwhich superheavy degrees of freedom do not explicitlyappear, but the coupling parameters implicitly repre-sent sums over these hidden degrees of freedom.

To see if this makes sense, let us suppose it is true,and ask what kinds of interactions we would expect onthis basis to find at ordinary energy. By "integratingout" the superhigh energy degrees of freedom in afundamental theory, we generally encounter a verycomplicated effective field theory —so complicated, infact, that it contains all interactions allowed by sym-metry principles. But where dimensional analysistells us that a coupling constant is a certain power ofsome mass, that ma, ss is likely to be a typical super-heavy mass, such as 10' GeV. The infinite variety ofnonrenormalizable interactions in the effective theoryhave coupling constants with the dimensionality of nega-tive powers of mass, so their effects are suppressedat ordinary energies by powers of energy divided bysuperheavy masses. Thus the only interactions that wecan detect at ordinary energies are those that are re-normalizable in the usual sense, plus any nonrenorrna-lizable interactions that produce effects which, althoughtiny, are somehow exotic enough to be seen.

One way that a very weak interaction could be de-tected is for it to be coherent and of long range, so thatit can add up and have macroscopic effects. It has beenshown" that the only particles which could produce suchforces are massless particles of spin 0, 1, or 2. Andfurthermore, Lorentz invariance alone is enough toshow that the long-range interactions of any particleof mass zero and spin 2 must be governed by generalrelativity. " Thus from this point of view we should notbe too surprised that gravitation is the only interactiondiscovered so far that does not seem to be described bya renormabzable field theory —it is almost the onlysuperweak intera. ction that could have been detected.And we should not be surprised to find that gravity iswell described by general relativity at macroscopicscales, even if we do not think that general relativityapplies at 10"GeV.

Nonrenorrnalizable effective interactions may also bedetected if they violate otherwise exact conservationlaws. The leading candidates for violation are baryonand lepton conservation. It is a remarkable consequenceof the SU(3) and SU(2) x U(l) gauge symmetries ofstrong, weak, and electromagnetic interactions, thatall renormalizable interactions among known particlesautomatically conserve baryon and lepton number.

Thus, the fact that ordinary rnatter seems-pretty sta-ble, that proton decay has not been seen, should notlead us to the conclusion that baryon and lepton cori-servation are fundamental conservation laws. To theaccuracy with which they have been verified, baryonand lepton conservation can be explained as dynamicalconsequences of other symmetries, in the sarge waythat strangeness conservation has been explained with-in QCD. But superheavy particles may exist, and theseparticles may have unusual SU(3) or SU(2) && U(1) trans-formation properties, and in this case, there is noreason why their interactions should conserve baryonor lepton number. I doubt that they would. Indeed, thefact that the universe seems to contain an excess ofbaryons over antibaryons should lead us to suspect thatbaryon nonconserving processes have actually oc-curred. If effects of a tiny none onservation of baryon orlepton number such as proton decay or neutrino massesare discovered experimentally, we will then be leftwith gauge symmetries as the only true internal sym-metries of nature, a conclusion that I would regard asmost satisfactory.

The idea of a new scale of superheavy masses hasarisen in another way. " If any sort of "grand unifica-tion" of strong and electroweak gauge couplings isto be possible, then one would expect all of the SU(3)and SU(2) && U(1) gauge coupling constants to be of com-parable magnitude. (In particular, if SU(3) and SU(2)x U(1) are subgroups of a larger simple group, thenthe ratios of the squa. red couplings are fixed as rationalnumbers of order unity. ") But this appears in con-tradiction with the obvious fact that the strong inter-actions are stronger than the weak and electromagneticinteractions. In 1974 Georgi, Quinn, and I suggestedthat the grand unification scale, at which the couplingsare all comparable, is at an enormous energy, and thatthe reason that the strong coupling is so much largerthan the electroweak couplings at ordinary energies isthat QCD is asymptotically free, so that its effectivecoupling constant rises slowly as the energy drops fromthe grand unification scale to ordinary values. Thechange of the strong couplings is very slow (likeI/din Z) so the grand unification scale must be enor-mous. %e found that for a fairly large class of theoriesthe grand unification scale comes out to be in the neigh-borhood of 10" GeV, an energy noi all that differentfrom the Planck energy of 10"GeV. The nucleon life-time is very difficult to estimate accurately, but wegave a representative value of 10"years, which maybe accessible experimentally in a few years. (Theseestimates have been improved in more detailed calcu-lations by several authors. )" We also calculated avalue for the mixing parameter sin'19 of about 0.2, notfar from the present experimental value' of 0,23~ 0.01.It will be an important task for future experiments onneutral currents to improve the precision with whichsin 8 is known, to see if it really agrees with this pre-diction.

In a grand unified theory, in order for elementaryscalar particles to be available to produce the spon-taneous breakdown of the electroweak gauge symmetryat a few hundred GeV, it is necessary forsuch particles io escape getting superlarge masses

Rev. Mod. Phys. , Vol. 52, No. 3, Juiy 1980

Steven N/einberg: Unified theory of weak and electromagnetic interactions

from the spontaneous breakdown of the grand unifiedgauge group. There is nothing impossible in this,but I have not been able to think of any reason why itshould happen. (The problem may be related to the oldmystery of why quantum corrections do not produce anenormous cosmological constant; in both cases, one isconcerned with an anomalously small "super-renor-malizable" term in the effective Lagrangian which hasto be adjusted to be zero. In the case of the cosmo-logical constant, the adjustment must be precise tosome fifty decimal places. ) With elementary scalarsof small or zero mass, enormous ratios of symmetrybreaking scales can arise quite naturally. " On theother hand, if there are no elementary scalars whichescape getting superlarge masses from the breakdownof the grand unified gauge group, then as I have al-ready mentioned, there must be extra strong forcesto bind the composite Goldstone and Higgs bosonsthat are associated with the spontaneous breakdown ofSU(2) x U(1). Such forces can occur rather naturally ingrand unified theories. To take one example, supposethat the grand gauge group breaks, not into SU(3)x SU(2)x U(1), but into SU(4)x SU(3)x SU(2)x U(1). SinceSU(4) is a bigger group than SU(3), its coupling constantrises with decreasing energy more rapidly than theQCD coupling, so the SU(4) force becomes strong at amuch higher energy than the few hundred MeV at whichthe QCD force becomes strong. Ordinary quarks andleptons would be neutral under SU(4), so they wouldnot feel this force, but other fermions might carrySU(4) quantum numbers, and so get rather large masses.One can even imagine a sequence of increasingly largesubgroups of the grand gauge group, which would fillin the vast energy range up to 10" or 10"GeV withparticle masses that are produced by these successivelystrong interacti ons.

If there are elementary scalars whose vacuum ex-pectation values are responsible for the masses ofordinary quarks and leptons, then these masses can beaffected in order o. by radiative corrections involvingthe superheavy vector bosons of the grand gauge group,and it will probably be impossible to explain the valueof quantities like m, /m„without a complete grand uni-fied theory. On the other hand, if there are no suchelementary scalars, then almost all the details of thegrand unified theory are forgotten by the effective fieldtheory that describes physics at ordinary energies, andit ought to be possible to calculate quark and leptonmasses purely in terms of processes at accessible en-ergies. Unfortunately, no one so far has been able tosee how in this way anything resembling the observedpattern of masses could arise. '

Putting aside all these uncertainties, suppose thatthere is a truly fundamental theory, characterized byan energy scale of order 10' to 10~ GeV, at whichstrong, electroweak, and gravitational interactions areall united. It might be a conventional renormalizablequantum field theory, but at the moment, if we includegravity, we do not see how this is possible. (1 leave thetopic of supersymmetry and supergravity for ProfessorSalam's talk. ) But if it is not renormalizable, whatthen determines the infinite set of coupling constants'that are needed to absorb all the ultraviolet divergences

of the theory~I think the answer must lie in the fact that the quan-

tum field theory, which was born just fifty years agofrom the marriage of quantum mechanics with rela-tivity, is a beautiful but not a very robust child. AsLandau and Kallen recognized long Bgo, quantum fieldtheory at superhigh energies is susceptible to all sortsof diseases —tachyons, ghosts, etc.—and it needsspecial medicine to survive. One way that a quantumfield theory can avoid these diseases is to be re-normalizable and asymptotically free, but there areother possibilities. For instance, even an infinite setof coupling constants may approach a nonzero fixedpoint as the energy at which they are measured goesto infinity. However, to require this behavior generallyimposes so many constraints on the couplings that thereare only a finite number of free parameters left"—justas for theories that are renormalizable in the usualsense. Thus, one way or another, I think that quantumfield theory is going to go on being very stubborn, re-fusing to allow us to describe all but a small number ofpossible worlds, among which, we hope, is ours.

I suppose that I tend to be optimistic about the futureof physics. And nothing makes me more optimistic thanthe discovery of broken symmetries. In the seventhbook of The AePublic, Plato describes prisoners whoare chained in a cave and can see only shadows thatthings outside cast on the cave wall. When releasedfrom the cave at first their eyes hurt, and for a whilethey think that the shadows they saw in the cave aremore real than the objects they now see. But eventuallytheir vision clears, and they can understand how beauti-ful the real world is. We are in such a cave, im-prisoned by the limitations on the sorts of experimentswe can. do. In particular, we can study matter only atrelatively low temperatures, where symmetries arelikely to be spontaneously broken, so that nature doesnot appear very simple or unified. We have not beenable to get out of this cave, but by looking long and hardat the shadows on the cave mall, we can at least makeout the shapes of symmetries, which though broken,are exact principles governing all phenomena, expres-sions of the beauty of the world outside.

It has only been possible here to give references to avery small part of the literature on the subjects dis-cussed in this talk. Additional references can be foundin the following reviews: E. 3. Abers and B. W. Lee,"Gauge Theories" (Phys. Rep. C 9, No. 1, 1973);W. Marciano and H. Pagels, "Quantum Chromodynamics"(Phys. Rep. C 36, No. 3, 1978); J. C. Taylor, GaugeTheories of W'eak Interactions (Cambridge University,1976).

REFERENCESM. A. Tuve, N. Heydenberg, and L. B. H~stad, Phys. Bev.50, 806 (1936); G. Breit, E. V. Condon, and B.D. Present,Phys. Bev. 50, 825 (1936); G. Breit and E. Feenberg, Phys.Bev. 50, 850 (1936).

M. Geol-Mann, Phys. Bev. 92, 833 (1953); T. Nakano andK. Nishijima, Prog. Theor. Phys. 3.0, 581 (1955).

T. D. I ee and C. N. Yang, Phys. Bev. 104, 254 (1956);C. 8. Wu et al. , Phys. Bev. M5, 1413 (1957); R. Garwin,L. Lederman, and M. heinrich, Phys. Bev. 3.05, 1415 (1957);J. I. Friedman and V. L. Telegdi, Phys. Bev. 105, 1681 (1957).

Rev. Mod. Phys. , Vol. 52, No. 3, July 1980

Steven Weinberg: Unified theory of weak and electromagnetic interactions

M. Gell-Mann, Cal. Tech. Synchotron Laboratory ReportCTSL-20 (1961), unpublished; Y. Ne'eman, Nucl. Phys. 26,222 (1961).

V. Fock, Z. Phys. 39, 226 (1927); H. Weyl, Z. Phys. 56, 330(1929). The name "gauge invariance" is based on an analogywith the earlier speculations of H. Weyl, in Raum, &eit,Materie, 3rd edition (Springer, 1920). Also see F. London, Z.Phys. 42, 375 (1927). (This history has been reviewed byC. N. Yang in a talk at City College, 1977.)

C. N. Yang and B. L. Mills, Phys. Bev. 96, 191 (1954).'J. Goldstone, Nuovo Cimento 19, 154 (1961).J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127,965 (1962).

9P. W. Higgs, Phys. Lett. 12, 132 (1964}; 13, 508 (1964);Phys. Rev. 145, 1156 (1966); T. W. B. Kibble, Phys. Rev.155, 1554 (1967); G. S. Guralnik, C. B.Hagen, andT. W. B. Kibble, Phys. Bev. Lett. 13, 585 (1964); F. Englertand B. Brout, Phys. Rev. Lett. 13, 321 (1964). Also seeP. W. Anderson, Phys. Bev. 130, 439 (1963).S. L. Adler, Phys. Rev. Lett. 14, 1051 (1965); Phys. Rev.140, B736 (1965); W. I. Weisberger, Phys. Rev. Lett. 14,1047 (1965); Phys. Rev. 143, 1302 (1966).

«M. Gell Ma~, Physics 1, 63 Q.964).Y. Nambu and G. Jona-Lasinio, Phys. Bev. 122, 345 (1961);124, 246 (1961); Y. Nambu and D. Lurie, Phys. Bev. 125,1429 {1962);Y. Nambu and E. Shrauner, Phys. Rev. 128, 862(1962). Also see M. Gell-Mann and M. Levy, Nuovo Cimento16, 705 (1960).

~3M. L. Goldberger, H. Miyazawa, and R. Oehme, Phys. Bev.99, 986 (1955).

~4M.. L. Goldberger and S. B. Treiman, Phys. Rev. 111, 354(1958).S. Weinberg, Phys. Rev. Lett. 16, 879 (1966); 17, 336 (1966);17, 616 {1966);18, 188 (1967); Phys. Rev. 166, 1568 (1967).

~6J. B.Oppenheimer, Phys. Bev. 35, 461 (1930); I. Wailer, Z.Phys. 59, 168 (1930); ibid. 62, 673 (1930).B. P. Feynman, Rev. Mod. Phys. $0, 367 (1948); Phys. Rev.74, 939, 1430 (1948); 76, 749, 769 (1949); 80, 440 (1950);J. Schwinger, Phys. Bev. 73, 146 (1948); 74, 1439 (1948);75, 651 (1949); 76, 790 (1949); 82, 664, 914 (1951};91, 713(1953); Proc. Nat. Acad. Sci. USA 37, 452 (1951); S. Tomo-naga, Prog. Theor. Phys. (Japan) 1, 27 (1946); Z. Koba,T. Tati, and S. Tomonaga, ibid. 2, 101 (1947); S. Kanazawaand S. Tomonaga, ibid. 3, 276 (1948); Z. Koba and S. Tomo-naga, ibid. 3, 290 (1948).

~ There had been earlier suggestions that infinities could beeliminated from quantum field theories in this way, byV. F. Weisskopf, K. Dan. Vidensk. Selsk. Mat. -Fys. Medd. 15(6} 1936, especially p. 34 and pp. 5—6; H. Kramers (unpub-lished) .

~ F. J. Dyson, Phys. Rev. 75, 486, 1736 (1949).OS. Weinberg, Phys. Rev. 106, 1301 (1957).S. Weinberg, Phys. Rev. 118, 838 (1960).A. Salam, Phys. Rev. 82, 217 (1951); 84, 426 (1951).S. Weinberg, Phys. Rev. Lett. 18, 507 (1967).For the nonrenormalizability of theories with intrinsicallybroken gauge symmetries, see A. Komar and A. Salam,Nucl. Phys. 21, 624 (1960); H. Umezawa and S. Kamefuchi,Nucl. Phys. 23, 399 (1961); S. Kamefuchi, L. O'Baifeartaigh,and A. Salam, Nucl. Phys. 28, 529 (1961); A. Salam, Phys.Rev. 127, 331 (1962); M. Veltman, Nucl. Phys. B 7, 637(1968); B 21, 288 (1970); D. Boulware, Ann. Phys. (N.Y.) 56,140 {1970).This work was briefly reported i.n reference 23, footnote 7.

6S. Weinberg, Phys. Bev. Lett. 19, 1264 (1967).A. Salam, in &Eernentary ParticEe Physics (Nobel SymposiumNo. 8), edited by N. Svartholm {Almquist and Wiksell, Stock-holm, 1968), p. 367.B. deWitt, Phys. Bev. Lett. 12, 742 (1964); Phys. Pev. 162,1195 (1967); L. D. Faddeev and V. N. Popov, Phys. Lett.B25, 29 (1967). Also see B.P. Feynman, Acta Phys. Pol.

24, 697 (1963); S. Mandelstam, Phys. Rev. 175, 1580, 1604(1968).See L. Stuller, MIT Ph. D. Thesis (1971), unpublished.My work with the unitarity gauge was reported in S. Wein-berg, Phys. Bev. Lett. 27, 1688 (1971), and described inmore detail in S. Weinberg, Phys. Bev. D 7, 1068 (1973).G. 't Hooft, Nucl. Phys. B 35, 167 (1971).B. W„Lee and J. Zinn-Justin, Phys. Rev. D 5, 3121, 3137,3155 (1972); G. 't Hooft and M. Veltman, Nucl. Phys. B 44,189 (1972); B 50, 318 (1972). There still remained the prob-lem of possible Adler-Bell-Jackiw anomalies, but these nicelycancelled; see D. J. Gross and R. Jackiw, Phys. Rev. D 6,477 (1972), and C. Bouchiat, J. Iliopoulos, and Ph. Meyer,Phys- Lett- 38B, 519 {1972)-C. Beechi, A. Bouet, and R. Stora, Commun. Math. Phys. 42,127 (1975).B. W. Lee, Phys. Bev. D 5, 823 (1972).G. Gamow and E. Teller, Phys. Bev. 51, 288 (1937); N. Kem-mer, Phys. Rev. 52, 906 (1937); G. Wentzel, Helv. Phys.Acta 10, 108 (1937); S. Bludman, Nuovo Cimento 9, 433(1958); J. Leite-Lopes, Nucl. Phys. 8, 234 (1958).S. L. Glashow, Nucl. Phys. 22, 519 (1961); A. Salam andJ. C. Ward, Phys. Lett. 13, 168 (1964).

3 S. Weinberg, Phys. Rev. 5, 1412 (1972).D. C. Cundy et al. , Phys. Lett. 31B, 478 (1970).The first published discovery of neutral currents was at theGargamelle Bubble Chamber at CERN: F. J. Hasert et al. ,Phys. Lett. B 46, 121, 138 {1973). Also see P. Musset, J.Phys. (Paris) 11/12 T34 (1973). Muonless events were seenat about the same time by the HPWF group at Fermilab, butwhen publication of their paper was delayed, they took theopportunity to rebuild their detector, and then did not atfirst find the same neutral current signal. The HPWF grouppublished evidence for neutral currents in A. Benvenuti et al. ,Phys. Bev. Lett. 32, 800 (1974).For a survey of the data see C. Baltay, Proceedings of the19th International Conference on High Energy Physics, To-kyo, 1978. For theoretical analyses, see L. F. Abbott andR. M. Barnett, Phys. Bev. D 19, 3230 (1979); P. Langacker,J.K. Kim, M. Levine, H. H. Williams, and D. P. Sidhu, "Neutri-no '79"Conference; and earlier references cited therein.

~C. Y. Prescott et aE. , Phys. Lett. B 77, 347 (1978).4 S. L. Glashow and H. L. Georgi, Phys. Bev. Lett. 28, 1494

{1972). Also see J. Schwinger, Ann. Phys. (N.Y.) 2, 407(1957).

43S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Bev. D 2,1285 (1970). This paper was cited in reference 37 as provid-ing a possible solution to the problem of strangeness chang-ing neutral currents. However, at that time I was skepticalabout the quark model, so in the calculations of reference 37baryons were incorporated in the theory by taking the protonsand neutrons to form an SU(2) doublet, with strange particlessimply ignored.

44H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973); D. J. Grossand F. Wilczek, Phys. Bev. Lett. 30, 1343 {1973).

4~Energy dependent effective coupling constants were intro-duced by M. Gell-Mann and F. E. Low, Phys. Bev. 95, 1300{1954}.

46E. D. Bloom et aE. , Phys. Rev. Lett. 23, 930 (1969);M. Breidenbach et al. , Phys. Bev. Lett. 23, 935 (1969).

4 S. Weinberg, Phys. Bev. D 8, 605 (1973).4 D. J. Gross and F. Wilczek, Phys. Rev. D 8, 3633 (1973);

S. Weinberg, Phys. Rev. Lett. 31, 494 (1973). A similar ideahad been proposed before the discovery of asymptotic free-dom by H. Fritzsch, M. Gell-Mann, and H. Leutwyler, Phys.X tt. B47, 365 0.973).

4~0. W. Greenberg, Phys. Bev. Lett. 13, 598 (1964); M. Y. Hanand Y. Nambu, Phys. Rev. j.39, B1006 (1965); W. A. Bar-deen, H. Fritzsch, and M. Gell-Mann, in Scale and Conforrn-aE Symmetry in Hadron Physics, edited by R. Gatto (Wiley,1973), p. 139; etc.

Rev. Mod. Phys. , Vol. 52, No. 3, July 1980

Steven M/einberg: Unified theory of weak and eteetromagnetic interactions 523

~OG. 't Hooft, Phys. Rev. Lett. 37, 8 (1976).5~Such "dynamical" mechanisms for spontaneous symmetry

breaking were first discussed by Y. Nambu and G. Jona-La-sinio, Phys. Bev. 122, 345 (1961); J.Schwinger, Phys. Bev.125, 397 (1962); 128, 2425 (1962); and in the context of,modern gauge theories by R. Jackiw and K. Johnson, Phys.Rev. D 8, 2386 {1973);J.M. Cornwall and R. E. Norton,Phys. Rev. D 8, 3338 (1973). The implications of dynamicalsymmetry breaking have been considered by S. Weinberg,Phys. Bev. D 13, 974 (1976); D 19, 1277 (1979); L. Susskind,Phys. Rev. D 20, 2619 (1979}.S. Weinberg, reference 51. The possibility of pseudo-Gold-stone bosons was originally noted in a different context byS. Weinberg, Phys. Bev. Lett. 29, 1698 (1972).S. Weinberg, reference 51. Models involving such interac-tions have also been discussed by L. Susskind, reference 51.S. steinberg, Phys. Bev. 135, B1049 (1964).S. Weinberg, Phys. Lett. 9, 357 (1964); Phys. Rev. B 138,988 (1965); Iectlxes in ~axtieles and Field Theory, editedby S. Deser and K. Ford (Prentice-Hall, 1965), p. 988; andreference 54. The program of deriving general relativityfrom quantum mechanics and special relativity was com-pleted by D. Boulware and S. Deser, Ann. Phys. {N.Y.) 89,173 (1975).- I understand that similar ideas were developedby R. Feynman in unpublished lectures at Cal. Tech.

56H. Georgi, H. Quinn, and S. Weinberg, Phys. Bev. Lett. 33,451 (1974).An example of a simple gauge group for weak and electro-magnetic interactions (for which sin = 4) was given byS. Weinberg, Phys. Rev. D 5, 1962 (1972). There are anumber of specific models of weak, electromagnetic, andstrong interactions based on simple gauge groups, includingthose of J. C. Pati and A. Salam, Phys. Rev. D 10, 275(1974); H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32,

438 (1974); H. Georgi, in I articles and Fields (American In-stitute of Physics, 1975); H. Fritzsch and P. Minkowski,~.Phys. (N.Y.) 93, 193 (1975); H. Georgi and D. V. Nan-opoulos, Phys. Lett. B 82, 392 (1979); F. Gursey, P. Ra-mond, and P. Sikivie, Phys. Lett. B 60, 177 (1975); F. Gur-sey and P. Sikivie, Phys. Rev. Lett. 36, 775 {1976);P. Ba-mond, Nucl. Phys. B 110, 214 {1976);etc. ; all these vxolatebaryon and lepton conservation

58A. Buras, J.Ellis, M. K. Gaillard, and D. U. Nanopoulos,Nucl. Phys. B 135, 66 (1978); D. Boss, Nucl. Phys. B 140, 1{1978);W. J. Marciano, Phys. Bev. D 20, 274 (1979);T. Goldman and D. Ross, CALT 68-704, to be published;C. Jar lskog and F. J.Yndurain, CERN pr epr int, to be pub-lished; M. Machacek, Harvard preprint HUTP-79/AO21, tobe published in Nucl. Phys. ; S. Weinberg, paper in prepara-tion. The phenemonology of nucleon decay has been dis-cussed in general terms by S. Weinberg, Phys. Rev. Lett. 43,1566 (1979); F.. Wilczek and A. Zee, Phys. Bev. Lett. 43,1571 (1979}.

59E. Gildener and S. Weinberg, Phys. Rev. D 13, 3333 (1976);S. Weinberg, Phys. Lett. B 82, 387 (1979). In general in thiscase there should exist at least one scalar particle with phys-ical mass of order 10 GeV. The spontaneous symmetrybreaking in models with zero bare scalar mass was first con-.sidered by S. Coleman and E. Weinberg, Phys. Rev. D 7,1888 (1973).This problem has been studied recently by S. Dimopoulos andL. Susskind, Nucl. Phys. B 155, 237 (1979); E. Eichten andK. Lane, Phys. Lett. , to be published; S. Weinberg, unpub-lished.

6~S. Weinberg, in General Relativity —&n Einstein Centena~SN~ey, edited by S. W. Hawking and W. Israel (CambridgeUniversity, 1979), Chap. 16.

Rev. Mod. Phys. , Vol. 52, No. 3, July 1980