University of Texas
Steven Weinberg
May 3, 1933–



Steven Weinberg


I was born in 1933, in New York City to Frederick and Eva Weinberg. My early inclination toward science received encouragement from my father and, by the time I was 15 or 16, my interests had focused on theoretical physics.

I received my undergraduate degree from Cornell in 1954, and then went for a year of graduate study to the Institute for Theoretical Physics in Copenhagen (now the Niels Bohr Institute). There, with the help of David Frisch and Gunnar Källén, I began to do research in physics. I then returned to the U.S. to complete my graduate studies at Princeton. My Ph.D thesis, with Sam Treiman as adviser, was on the application of renormalization theory to the effects of strong interactions in weak interaction processes.

After receiving my PhD in 1957, I worked at Columbia and then, from 1959 to 1966, at Berkeley. My research during this period was on a wide variety of topics—high energy behavior of Feynman graphs, second-class weak interaction currents, broken symmetries, scattering theory, muon physics, etc.—topics chosen in many cases because I was trying to teach myself some area of physics. My active interest in astrophysics dates from 1961–1962; I wrote some papers on the cosmic population of neutrinos and then began to write a book, Gravitation and Cosmology, which was eventually completed in 1971. Late in 1965, I began my work on current algebra and the application to the strong interactions of the idea of spontaneous symmetry breaking.

From 1966 to 1969, on leave from Berkeley, I was Loeb Lecturer at Harvard and then visiting professor at M.I.T. In 1969, I accepted a professorship in the physics department at M.I.T., then chaired by Viki Weisskopf. It was while I was a visitor to M.I.T. in 1967 that my work on broken symmetries, current algebra, and renormalization theory turned in the direction of the unification of weak and electromagnetic interactions. In 1973, when Julian Schwinger left Harvard, I was offered and accepted his chair there as Higgins Professor of Physics together with an appointment as Senior Scientist at the Smithsonian Astrophysical Observatory.

My work during the 1970s has been mainly concerned with the implications of the unified theory of weak and electromagnetic interactions, with the development of the related theory of strong interactions known as quantum chromodynamics, and with steps toward the unification of all interactions.

In 1982, I moved to the physics and astronomy departments of the University of Texas at Austin, as Josey Regental Professor of Science. I met my wife, Louise, when we were undergraduates at Cornell, and we were married in 1954. She is now a professor of law. Our daughter, Elizabeth was born in Berkeley in 1963.

Weinberg Photo Album at end of entry.

Awards and Honors

Honorary Doctor of Science degrees, University of Chicago, Knox College, City University of New York, University of Rochester, Yale University
American Academy of Arts and Sciences, elected 1968
National Academy of Sciences, elected 1972
J. R. Oppenheimer Prize, 1973
Richtmeyer Lecturer of Am. Ass'n. of Physics Teachers, 1974
Scott Lecturer, Cavendish Laboratory, 1975
Dannie Heineman Prize for Mathematical Physics, 1977
Silliman Lecturer, Yale University, 1977
Am. Inst. of Physics-U.S. Steel Foundation Science Writing Award, 1977, for authorship of The First Three Minutes (1977)
Lauritsen Lecturer, Cal. Tech., 1979
Bethe Lecturer, Cornell Univ., 1979
Elliott Cresson Medal (Franklin Institute), 1979
Nobel Prize in Physics, 1979

Awards and Honors since 1979
Honorary Doctoral degrees, Clark University, City University of New York, Dartmouth College, Weizmann Institute, Clark University, Washington College, Columbia University
Elected to American Philosophical Society, Royal Society of London (Foreign Honorary Member), Philosophical Society of Texas
Henry Lecturer, Princeton University, 1981
Cherwell-Simon Lecturer, University of Oxford, 1983
Bampton Lecturer, Columbia University, 1983
Einstein Lecturer, Israel Academy of Arts and Sciences, 1984
McDermott Lecturer, University of Dallas, 1985
Hilldale Lecturer, University of Wisconsin, 1985
Clark Lecturer, University of Texas at Dallas, 1986
Brickweede Lecturer, Johns Hopkins University, 1986
Dirac Lecturer, University of Cambridge, 1986
Klein Lecturer, University of Stockholm, 1989
James Madison Medal of Princeton University, 1991
National Medal of Science, 1991

From Nobel Lectures, Physics 1971-1980, Editor Stig Lundqvist, World Scientific Publishing Co., Singapore, 1992
This autobiography/biography was written at the time of the award and first published in the book series Les Prix Nobel. It was later edited and republished in Nobel Lectures.


Nobel Lecture, December 8, 1979 by STEVEN WEINBERG Lyman Laboratory of Physics Harvard University and Harvard-Smithsonian Center for Astrophysics Cambridge, Mass., USA.

Our job in physics is to see things simply, to understand a great many complicated phenomena in a unified way, 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 neutrino reactions. But even in the dark times between experimental breakthroughs, there always continues a steady evolution of theoretical ideas, leading almost imperceptibly to changes in previous beliefs. In this talk, I want to discuss the development of two lines of thought in theoretical physics. One of them is the slow growth in our understanding of symmetry and, in particular, broken or hidden symmetry. The other is the old struggle to come to terms with the infinities in quantum field theories. To a remarkable degree, our present detailed theories of elementary particle interactions can be understood deductively, as consequences of symmetry principles and of a principle of renormalizability which is invoked to deal with the infinities. I will also briefly describe how the convergence of these lines of thought led to my own work on the unification of weak and electromagnetic interactions. For the most part, my talk will center on my own gradual education in these matters because that is one subject on which I can speak with some confidence. With rather less confidence, I will also try to look ahead and suggest what role these lines of thought may play in the physics of the future.

Symmetry principles made their appearance in twentieth century physics in 1905 with Einstein’s identification of the invariance group of space and time. With this as a precedent, symmetries took on a character in physicists’ minds as a priori principles of universal validity, expressions of the simplicity of nature at its deepest level. So it was painfully difficult in the 1930s to realize that there are internal symmetries, such as isospin conservation, [1] having nothing to do with space and time, symmetries which are far from self-evident, and that only govern what are now called the strong interactions. The 1950s saw the discovery of another internal symmetry - the conservation of strangeness [2] —which is not obeyed by the weak interactions, and even one of the supposedly sacred symmetries of space-time—parity—was also found to be violated by weak interactions. [3] Instead of moving toward unity, physicists were learning that different interactions are apparently governed by quite different symmetries. Matters became yet more confusing with the recognition in the early 1960s of a symmetry group—the “eightfold way” —which is not even an exact symmetry of the strong interactions. [4]

These are all “global” symmetries, for which the symmetry transformations do not depend on position in space and time. It had been recognized [5] in the 1920s that quantum electrodynamics has another symmetry of a far more powerful kind, a “local” symmetry under transformations in which the electron field suffers a phase change that can vary freely from point to point in space-time, and the electromagnetic vector potential undergoes a corresponding gauge transformation. Today, this would be called a U(1) gauge symmetry, because a simple phase change can be thought of as multiplication by a 1 x 1 unitary matrix. The extension to more complicated groups was made by Yang and Mills [6] in 1954 in a seminal paper in which they showed how to construct an SU(2) gauge theory of strong interactions. (The name “SU(2)” means that the group of symmetry transformations consists of 2 x 2 unitary matrices that are “special,” in that they have determinant unity). But here, again, it seemed that the symmetry, if real at all, would have to be approximate because at least on a naive level gauge invariance requires that vector bosons like the photon would have to be massless, and it seemed obvious that the strong interactions are not mediated by massless particles. The old question remained: if symmetry principles are an expression of the simplicity of nature at its deepest level, then how can there be such a thing as an approximate symmetry? Is nature only approximately simple?

Some time in 1960 or early 1961, I learned of an idea which had originated earlier in solid state physics and had been brought into particle physics by those like Heisenberg, Nambu, and Goldstone, who had worked in both areas. It was the idea of “broken symmetry,” that the Hamiltonian and commutation relations of a quantum theory could possess an exact symmetry, and that the physical states might nevertheless not provide neat representations of the symmetry. In particular, a symmetry of the Hamiltonian might turn out to be not a symmetry of the vacuum.

As theorists sometimes do, I fell in love with this idea. But, as often happens with love affairs, at first I was rather confused about its implications. I thought (as turned out, wrongly) that the approximate symmetries— parity, isospin, strangeness, the eight-fold way— might really be exact a priori symmetry principles, and that the observed violations of these symmetries might somehow be brought about by spontaneous symmetry breaking. It was, therefore, rather disturbing for me to hear of a result of Goldstone, [7] that, in at least one simple case, the spontaneous breakdown of a continuous symmetry like isospin would necessarily entail the existence of a massless spin zero particle - what would today be called a “Goldstone boson.” It seemed obvious that there could not exist any new type of massless particle of this sort which would not already have been discovered.

I had long discussions of this problems with Goldstone at Madison in the summer of 1961, and then, with Salam while I was his guest at Imperial College in 196l–1962. The three of us soon were able to show that Goldstone bosons must, in fact, occur whenever a symmetry like isospin or strangeness is spontaneously broken and that their masses then remain zero to all orders of perturbation theory. I remember being so discouraged by these zero masses that when we wrote our joint paper on the subject, [8] I added an epigraph to the paper to underscore the futility of supposing that anything could be explained in terms of a non-invariant vacuum state: it was Lear’s retort to Cordelia, “Nothing will come of nothing: speak again.” Of course, The Physical Review protected the purity of the physics literature, and removed the quote. Considering the future of the non-invariant vacuum in theoretical physics, it was just as well.

There was actually an exception to this proof pointed out soon afterwards by Higgs, Kibble, and others. [9] They showed that if the broken symmetry is a local, gauge symmetry like electromagnetic gauge invariance then, although the Goldstone bosons exist formally and are in some sense real, they can be eliminated by a gauge transformation, so that they do not appear as physical particles. The missing Goldstone bosons appear instead as helicity zero states of the vector particles, which thereby acquire a mass.

I think that, at the time, physicists who heard about this exception generally regarded it as a technicality. This may have been because of a new development in theoretical physics which suddenly seemed to change the role of Goldstone bosons from that of unwanted intruders to that of welcome friends.

In 1964, Adler and Weisberger [10] independently derived sum rules which gave the ratio gA/gV of axial-vector to vector coupling constants in beta decay in terms of pion-nucleon cross sections. One way of looking at their calculation, (perhaps the most common way at the time), was as an analogue to the old dipole sum rule in atomic physics: a complete set of hadronic states is inserted in the commutation relations of the axial vector currents. This is the approach memorialized in the name of “current algebra.” [11] But there was another way of looking at the Adler-Weisberger sum rule. One could suppose that the strong interactions have an approximate symmetry, based on the group SU(2) x SU(2), and that this symmetry is spontaneously broken, giving rise among other things to the nucleon masses. The pion is then identified as (approximately) a Goldstone boson, with small non-zero mass, an idea that goes back to Nambu. [12] Although the SU(2) X SU(2) symmetry is spontaneously broken, it still has a great deal of predictive power, but its predictions take the form of approximate formulas, which give the matrix elements for low energy pionic reactions. In this approach, the Adler-Weisberger sum rule is obtained by using the predicted pion nucleon scattering lengths in conjunction with a well-known sum rule [13] which, years earlier, had been derived from the dispersion relations for pion-nucleon scattering.

In these calculations, one is really using not only the fact that the strong interactions have a spontaneously broken approximate SU(2) X SU(2) symmetry, but also that the currents of this symmetry group are, up to an overall constant, to be identified with the vector and axial vector currents
of beta decay. (With this assumption gA/gV gets into the picture through the Goldberger-Treiman relation, [14] which gives gA/gV in terms of the pion decay constant and the pion nucleon coupling.) Here, in this relation between the currents of the symmetries of the strong interactions and the physical currents of beta decay, there was a tantalizing hint of a deep connection between the weak interactions and the strong interactions. But this connection was not really understood for almost a decade.

I spent the years 1965-67 happily developing the implications of spontaneous symmetry breaking for the strong interactions. [15] It was this work that led to my 1967 paper on weak and electromagnetic unification. But before I come to that I have to go back in history and pick up one other line of thought having to do with the problem of infinities in quantum field theory.

I believe that it was Oppenheimer and Waller in 1930 [16] who independently first noted that quantum field theory when, pushed beyond the lowest approximation, yields ultraviolet divergent results for radiative self energies. Professor Waller told me last night that when he described this result to Pauli, Pauli did not believe it. It must have seemed that these infinities would be a disaster for the quantum field theory that had just been developed by Heisenberg and Pauli in 1929–1930. And indeed, these infinities did lead to a sense of discouragement about quantum field theory, and many attempts were made in the 1930s and early 1940s to find alternatives. The problem was solved (at least for quantum electrodynamics) after the war, by Feynman, Schwinger, and Tomonaga [17] and Dyson [19]. It was found that all infinities disappear if one identifies the observed finite values of the electron mass and charge, not with the parameters m and e appearing in the Lagrangian, but with the electron mass and charge that are calculated from m and e, when one takes into account the fact that the electron and photon are always surrounded with clouds of virtual photons and electron-positron pairs [18]. Suddenly all sorts of calculations became possible and gave results in spectacular agreement with experiment.

But even after this success, opinions differed as to the significance of the ultraviolet divergences in quantum field theory. Many thought—and some still do think—that what had been done was just to sweep the real problems under the rug. And it soon became clear that there was only a limited class of so-called “renormalizable” theories in which the infinities could be eliminated by absorbing them into a redefinition, or a “renormalization,” of a finite number of physical parameters. (Roughly speaking, in renormalizable theories, no coupling constants can have the dimensions of negative powers of mass. But every time we add a field or a space-time derivative to an interaction, we reduce the dimensionality of the associated coupling constant. So only a few simple types of interaction can be renormalizable.) In particular, the existing Fermi theory of weak interactions clearly was not renormalizable. (The Fermi coupling constant has the dimensions of [mass]-2.) The sense of discouragement about quantum field theory persisted into the 1950s and 1960s.

I learned about renormalization theory as a graduate student, mostly by reading Dyson’s papers. [19] From the beginning, it seemed to me to be a wonderful thing that very few quantum field theories are renormalizable. Limitations of this sort are, after all, what we most want, not mathematical methods which can make sense of an infinite variety of physically irrelevant theories, but methods which carry constraints, because these constraints may point the way toward the one true theory. In particular, I was impressed by the fact that quantum electrodynamics could, in a sense, be derived from symmetry principles and the constraints of renormalizability; the only Lorentz invariant and gauge invariant renormalizable Lagrangian for photons and electrons is precisely the original Dirac Lagrangian of QED. Of course, that is not the way Dirac came to his theory. He had the benefit of the information gleaned in centuries of experimentation on electromagnetism and, in order to fix the final form of his theory, he relied on ideas of simplicity (specifically, on what is sometimes called minimal electromagnetic coupling). But we have to look ahead to try to make theories of phenomena which have not been so well studied experimentally, and we may not be able to trust purely formal ideas of simplicity. I thought that renormalizability might be the key criterion, which also, in a more general context, would impose a precise kind of simplicity on our theories and help us to pick out the one true physical theory out of the infinite variety of conceivable quantum field theories. As I will explain later, I would say this a bit differently today, but I am more convinced than ever that the use of renormalizability as a constraint on our theories of the observed interactions is a good strategy. Filled with enthusiasm for renormalization theory, I wrote my PhD thesis under Sam Treiman in 1957 on the use of a limited version of renormalizability to set constraints on the weak interactions, [20] and a little later, I worked out a rather tough little theorem [21] which completed the proof by Dyson [19] and Salam [22] that ultraviolet divergences really do cancel out to all orders in nominally renormalizable theories. But none of this seemed to help with the important problem of how to make a renormalizable theory of weak interactions.

Now, back to 1967. I had been considering the implications of the broken SU(2) x SU(2) symmetry of the strong interactions, and I thought of trying out the idea that perhaps the SU(2) x SU(2) symmetry was a “local,” not merely a “global,” symmetry. That is, the strong interactions might be described by something like a Yang-Mills theory, but, in addition to the vector Ú mesons of the Yang-Mills theory, there would also be axial vector Al mesons. To give the Ú meson a mass, it was necessary to insert a common Ú and Al mass term in the Lagrangian, and the spontaneous breakdown of the SU(2) x SU(2) symmetry would then split the Ú and Al by something like the Higgs mechanism, but since the theory would not be gauge invariant the pions would remain as physical Goldstone bosons. This theory gave an intriguing result, that the A1/Ú mass ratio should be √2 and in trying to understand this result without relying on perturbation theory, I discovered certain sum rules, the “spectral function sum rules,” [23] which turned out to have variety of other uses. But the SU(2) x SU(2) theory was not gauge invariant, and hence it could not be renormalizable, [24] so I was not too enthusiastic about it. [25] Of course, if I did not insert the A1-Ú mass term in the Lagrangian, then the theory would be gauge invariant and renormalizable, and the Al would be massive. But then there would be no pions and the Ú mesons would be massless, in obvious contradiction (to say the least) with observation.

At some point in the fall of 1967, I think, while driving to my office at M.I.T., it occurred to me that I had been applying the right ideas to the wrong problem. It is not the Ú mesons that are massless: it is the photon. And its partner is not the Al, but the massive intermediate boson, which since the time of Yukawa had been suspected to be the mediator of the weak interactions. The weak and electromagnetic interactions could then be described [26] in a unified way in terms of an exact but spontaneously broken gauge symmetry. [Of course, not necessarily SU(2) X SU(2)]. And this theory would be renormalizable like quantum electrodynamics because it is gauge invariant like quantum electrodynamics.

It was not difficult to develop a concrete model which embodied these ideas. I had little confidence then in my understanding of strong interactions, so I decided to concentrate on leptons. There are two left-handed electron-type leptons, the Óe and eL and one right-handed, electron-type lepton, the eR so I started with the group U(2) X U(1): all unitary 2 x 2 matrices acting on the left-handed e-type leptons, together with all unitary 1 X 1 matrices acting on the right-handed e-type lepton. Breaking up U(2) into unimodular transformations and phase transformations, one could say that the group was SU(2) X U( 1) X U( 1). But then one of the U(l)‘s could be identified with ordinary lepton number and since lepton number appears to be conserved and there is no massless vector particle coupled to it, I decided to exclude it from the group. This left the four-parameter group SU(2) x U( 1). The spontaneous breakdown of SU(2) x U( 1) to the U(1) of ordinary electromagnetic gauge invariance would give masses to three of the four vector gauge bosons: the charged bosons W±, and a neutral boson that I called the Z0. The fourth boson would automatically remain massless, and could be identified as the photon. Knowing the strength of the ordinary charged current weak interactions like beta decay which are mediated by W±, the mass of the W± was then determined as about 40 GeV/sin(ı) where (ı) is the Á-Z0 mixing angle.

To go further, one had to make some hypothesis about the mechanism for the breakdown of SU (2) x U (1). The only kind of field in a renormalizable SU(2) X U(1) theory whose vacuum expectation values could give the electron a mass is a spin zero SU(2) doublet (º+, º0 ), so for simplicity I assumed that these were the only scalar fields in the theory. The mass of the Z0 was then determined as about 80 GeV/sin (2ı). This fixed the strength of the neutral current weak interactions. Indeed, just as in QED, once one decides on the menu of fields in the theory, all details of the theory are completely determined by symmetry principles and renormalizability, with just a few free parameters: the lepton charge and masses, the Fermi coupling constant of beta decay, the mixing angle ı, and the mass of the scalar particle. (It was of crucial importance to impose the constraint of renormalizability; otherwise, weak interactions would receive contributions from SU(2)xU(I) - invariant four-fermion couplings as well as from vector boson exchange, and the theory would lose most of its predictive power.) The naturalness of the whole theory is well-demonstrated by the fact that much the same theory was independently developed [27] by Salam in 1968.

The next question now was renormalizability. The Feynman rules for Yang-Mills theories with unbroken gauge symmetries had been worked out [28] by deWitt, Faddeev and Popov and others, and it was known that such theories are renormalizable. But in 1967, I did not know how to prove that this renormalizability was not spoiled by the spontaneous symmetry breaking. I worked on the problem on and off for several years, partly in collaboration with students, [29] but I made little progress. With hindsight, my main difficulty was that, in quantizing the vector fields, I adopted a gauge now known as the unitarity gauge [30]: this gauge has several wonderful advantages, it exhibits the true particle spectrum of the theory, but it has the disadvantage of making renormalizability totally obscure.

Finally, in 1971, ‘t Hooft [31] showed in a beautiful paper how the problem could be solved. He invented a gauge, like the “Feynman gauge” in QED, in which the Feynman rules manifestly lead to only a finite number of types of ultraviolet divergence. It was also necessary to show that these infinities satisfied essentially the same constraints as the Lagrangian itself, so that they could be absorbed into a redefinition of the parameters of the theory. (This was plausible, but not easy to prove, because a gauge invariant theory can be quantized only after one has picked a specific gauge, so it is not obvious that the ultraviolet divergences satisfy the same gauge invariance constraints as the Lagrangian itself.) The proof was subsequently completed [32] by Lee and Zinn-Justin and by ‘t Hooft and Veltman. More recently, Becchi, Rouet and Stora [33] have invented an ingenious method for carrying out this sort of proof by using a global supersymmetry of gauge theories which is preserved even when we choose a specific gauge.

I have to admit that, when I first saw ‘t Hooft’s paper in 1971, I was not convinced that he had found the way to prove renormalizability. The trouble was not with ‘t Hooft, but with me: I was simply not familiar enough with the path integral formalism on which ‘t Hooft’s work was based, and I wanted to see a derivation of the Feynman rules in ‘t Hooft’s gauge from canonical quantization. That was soon supplied (for a limited class of gauge theories) by a paper of Ben Lee, [34] and after Lee’s paper, I was ready to regard the renormalizability of the unified theory as essentially proved.

By this time, many theoretical physicists were becoming convinced of the general approach that Salam and I had adopted: that is, the weak and electromagnetic interactions are governed by some group of exact local gauge symmetries; this group is spontaneously broken to U(l), giving mass to all the vector bosons except the photon; and the theory is renormalizable. What was not so clear was that our specific simple model was the one chosen by nature. That, of course, was a matter for experiment to decide.

It was obvious even back in 1967 that the best way to test the theory would be by searching for neutral current weak interactions, mediated by the neutral intermediate vector boson, the Z0. Of course, the possibility of neutral currents was nothing new. There had been speculations [35] about possible neutral currents as far back as 1937 by Gamow and Teller, Kemmer, and Wentzel, and again in 1958 by Bludman and Leite-Lopes. Attempts at a unified weak and electromagnetic theory had been made
[36] by Glashow and Salam and Ward in the early 1960’s, and these had neutral currents with many of the features that Salam and I encountered in developing the 1967-68 theory. But, since one of the predictions of our theory was a value for the mass of the Z0, it made a definite prediction of the strength of the neutral currents. More important, now we had a comprehensive quantum field theory of the weak and electromagnetic interactions that was physically and mathematically satisfactory in the same sense as was quantum electrodynamics—a theory that treated photons and intermediate vector bosons on the same footing, that was based on an exact symmetry principle, and that allowed one to carry calculations to any desired degree of accuracy. To test this theory, it had now become urgent to settle the question of the existence of the neutral currents.

Late in 1971, I carried out a study of the experimental possibilities. [37] The results were striking. Previous experiments had set upper bounds on the rates of neutral current processes which were rather low, and many people had received the impression that neutral currents were pretty well ruled out, but I found that, in fact, the 1967-68 theory predicted quite low rates, low enough, in fact, to have escaped clear detection up to that time. For instance, experiments [38] a few years earlier had found an upper bound of 0.12 ± 0.06 on the ratio of a neutral current process, the elastic scattering of muon neutrinos by protons, to the corresponding charged current process, in which a muon is produced. I found a predicted ratio of 0.15 to 0.25, depending on the value of the Z0 - Á mixing angle ı. So there was every reason to look a little harder.

As everyone knows, neutral currents were finally discovered [39] in 1973. There followed years of careful experimental study on the detailed properties of the neutral currents. It would take me too far from my subject to survey these experiments, [40] so I will just say that they have
confirmed the 1967-68 theory with steadily improving precision for neutrino-nucleon and neutrino electron neutral current reactions, and since the remarkable SLAC-Yale experiment [41] last year, for the electron-nucleon neutral current as well.

This is all very nice. But I must say that I would not have been too disturbed if it had turned out that the correct theory was based on some other spontaneously broken gauge group, with very different neutral currents. One possibility was a clever SU(2) theory proposed in 1972 by Georgi and Glashow, [42] which has no neutral currents at all. The important thing to me was the idea of an exact spontaneously broken gauge symmetry, which connects the weak and electromagnetic interactions, and allows these interactions to be renormalizable. Of this, I was convinced, if only because it fitted my conception of the way that nature ought to be.

There were two other relevant theoretical developments in the early 1970s, before the discovery of neutral currents, that I must mention here. One is the important work of Glashow, Iliopoulos, and Maiani on the charmed quark. [43] Their work provided a solution to what, otherwise, would have been a serious problem, that of neutral strangeness changing currents. I leave this topic for Professor Glashow’s talk. The other theoretical development has to do specifically with the strong interactions, but it will take us back to one of the themes of my talk, the theme of symmetry.

In 1973, Politzer and Gross and Wilczek discovered [44] a remarkable property of Yang-Mills theories which they called “asymptotic freedom” —the effective coupling constant [45] decreases to zero as the characteristic energy of a process goes to infinity. It seemed that this might explain the experimental fact that the nucleon behaves in high-energy, deep inelastic, electron scattering as if it consists of essentially free quarks. [46] But there was a problem. In order to give masses to the vector bosons in a gauge theory of strong interactions, one would 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 electromagnetic interactions, the fundamental weak coupling is of the same order as the electronic charge, e, so the effects of virtual intermediate vector bosons would introduce much too large violations of parity and strangeness conservation, of order 1/137, into the strong interactions of the scalars with each other and with the quarks. [47] At some point in the spring of 1973, it occurred to me (and independently to Gross and Wilczek) that one could do away with strongly interacting scalar fields altogether, allowing the strong interaction gauge symmetry to remain unbroken so that the vector bosons, or “gluons”, are massless, and relying on the increase of the strong forces with increasing distance to explain why quarks as well as the massless gluons are not seen in the laboratory. [48] Assuming no strongly interacting scalars, three “colors” of quarks (as indicated by earlier work of several authors [49]), and an SU(3) gauge group, one then had a specific theory of strong interactions, the theory now generally known as quantum chromodynamics.

Experiments since then have increasingly confirmed QCD as the correct theory of strong interactions. What concerns me here, though, is its impact on our understanding of symmetry principles. Once again, the constraints of gauge invariance and renormalizability proved enormously powerful. These constraints force the Lagrangian to be so simple that the strong interactions in QCD must conserve strangeness, charge conjugation, and (apart from problems [50] having to do with instantons) parity. One does not have to assume these symmetries as a priori principles; there is simply no way that the Lagrangian can be complicated enough to violate them. With one additional assumption, that the u and d quarks have relatively small masses, the strong interactions must also satisfy the approximate SU(2) X SU(2) symmetry of current algebra, which, when spontaneously broken, leaves us with isospin. If the s quark mass is also not too large, then one gets the whole eight-fold way as an approximate symmetry of the strong interactions. And the breaking of the SU(3)xSU(3) symmetry by quark masses has just the (3,3)+(3,3) form required to account for the pion-pion scattering lengths [15] and Gell-Mann-Okubo mass formulas. Furthermore, with weak and electromagnetic interactions also described by a gauge theory, the weak currents are necessarily just the currents associated with these strong interaction symmetries. In other words, pretty much the whole pattern of approximate symmetries of strong, weak, and electromagnetic interactions that puzzled us so much in the 1950s and 1960s now stands explained as a simple consequence of strong, weak, and electromagnetic gauge invariance, plus renormalizability. Internal symmetry is now at the point where space-time symmetry was in Einstein’s day. All the approximate internal symmetries are explained dynamically. On a fundamental level, there are no approximate or partial symmetries; there are only exact symmetries which govern all interactions.

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

We are still confronted with the question whether the scalar particles that are responsible for the spontaneous breakdown of the electroweak gauge symmetry SU(2) X U(1) are really elementary. If they are, then spin zero semi-weakly decaying “Higgs bosons” should be found at energies comparable with those needed to produce the intermediate vector bosons. On the other hand, it may be that the scalars are composites. [51] The Higgs bosons would then be indistinct broad states at very high mass, analogous to the possible s-wave enhancement in π-π scattering. There would probably also exist lighter, more slowly decaying, scalar particles of a rather different type, known as pseudo-Goldstone bosons. [52] And there would have to exist a new class of “extra strong” interactions [53] to provide the binding force, extra strong in the sense that asymptotic freedom sets in not at a few hundred MeV, as in QCD, but at a few hundred GeV. This “extra strong” force would be felt by new families of fermions, and would give these fermions masses of the order of several hundred GeV. We shall see.

Of the four (now three) types of interactions, only gravity has resisted incorporation into a renormalizable quantum field theory. This may just mean that we are not being clever enough in our mathematical treatment of general relativity. But there is another possibility that seems to me quite plausible. The constant of gravity defines a unit of energy known as the Planck energy, about 1019 GeV. This is the energy at which gravitation becomes effectively a strong interaction, so that at this energy, one can no longer ignore its ultraviolet divergences. It may be that there is a whole world of new physics with unsuspected degrees of freedom at these enormous energies, and that general relativity does not provide an adequate framework for understanding the physics of these super-high energy degrees of freedom. When we explore gravitation or other ordinary phenomena, with particle masses and energies no greater than a TeV or so, we may be learning only about an “effective” field theory; that is, one in which superheavy degrees of freedom do not explicitly appear, but the coupling parameters implicitly represent 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 on this basis to find at ordinary energy. By “integrating out” the super-high energy degrees of freedom in a fundamental theory, we generally encounter a very complicated effective field theory—so complicated, in fact, that it contains all interactions allowed by symmetry principles. But where dimensional analysis tells us that a coupling constant is a certain power of some mass, that mass is likely to be a typical superheavy mass, such as 1019 GeV. The infinite variety of nonrenormalizable interactions in the effective theory have coupling constants with the dimensionality of negative powers of mass, so their effects are suppressed at ordinary energies by powers of energy divided by super-heavy masses. Thus, the only interactions that we can detect at ordinary energies are those that are renormalizable in the usual sense, plus any nonrenormalizable interactions that produce effects which, although tiny, are somehow exotic enough to be seen.

One way that a very weak interaction could be detected is for it to be coherent and of long range, so that it can add up and have macroscopic effects. It has been shown [54] that the only particles whose exchange could produce such forces are massless particles of spin 0, 1, or 2. And furthermore, Lorentz’s invariance alone is enough to show that the long-range interactions produced by any particle of mass zero and spin 2 must be governed by general relativity. [55] Thus, from this point of view, we should not be too surprised that gravitation is the only interaction discovered so far that does not seem to be described by a renormalizable field theory - it is almost the only super-weak interaction that could have been detected. And we should not be surprised to find that gravity is well described by general relativity at macroscopic scales, even if we do not think that general relativity applies at 1019 GeV.

Non-renormalizable effective interactions may also be detected if they violate otherwise exact conservation laws. The leading candidates for violation are baryon and lepton conservation. It is a remarkable consequence of the SU(3) and SU(2) x U( 1) gauge symmetries of strong, weak, and electromagnetic interactions, that all renormalizable interactions among known particles automatically conserve baryon and lepton number. Thus, the fact that ordinary matter seems pretty stable, that proton decay has not been seen, should not lead us to the conclusion that baryon and lepton conservation are fundamental conservation laws. To the accuracy with which they have been verified, baryon and lepton conservation can be explained as dynamical consequences of other symmetries, in the same way that strangeness conservation has been explained within QCD. But superheavy particles may exist, and these particles may have unusual SU(3) or SU(2) x SU(1) transformation properties, and in this case, there is no reason why their interactions should conserve baryon or lepton number. I doubt that they would. Indeed, the fact that the universe seems to contain an excess of baryons over anti-baryons should lead us to suspect that baryon non-conserving processes have actually occurred. If effects of a tiny non-conservation of baryon or lepton number such as proton decay or neutrino masses are discovered experimentally, we will then be left with gauge symmetries as the only true internal symmetries of nature, a conclusion that I would regard as most satisfactory.

The idea of a new scale of superheavy masses has arisen in another way. [56] If any sort of “grand unification” of strong and electroweak gauge couplings is to be possible, then one would expect all of the SU(3) and SU(2) x U(1) gauge coupling constants to be of comparable magnitude. (In particular, if SU(3) and SU(2) x U(1) are subgroups of a larger simple group, then the ratios of the squared couplings are fixed as rational numbers of order unity.[57]) But this appears in contradiction with the obvious fact that the strong interactions are stronger than the weak and electromagnetic interactions. In 1974, Georgi, Quinn and I suggested that the grand unification scale, at which the couplings are comparable, is at an enormous energy, and that the reason that the strong coupling is so much larger than the electroweak couplings at ordinary energies is that QCD is asymptotically free, so that its effective coupling constant rises slowly as the energy drops from the grand unification scale to ordinary values. The change of the strong couplings is very slow (like 1/√ln E) so the grand unification scale must be enormous. We found that for a fairly large class of theories, the grand unification scale comes out to be in the neighborhood of 1016 GeV, an energy not all that different from the Planck energy of 1019 GeV. The nucleon lifetime is very difficult to estimate accurately, but we gave a representative value of 1032 years, which may be accessible experimentally in a few years. (These estimates have been improved in more detailed calculations by several authors.) [58] We also calculated a value for the mixing parameter of about 0.2, not far from the present experimental of 0.23±0.01. It will be an important task for future experiments on neutral currents to improve the precision with which is known to see if it really agrees with this prediction.

In a grand unified theory, in order for elementary scalar particles to be available to produce the spontaneous breakdown of the electroweak gauge symmetry at a few hundred GeV, it is necessary for such particles to escape getting super-large masses from the spontaneous breakdown of the grand unified gauge group. There is nothing impossible in this, but I have not been able to think of any reason why it should happen. (The problem may be related to the old mystery of why quantum corrections do not produce an enormous cosmological constant; in both cases, one is concerned with an anomalously small “super-renormalizable” term in the effective Lagrangian which has to be adjusted to be zero. In the case of the cosmological constant, the adjustment must be precise to some fifty decimal places.) With elementary scalars of small or zero bare mass, enormous ratios of symmetry breaking scales can arise quite naturally [59]. On the other hand, if there are no elementary scalars which escape getting superlarge masses from the breakdown of the grand unified gauge group then as I have already mentioned, there must be extra strong forces to bind the composite Goldstone and Higgs bosons that are associated with the spontaneous breakdown of SU(2) x U(1). Such forces can occur rather naturally in grand unified theories. To take one example, suppose that the grand gauge group breaks, not into SU(3) x SU(2) x U(l), but into SU(4) x SU(3) x SU(2) x U(1). Since SU(4) is a bigger group than SU(3), its coupling constant rises with decreasing energy more rapidly than the QCD coupling, so the SU(4) force becomes strong at a much higher energy than the few hundred MeV at which the QCD force becomes strong. Ordinary quarks and leptons would be neutral under SU(4), so they would not feel this force, but other fermions might carry SU(4) quantum numbers, and so get rather large masses. One can even imagine a sequence of increasingly large subgroups of the grand gauge group, which would fill in the vast energy range up to 1015 or 1019 GeV with particle masses that are produced by these successively stronger interactions.

If there are elementary scalars whose vacuum expectation values are responsible for the masses of ordinary quarks and leptons, then these masses can be affected in order α by radiative corrections involving the superheavy vector bosons of the grand gauge group, and it will probably be impossible to explain the value of quantities like me/mu a complete grand unified theory. On the other hand, if there are no such elementary scalars, then almost all the details of the grand unified theory are forgotten by the effective field theory that describes physics at ordinary energies, and it ought to be possible to calculate quark and lepton masses purely in terms of processes at accessible energies. Unfortunately, no one so far has been able to see how, in this way, anything resembling the observed pattern of masses could arise. [60]

Putting aside all these uncertainties, suppose that there is a truly fundamental theory, characterized by an energy scale of order 1016 to 1019 GeV, at which strong, electroweak, and gravitational interactions are all united. It might be a conventional renormalizable quantum field theory but at the moment, if we include gravity, we do not see how this is possible. (I leave the topic of supersymmetry and supergravity for Professor Salam’s talk.) But if it is not renormalizable, what then 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 quantum field theory, which was born just fifty years ago from the marriage of quantum mechanics with relativity, is a beautiful but not very robust child. As Landau and Kallen recognized long ago, quantum field theory at superhigh energies is susceptible to all sorts of diseases—tachyons, ghosts, etc. and it needs special medicine to survive. One way that a quantum field theory can avoid these diseases is to be renormalizable and asymptotically free, but there are other possibilities. For instance, even an infinite set of coupling constants may approach a non-zero fixed point as the energy at which they are measured goes to infinity. However, to require this behavior generally imposes so many constraints on the couplings that there are only a finite number of free parameters left[6 1] —just as for theories that are renormalizable in the usual sense. Thus, one way or another, I think that quantum field theory is going to go on being very stubborn, refusing to allow us to describe all but a small number of possible worlds, among which, we hope, is ours.

I suppose that I tend to be optimistic about the future of physics. And nothing makes me more optimistic than the discovery of broken symmetries. In the seventh book of the Republic, Plato describes prisoners who are chained in a cave and can see only shadows that things outside cast on the cave wall. When released from the cave, at first their eyes hurt, and, for a while, they think that the shadows they saw in the cave are more real than the objects they now see. But eventually their vision clears, and they can understand how beautiful the real world is. We are in such a cave, imprisoned by the limitations on the sorts of experiments we can do. In particular, we can study matter only at relatively low temperatures, where symmetries are likely to be spontaneously broken, so that nature does not appear very simple or unified. We have not been able to get out of this cave, but by looking long and hard at the shadows on the cave wall, we can at least make out the shapes of symmetries, which though broken, are exact principles governing all phenomena, expressions of the beauty of the world outside.

It has only been possible here to give references to a very small part of the literature on the subjects discussed in this talk. Additional references can be found in the following reviews:.

Abers, E.S. and Lee, B.W., Gauge Theories (Physics Reports 9C, No. 1, 1973).

Taylor, J.C., Gauge Theories of Weak Interactions (Cambridge Univ. Press, 1976).


1. Tuve, M. A., Heydenberg, N. and Hafstad, L. R. Phys. Rev. 50, 806 (1936); Breit, G., Condon, E. V. and Present, R. D. Phys. Rev. 50, 825 (1936); Breit, G. and Feenberg, E. Phys. Rev. 50, 850 (1936).
2. Gell-Mann, M. Phys. Rev. 92, 833 (1953); Nakano. T. and Nishijima, K. Prog. Theor. Phys. 10, 581 (1955).
3. Lee, T. D. and Yang, C. N. Phys. Rev. 104, 254 (1956); Wu. C. S. Phys. Rev. 105, 1413 (1957); Garwin, R., Lederman, L. and Weinrich, M. Phys. Rev. 105, 1415 (1957); Friedman, J. I. and Telegdi V. L. Phys. Rev. 105, 1681 (1957).
4. Gell-Mann, M. Cal. Tech. Synchotron Laboratory Report CTSL-20 (1961). unpublished; Ne’eman, Y. Nucl. Phys. 26, 222 (1961).
5. Fock, V. Z. f. Physik 39, 226 (1927); Weyl, H. Z. f. Physik 56, 330 (1929). The name “gauge invariance” is based on an analogy with the earlier speculations of Weyl, H. in Raum, Zeit, Materie, 3rd edn, (Springer, 1920). Also see London, F. Z. f. Physik 42, 375 (1927). (This history has been reviewed by Yang, C. N. in a talk at City College, (1977).)
6. Yang, C. N. and Mills, R. L. Phys. Rev. 96, 191 (1954).
7. Goldstone, J. Nuovo Cimento 19, 154 (1961).
8. Goldstone, J., Salam, A. and Weinberg, S. Phys. Rev. 127, 965 (1962).
9. Higgs, P. W. Phys. Lett. 12, 132 (1964); 13, 508 (1964); Phys. Rev. 145, 1156 (1966); Kibble, T. W. B. Phys. Rev. 155, 1554 (1967); Guralnik, G. S., Hagen, C. R. and Kibble, T. W. B. Phys. Rev. Lett. 13, 585 (1964); Englert, F. and Brout, R. Phys. Rev. Lett. 13, 32 1 (1964); Also see Anderson, P. W. Phys. Rev. 130, 439 (1963).
10. Adler, S. L. Phys. Rev. Lett. 14, 1051 (1965); Phys Rev. 140, B736 (1965); Weisberger, W.
Phys. Rev. Lett. 14, 1047 (1965); Phys Rev. 143, 1302 (1966).
11. Gell-Mann, M. Physics I, 63 (1964).
12. Nambu, Y. and Jona-Lasinio, G. Phys. Rev. 122, 345 (1961); 124, 246 (1961); Nambu, Y, and Lurie, D. Phys. Rev. 125, 1429 (1962); Nambu. Y. and Shrauner, E. Phys. Rev. 128, 862 (1962); Also see Gell-Mann, M. and Levy, M., Nuovo Cimento 16, 705 (1960).
13. Goldberger, M. L., Miyazawa, H. and Oehme, R. Phys Rev. 99, 986 (1955).
14. Goldberger, M. L., and Treiman, S. B. Phys. Rev. 111, 354 (1958).
15 .Weinberg, S. Phys. Rev. Lett. 16, 879 (1966); 17, 336 (1966); 17, 616 (1966); 18, 188 (1967); Phys Rev.166, 1568 (1967).
16. Oppenheimer, J, R. Phys. Rev. 35, 461 (1930); Waller, I. Z. Phys. 59, 168 (1930); ibid., 62, 673 (1930).
17. Feynman, R. P. Rev. Mod. Phys. 20, 367 (1948); Phys. Rev. 74, 939, 1430 (1948); 76, 749, 769 (1949); 80, 440 (1950); Schwinger, J. Phys. Rev. 73, 146 (1948); 74, 1439 (1948); 75, 651 (1949); 76, 790 (1949); 82, 664, 914 (1951);91, 713 (1953); Proc. Nat. Acad. Sci.37, 452 (1951); Tomonaga, S. Progr. Theor. Phys. (Japan) I, 27 (1946); Koba, Z., Tati, T. and Tomonaga, S. ibid. 2, 101 (1947); Kanazawa, S. and Tomonaga, S. ibid. 3, 276 (1948); Koba, Z. and Tomonaga, S. ibid 3, 290 (1948).
18. There had been earlier suggestions that infinities could be eliminated from quantum field theories in this way, by Weisskopf, V. F. Kong. Dansk. Vid. Sel. Mat.-Fys. Medd. 15,(6) 1936, especially p. 34 and pp. 5-6; Kramers,.H. (unpublished).
19. Dyson, F. J. Phys. Rev. 75, 486, 1736 (1949).
20. Weinberg, S. Phys. Rev. 106, 1301 (1957).
21. Weinberg, S. Phys. Rev. 118, 838 (1960).
22. Salam, A. Phys. Rev. 82, 217 (1951); 84, 426 (1951).
23. Weinberg, S. Phys. Rev. Lett. 18, 507 (1967).
24. For the non-renormalizability of theories with intrinsically broken gauge symmetries, see Komar, A. and Salam, A. Nucl. Phys. 21, 624 (1960); Umezawa, H. and Kamefuchi, S. Nucl. Phys. 23, 399 (1961); Kamefuchi, S., O’Raifeartaigh, L. and Salam, A. Nucl. Phys. 28, 529 (1961); Salam, A. Phys. Rev. 127, 331 (1962); Veltman, M. Nucl. Phys. B7, 637 (1968); B21, 288 (1970); Boulware, D. Ann. Phys. (N, Y,)56, 140 (1970).
25. This work was briefly reported in reference 23, footnote 7.
26. Weinberg, S. Phys. Rev. Lett. 19, 1264 (1967).
27. Salam, A. In Elementary Particle Physics (Nobel Symposium No. 8), ed. by Svartholm, N. (Almqvist and Wiksell, Stockholm, 1968), p. 367.
28. deWitt, B. Phys. Rev. Lett. 12, 742 (1964); Phys. Rev. 162, 1195 (1967); Faddeev L. D., and Popov, V. N. Phys. Lett. B25, 29 (1967); Also see Feynman, R. P. Acta. Phys. Vol. 24, 697 (1963); Mandelstam, S. Phys. Rev. 175, 1580, 1604 (1968).
29. See Stuller, I.. M. I. T., Thesis, PhD (1971), unpublished.
30. My work with the unitarity gauge was reported in Weinberg, S. Phys. Rev. Lett. 27, 1688 (1971 ), and described in more detail in Weinberg, S. Phys. Rev. D7, 1068 (1973).
31. ‘t Hooft, G. Nucl. Phys. B35, 167 (1971).
32. Lee, B. W. and Zinn-Justin, J. Phys. Rev. D5, 3121, 3137, 3155 (1972); ‘t Hooft, G. and Veltman, M. Nucl. Phys. 844, 189 (1972), B50, 318 (1972). There still remained the problem of possible Adler-Bell-Jackiw anomalies, but these nicely cancelled; see D. J. Gross and R. Jackiw, Phys. Rev. D6, 477 (1972) and C. Bouchiat, J. lliopoulos, and Ph. Meyer, Phys. Lett. 388, 519 (1972).
33. Beechi, C., Rouet, A. and Stora R. Comm. Math. Phys. 42, 127 (1975).
34. Lee, B. W. Phys. Rev. D5, 823 (1972).
35. Gamow, G. and Teller, E. Phys. Rev. 51, 288 (1937); Kemmer, N. Phys. Rev. 52, 906 (1937); Wentrel, G. Helv. Phys. Acta. 10, 108 (1937); Bludman, S. Nuovo Cimento 9, 433 (1958); Leite-Lopes, J. Nucl. Phys. 8, 234 (1958).
36. Glashow, S. L. Nucl. Phys. 22, 519 (1961); Salam, A. and Ward, J. C. Phys. Lett. 13, 168 (1964).
37. Weinberg, S. Phys. Rev. 5, 1412 (1972).
38. Cundy, D. C. et al., Phys. Lett. 31B, 478 (1970).
39. The first published discovery of neutral currents was at the Gargamelle Bubble Chamber at CERN: Hasert, F. J. et al., Phys. Lett. 468, 121, 138 (1973). Also see Musset, P. Jour. de Physique 11 /12 T34 (1973). Muonless events were seen at about the same time by the HPWF group at Fermilab, but when publication of their paper was delayed, they took the opportunity to rebuild their detector, and then did not at first find the same neutral current signal. The HPWF group published evidence for neutral currents in Benvenuti, A. et al., Phys. Rev. Lett. 52, 800 (1974).
40. For a survey of the data see Baltay, C. Proceedings of the 19th International Conference on High Energy Physics, Tokyo, 1978. For theoretical analyses, see Abbott, L. F. and Barnett,
R. M. Phys. Rev. D19, 3230 (1979); Langacker, P., Kim, J. E., Levine, M., Williams, H. H. and Sidhu, D. P. Neutrino Conference ‘79; and earlier references cited therein.
41. Prescott, C. Y., Phys. Lett. 778, 347 (1978).
42. Glashow, S. L. and Georgi, H. L. Phys. Rev. Lett. 28, 1494 (1972). Also see Schwinger, J. Annals of Physics (N. Y.)2, 407 (1957).
43. Glashow, S. L., Iliopoulos, J. and Maiani, L. Phys. Rev. D2, 1285 (1970). This paper was cited in ref. 37 as providing a possible solution to the problem of strangeness changing neutral currents. However, at that time I was skeptical about the quark model, so in the calculations of ref. 37 baryons were incorporated in the theory by taking the protons and neutrons to form an SU(2) doublet, with strange particles simply ignored.
44. Politzer, H. D. Phys. Rev. Lett. 30, 1346 (1973); Gross, D. J. and Wilczek, F. Phys. Rev. Lett. 30, 1343 (1973).
45. Energy dependent effective couping constants were introduced by Gell-Mann, M. and Low, F. E. Phys. Rev. 95, 1300 (1954).
46. Bloom, E. D., Phys. Rev. Lett. 23, 930 (1969); Breidenbach, M., Phys. Rev. Lett. 23, 935 (1969).

47. Weinberg, S. Phys. Rev. D8, 605 (1973).
48. Gross, D. J. and Wilczek, F. Phys. Rev. D8, 3633 (1973); Weinberg, S. Phys. Rev. Lett. 31, 494 (1973). A similar idea had been proposed before the discovery of asymptotic freedom by Fritzsch, H., Gell-Mann, M. and Leutwyler, H. Phys. Lett. 478, 365 (1973).
49. Greenberg, O. W. Phys. Rev. Lett. 13, 598 (1964); Han, M. Y. and Nambu, Y. Phys. Rev. 139, B1006 (1965); Bardeen, W. A., Fritzsch, H. and Gell-Mann, M. in Scale and Conforma1 Symmetry in Hadron Physics, ed. by Gatto, R. (Wiley, 1973), p. 139; etc.
50. ‘t Hooft, G. Phys. Rev. Lett. 37, 8 (1976).
51. Such “dynamical” mechanisms for spontaneous symmetry breaking were first discussed by Nambu, Y. and Jona-Lasinio, G. Phys. Rev. 122, 345 (1961); Schwinger, J. Phys. Rev. 125, 397 (1962); 128, 2425 (1962); and in the context of modern gauge theories by Jackiw, R. and Johnson, K. Phys. Rev. D8, 2386 (1973); Cornwall, J. M. and Norton, R. E. Phys. Rev. D8, 3338 (1973). The implications of dynamical symmetry breaking have been considered by Weinberg, S. Phys. Rev. D13, 974 (1976); D19, 1277 (1979); Susskind, L. Phys. Rev. D20, 2619 (1979).
52. Weinberg, S. ref 51, The possibility of pseudo-Goldstone bosons was originally noted in a different context by Weinberg, S. Phys. Rev. Lett. 29, 1698 (1972).
53. Weinberg, S. ref. 51. Models involving such interactions have also been discussed by Susskind, L. ref. 51.
54. Weinberg, S. Phys. Rev. 135, B1049 (1964).
55. Weinberg. S. Phys. Lett. 9, 357 (1964); Phys. Rev. 8138, 988 (1965); Lectures in Particles and Field Theory, ed. by Deser, S. and Ford, K. (Prentice-Hall, 1965), p. 988; and ref. 54. The program of deriving general relativity from quantum mechanics and special relativity was completed by Boulware, D. and Deser, S. Ann. Phys. 89, 173 (1975). I understand that similar ideas were developed by Feynman, R. in unpublished lectures at Cal. Tech.
56. Georgi, H., Quinn, H. and Weinberg, S. Phys. Rev. Lett. 33, 45 1 (1974).
57. An example of a simple gauge group for weak and electromagnetic interactions (for which sin 2 ı =1/4, was given by S. Weinberg, Phys. Rev. D5, 1962 (1972). There are a number of specific models of weak, electromagnetic, and strong interactions based on simple gauge groups, including those of Pati, J. C. and Salam, A. Phys. Rev. D10, 275 (1974); Georgi, H. and Glashow, S. L. Phys. Rev. Lett. 32, 438 (1974); Georgi, H. in Particles and Fields (American Institute of Physics, 1975); Fritzsch, H. and Minkowski, P. Ann. Phys. 93, 193 (1975); Georgi, H. and Nanopoulos, D. V. Phys. Lett. 82B, 392 (1979); Gürsey, F. Ramond, P. and Sikivie, P. Phys. Lett. B60, 177 (1975); Gürsey, F. and Sikivie, P. Phys. Rev. Lett. 36, 775 (1976); Ramond, P. Nucl. Phys, B110, 214 (1976); etc; all these violate baryon and lepton conservation, because they have quarks and leptons in the same multiplet; see Pati, J. C. and Salam, A. Phys. Rev. Lett. 31, 661 (1973); Phys. Rev. D8, 1240 (1973).
58. Buras, A., Ellis, J., Gaillard, M. K. and Nanopoulos, D. V. Nucl. Phys. B135, 66 (1978); Ross, D. Nucl. Phys. B140, 1 (1978); Marciano, W. J. Phys. Rev. D20, 274 (1979); ‘Goldman, T. and Ross, D. CALT 68-704, to be published; Jarlskog, C. and Yndurain, F. J. CERN preprint, to be published. Machacek, M. Harvard preprint HUTP-79/AO21, to be published in Nuclear Physics; Weinberg, S. paper in preparation. The phenomenonology of nucleon decay has been discussed in general terms by Weinberg, S. Phys. Rev. Lett. 43, 1566 (1979); Wilczek, F. and Zee, A. Phys. Rev. Lett. 43, 1571 (1979).
59. Gildener, E. and Weinberg, S. Phys. Rev. D13, 3333 (1976); Weinberg, S. Phys. Letters 82B, 387 (1979). In general there should exist at least one scalar particle with physical mass of order 10 GeV. The spontaneous symmetry breaking in models with zero bare scalar mass was first considered by Coleman, S. and Weinberg, E., Phys. Rev. D 7, 1888 (1973).
60. This problem has been studied recently by Dimopoulos, S. and Susskind, L. Nucl. Phys. B155, 237 (1979); Eichten, E. and Lane, K. Physics Letters, to be published; Weinberg, S. unpublished.
61. Weinberg, S. in General Relativity -An Einstein Centenary Survey, ed. by Hawking, S. W. and Israel, W. (Cambridge Univ. Press, 1979), Chapter-16.

Thanks to the Nobel Foundation for permission to publish this talk on this website.
© The Nobel Foundation 1979

Louise Weinberg is holder of the Bates Chair and Professor of Law at the University of Texas School of Law. Weinberg teaches and writes in Constitutional Law and Federal Courts. She received her undergraduate degree summa cum laude from Cornell, was elected to Phi Beta Kappa, holds two Harvard Law degrees, and clerked for Judge Wyzanski. She practiced in Boston as an associate in litigation with Bingham Dana & Gould, now Bingham McCutchen. She has taught at Harvard, Brandeis, and Stanford, and has received the Texas Exes' Excellence in Teaching Award. She is a member of the American Law Institute, and currently serves as an invited Adviser to the projected ALI Restatement (Third) of Conflict of Laws. A frequently invited public speaker, she has served as a Forum Fellow of the World International Forum, Davos. Professor Weinberg was Chair in 2013–2014 of the Association of American Law Schools Section on Conflict of Laws, and has chaired three different AALS Sections, thrice chairing the Section on Federal Courts, twice chairing the Section on Conflict of Laws, and chairing the Section on Admiralty. Recently she appeared in the Public Broadcasting System's four-part series, The Supreme Court.

In the field of Constitutional Law Weinberg's writings include Luther v. Borden, A Taney-Court Mystery Solved (forthcoming 2017);A General Theory of Governance: Due Process and Lawmaking Power, William & Mary Law Review (2013); Unlikely Beginnings of Modern Constitutional Thought, University of Pennsylvania Journal of Constitutional Law (2012); The McReynolds Mystery Solved, University of Denver Law Review (2011); An Almost Archeological Dig: Substantive Due Process, An Early View, Constitutional Commentary (2010); Dred Scott and the Crisis of 1860, Symposium, Chicago-Kent Law Review (2007); Our Marbury, Virginia Law Review (2003); and When Courts Decide Elections: The Constitutionality of Bush v. Gore, Symposium, Boston University Law Review (2002).

In the field of Federal Courts, Weinberg is author of Federal Courts: Judicial Federalism and Judicial Power (1994). Her recent work in the field includes Back to the Future: The New General Common Law, Symposium, Journal of Maritime Law and Commerce (2004); Of Sovereignty and Union: The Legends of Alden, Notre Dame Law Review (2001); and The Article III Box, Symposium, Texas Law Review (2000).

In the field of Conflict of Laws, Weinberg is co-author of The Conflict of Laws (2002). Her work in this field includes A Radical Transformation for Conflicts Restatements, Symposium, Illinois Law Review (2015, pub. 2016 ); What We Don't Talk About When We Talk About Extraterritoriality, Symposium, Cornell Law Review (2015); and Theory Wars in the Conflict of Laws, Michigan Law Review (2005).

In the field of Legal Theory and Jurisprudence, Weinberg's writings include Of Theory and Theodicy: The Problem of Immoral Law, in Law and Justice in a Multistate World (2002) and Choosing Law, Giving Justice, Symposium, Louisiana Law Review (2000).

Weinberg is author of such classic articles as Federal Common Law, Northwestern Law Review (1989) and The New Judicial Federalism, Stanford Law Review (1977), and such provocative essays as Holmes' Failure, Michigan Law Review (1997) and Against Comity, Georgetown Law Journal (1991). She is a contributor to legal encyclopedias for the Oxford and Yale University Presses. Her pieces for the general public have appeared in The American Scholar, The Public Interest, and Daedalus.


Steven Weinberg Photo Album

Steven Weinberg lecturing, University of Texas at Austin
Steven Weinberg, Bronx High School of Science, Observatory 1950
(Photo courtesy of Dr. Ben Forsyth, classmate)
Steven Weinberg, photo from 2019 Physics Today, "Today in History" article
Steven Weinberg

Steven Weinberg


Louise and Steven Weinberg with Queen Beatrix in 1983
Weinberg quotes are legendary.
Weinberg Quote
Steven Weinberg, University of Texas at Austin
Steven Weinberg, University of Texas at Austin


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