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PERSPECTIVE

Asymptotic freedom: From paradox to paradigm

Frank Wilczek*

A Pair of Paradoxes plethora of new particles was discovered have electric charges whose magnitudes n theoretical , paradoxes are in this way. Although these particles, are fractions (2͞3or1͞3) of what ap- good. That’s paradoxical, since a generically called , are unstable, pears to be the basic unit, namely the paradox appears to be a contradic- they otherwise behave in ways that magnitude of carried by an elec- tion, and contradictions imply seri- broadly resemble the way and tron or . All other observed electric Ious error. But Nature cannot realize behave. So the character of the charges are known, with great accuracy, contradictions. When our physical theo- subject changed. It was no longer natu- to be whole-number multiples of this ries lead to paradox we must find a way ral to think of it as simply as the study unit. Also, identical did not ap- out. Paradoxes focus our attention, and of a new that binds protons and pear to obey the normal rules of quan- we think harder. neutrons into atomic nuclei. Rather, a tum statistics. These rules would require When and I began the new world of phenomena had come into that, as 1͞2 particles, quarks should work that led to this Nobel Prize (1–3)† view. This world contained many unex- be , with antisymmetric wave in 1972, we were driven by paradoxes. pected new particles that could trans- functions. The pattern of observed bary- In resolving the paradoxes, we were led form into one another in a bewildering ons cannot be understood using anti- to discover a new dynamical principle, variety of ways. Reflecting this change symmetric wave functions; it requires . This principle, in in perspective, there was a change in symmetric wave functions. turn, has led to an expanded conception terminology. Instead of the nuclear The atmosphere of weirdness and pe- of fundamental particles, a new under- force, physicists came to speak of the culiarity surrounding quarks thickened standing of how gets its mass, a strong . into paradox when J. Friedman, H. Ken- new and much clearer picture of the In the early 1960s, Murray Gell-Mann dall, R. Taylor, and their collaborators early , and new ideas about the and made a great ad- at the Stanford Linear Accelerator unity of Nature’s . Today I’d like vance in the theory of the strong inter- (SLAC) used energetic to poke to share with you the story of these action by proposing the concept of into the inside of protons (J. Friedman, ideas. quarks. If you imagined that hadrons H. Kendall, and R. Taylor received the were not fundamental particles, but Nobel Prize for this work in 1990). They Paradox 1: Quarks Are Born Free, but Every- rather that they were assembled from a discovered that there are indeed entities where They Are in Chains. The first para- few more basic types, the quarks, pat- that look like quarks inside protons. dox was phenomenological. terns clicked into place. The dozens of Surprisingly, though, they found that Near the beginning of the 20th cen- observed hadrons could be understood, when quarks are hit hard they seem to tury, after pioneering experiments by at least roughly, as different ways of move (more accurately, to transport en- Rutherford, Geiger, and Marsden, phys- putting together just three kinds (‘‘fla- ergy and ) as if they were icists discovered that most of the mass vors’’) of quarks. You can have a given free particles. Before the experiment, and all of the positive charge inside an set of quarks in different spatial , most physicists had expected that what- is concentrated in a tiny central or with their spins aligned in different ever caused the of nucleus. In 1932, Chadwick discovered ways. The of the configuration quarks would also cause quarks to radi- neutrons, which together with protons will depend on these things, and so could be considered as the ingredients there will be a number of states with ate energy abundantly, and thus rapidly out of which atomic nuclei could be different , giving rise to particles to dissipate their motion, when they got constructed. But the known forces, grav- with different masses, according to m ϭ violently accelerated. ity and , were insuffi- E͞c2. It is analogous to the way we un- At a certain level of sophistication, cient to bind protons and neutrons derstand the spectrum of excited states this association of with forces tightly together into objects as small as of an atom, as arising from different appears inevitable, and profound. In- the observed nuclei. Physicists were con- orbits and spin alignments of . deed, the connection between forces fronted with a new force, the most pow- (For electrons in , the interaction and radiation is associated with some erful in Nature. Understanding this new energies are relatively small, however, of the most glorious episodes in the his- force became a major challenge in fun- and the effect of these energies on the tory of physics. In 1864, Maxwell pre- damental physics. overall mass of the atoms is insignifi- For many years, physicists gathered cant.) The rules for using quarks to model data to address that challenge, basically seemed quite weird, however. *E-mail: [email protected] by bashing protons and neutrons to- Quarks were supposed to hardly notice Adapted from Les Prix Nobel, 2004. © 2004 by the Nobel Foundation gether and studying what came out. The one another when they were close to- Editor’s Note: This article is a version of ’s results that emerged from these studies, gether, but if you tried to isolate one, you Nobel Lecture, ‘‘Asymptotic Freedom: From Paradox to Par- however, were complicated and hard to found that you could not. People looked adigm.’’ The 2004 was awarded to Drs. interpret. very hard for individual quarks, but with- Wilczek, H. David Politzer, and David J. Gross for their What you would expect, if the parti- out success. Only bound states of a discovery of asymptotic freedom in the theory of the strong interaction. The Nobel Foundation graciously has granted cles were really fundamental (indestruc- and an antiquark——or bound us permission to reprint this article. The Nobel Lectures tible), would be the same particles you states of three quarks——are provide examples of successful approaches to major scien- started with, coming out with just their observed. This experimental regularity tific problems. However, in recent years, these lectures have trajectories changed. was elevated into The Principle of Con- rarely been read, perhaps because of the difficulty in ob- taining the collections. By reprinting this lecture, we hope Instead, the outcome of the collisions finement. But giving it a dignified name to broaden their exposure. was often many particles. The final state didn’t make it less weird. †In view of the nature and scope of this write-up, its foot- might contain several copies of the orig- There were other peculiar things noting will be . Our major original papers (1–3) are inals, or different particles altogether. A about quarks. They were supposed to carefully referenced.

www.pnas.org͞cgi͞doi͞10.1073͞pnas.0501642102 PNAS ͉ June 14, 2005 ͉ vol. 102 ͉ no. 24 ͉ 8403–8413 Downloaded by guest on September 23, 2021 dicted the existence of electromagnetic conserved numbers, the extra are predicted to occur very rarely in radiation—including, but not limited to, particles must be accompanied by an thermal equilibrium, and cause no prob- ordinary light—as a consequence of his equal number of antiparticles. (Dirac lem. But quantum fluctuations are much consistent and comprehensive formula- was led to predict the existence of more efficient than are thermal fluctua- tion of electric and magnetic forces. antiparticles through a sequence of inge- tions at exciting the high-energy modes, Maxwell’s new radiation was subse- nious interpretations and reinterpreta- in the form of virtual particles, and so quently generated and detected by tions of the elegant relativistic wave those modes come back to haunt us. For Hertz, in 1883 (and over the 20th cen- equation he invented, rather than by example, they give a divergent contribu- tury its development has revolutionized heuristic reasoning of the sort I’ve pre- tion to the energy of empty space, the the way we manipulate matter and com- sented. The inevitability and generality so-called zero-point energy. municate with one another). Much later, of his conclusions, and their direct rela- theory was devel- in 1935, Yukawa predicted the existence tionship to basic principles of quantum oped to deal with this sort of difficulty. of based on his analysis of nuclear mechanics and , are The central observation that is exploited forces, and they were subsequently dis- only clear in retrospect.) in renormalization theory is that, al- covered in the late 1940s; the existences The second and third of these prizes though with high-energy of many other hadrons were predicted were to R. Feynman, J. Schwinger, and virtual particles appear to produce di- successfully by using a generalization S.-I. Tomonaga (in 1965) and to G. ’t vergent corrections, they do so in a very these ideas. (For experts: I have in mind Hooft and M. Veltman (in 1999), respec- structured way. That is, the same correc- the many resonances that were first seen tively. The main problem that all these tions appear over and over again in the in partial wave analyses, and then later authors, in one way or another, addressed calculations of many different physical in production.) More recently, the exis- is the problem of UV divergences. processes. For example, in quantum tence of W and Z , and of color When special relativity is taken into electrodynamics (QED), exactly two in- , and their properties, was in- account, quantum theory must allow for dependent divergent expressions appear, ferred before their experimental discov- fluctuations in energy over brief inter- one of which occurs when we calculate ery. Those discoveries were, in 1972, vals of time. This is a generalization of the correction to the mass of the elec- still ahead of us, but they serve to con- the complementarity between momen- tron, the other of which occurs when we firm, retroactively, that our concerns tum and position that is fundamental for calculate the correction to its charge. To were worthy ones. Powerful interactions ordinary, nonrelativistic quantum me- make the calculation mathematically ought to be associated with powerful chanics. Loosely speaking, energy can well defined, we must artificially exclude radiation. When the most powerful in- be borrowed to make evanescent virtual the highest energy modes, or dampen teraction in nature, the strong interac- particles, including particle–antiparticle their interactions, a procedure called tion, did not obey this rule, it posed a pairs. Each pair passes away soon after applying a cutoff, or regularization. In sharp paradox. it comes into being, but new pairs are the end we want to remove the cutoff, constantly boiling up, to establish an but at intermediate stages we need to Paradox 2: Special Relativity and Quantum equilibrium distribution. In this way, the leave it in, so as to have well defined Mechanics both Work. The second para- wave function of (superficially) empty (finite) mathematical expressions. If we dox is more conceptual. Quantum me- space becomes densely populated with chanics and special relativity are two virtual particles, and empty space comes are willing to take the mass and charge great theories of 20th-century physics. to behave as a dynamical medium. of the from experiment, we can Both are very successful. But these two The virtual particles with very high identify the formal expressions for these theories are based on entirely different energy create special problems. If you quantities, including the potentially di- ideas, which are not easy to reconcile. calculate how much the properties of vergent corrections, with their measured In particular, special relativity puts real particles and their interactions are values. Having made this identification, space and time on the same footing, but changed by their interaction with virtual we can remove the cutoff. We thereby treats them very particles, you tend to get divergent an- obtain well defined answers, in terms of differently. This leads to a creative ten- swers, due to the contributions from the measured mass and charge, for ev- sion, whose resolution has led to three virtual particles of very high energy. erything else of interest in QED. previous Nobel Prizes (and ours is This problem is a direct descendant of Feynman, Schwinger, and Tomonoga another). the problem that triggered the introduc- developed the technique for writing The first of these prizes went to tion of quantum theory in the first down the corrections due to interactions P. A. M. Dirac in 1933. Imagine a parti- place, i.e., the ‘‘UV catastrophe’’ of with any finite number of virtual parti- cle moving on average at very nearly the black body radiation theory, addressed cles in QED, and showed that renormal- , but with an uncertainty by Planck. There the problem was that ization theory worked in the simplest in position, as required by quantum the- high-energy modes of the electromag- cases. (I’m being a little sloppy in my ory. Evidently, there will be some prob- netic are predicted, classically, to terminology; instead of saying the num- ability for observing this particle to occur as thermal fluctuations, to such ber of virtual particles, it would be more move a little faster than average, and an extent that equilibrium at any finite proper to speak of the number of inter- therefore faster than light, which special temperature requires that there is an nal loops in a Feynman graph.) Free- relativity won’t permit. The only known infinite amount of energy in these man Dyson supplied a general proof. way to resolve this tension involves in- modes. The difficulty came from the This was intricate work that required troducing the idea of antiparticles. Very possibility of small-amplitude fluctua- new mathematical techniques. ’t Hooft roughly speaking, the required uncer- tions with rapid variations in space and and Veltman showed that renormaliza- tainty in position is accommodated by time. The element of discreteness intro- tion theory applied to a much wider allowing for the possibility that the act duced by quantum theory eliminates the class of theories, including the sort of of measurement can involve the creation possibility of very small-amplitude fluc- spontaneously broken gauge theories of several particles, each indistinguish- tuations, because it imposes a lower that had been used by Glashow, Salam, able from the original, with different bound on their size. The (relatively) and Weinberg to construct the (now) positions. To maintain the balance of large-amplitude fluctuations that remain of electroweak interac-

8404 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0501642102 Wilzcek Downloaded by guest on September 23, 2021 tions. Again, this was intricate and theory, despite substantial pragmatic subjected them to violent acceleration. highly innovative work. success, self-destructed logically due to This too can be understood from the This brilliant work, however, still did catastrophic screening. physics of antiscreening. According to not eliminate all of the difficulties. A antiscreening, the of a very profound problem was identified by Paradox Lost: Antiscreening, or quark, viewed up close, is small. It Landau (4). Landau argued that virtual Asymptotic Freedom builds up its power to drive the strong particles would tend to accumulate These paradoxes were resolved by our interaction by accumulating a growing around a real particle as long as there discovery of asymptotic freedom. We cloud at larger distances. Since the was any uncancelled influence. This is found that some very special quantum power of its intrinsic color charge is called screening. The only way for this field theories actually have antiscreen- small, the quark is actually only loosely screening process to terminate is for the ing. We called this property asymptotic attached to its cloud. We can jerk it source plus its cloud of virtual particles freedom, for reasons that will soon be away from its cloud, and it will—for a to cease to be of interest to additional clear. Before describing the specifics of short while—behave almost as if it had virtual particles. But then, in the end, the theories, I’d like to indicate in a no color charge, and no strong interac- no uncancelled influence would re- rough, general way how the phenome- tion. As the virtual particles in space main—and no interaction! Thus, all of non of antiscreening allows us to resolve respond to the altered situation, they the brilliant work in QED and more our paradoxes. rebuild a new cloud, moving along with general field theories represented, ac- Antiscreening turns Landau’s problem the quark, but this process does not in- cording to Landau, no more than a tem- on its head. In the case of screening, a volve significant radiation of energy and porary fix. You could get finite results source of influence—let us call it momentum. That, according to us, was for the effect of any particular number charge, understanding that it can repre- why you could analyze the most salient of virtual particles, but when you tried sent something quite different from aspects of the SLAC experiments—the to sum the whole thing up, to allow for —induces a canceling inclusive cross sections, which only keep the possibility of an arbitrary number of cloud of virtual particles. From a large track of overall energy-momentum virtual particles, you would get non- charge, at the center, you get a small flow—as if the quarks were free parti- sense—either infinite answers, or no observable influence far away. Anti- cles, although in fact they are strongly interaction at all. screening, or asymptotic freedom, im- interacting and ultimately confined. Landau and his school backed up this plies instead that a charge of intrinsically Thus, both of our paradoxes, nicely intuition with calculations in many small magnitude catalyzes a cloud of dovetailed, get resolved together different quantum field theories. They virtual particles that enhances its power. through antiscreening. showed, in all of the cases they calcu- I like to think of it as a thundercloud The theories that we found to display lated, that screening in fact occurred, that grows thicker and thicker as you asymptotic freedom are called non- and that it doomed any straightforward move away from the source. Abelian gauge theories, or Yang–Mills attempt to perform a complete, consis- Since the virtual particles themselves theories (5). They form a vast generali- tent calculation by adding up the contri- carry charge, this growth is a self- zation of electrodynamics. They postu- butions of more and more virtual reinforcing, runaway process. The situa- late the existence of several different particles. We can sweep this problem tion appears to be out of control. In kinds of charge, with complete symme- under the rug in QED or in electroweak particular, energy is required to build up try among them. So instead of one en- theory, because the answers including the thundercloud, and the required en- tity, ‘‘charge,’’ we have several ‘‘colors.’’ only a small finite number of virtual ergy threatens to diverge to infinity. If Also, instead of one , we have a particles provide an excellent fit to ex- that is the case, then the source could family of color gluons. periment, and we make a virtue of ne- never be produced in the first place. The color gluons themselves carry cessity by stopping there. But for the We’ve discovered a way to avoid Landau’s color charges. In this respect the strong interaction, that pragmatic ap- disease—by banishing the patients! non-Abelian theories differ from proach seemed highly questionable, be- At this point our first paradox, the electrodynamics, where the photon is cause there is no reason to expect that confinement of quarks, makes a virtue electrically neutral. Thus gluons in lots of virtual particles won’t come into of theoretical necessity. For it suggests non-Abelian theories play a much more play, when they interact strongly. that there are in fact sources—specifi- active role in the dynamics of these the- Landau thought that he had destroyed cally, quarks—that cannot exist on their ories than do photons in electrodynam- as a way of recon- own. Nevertheless, Nature teaches us, ics. Indeed, it is the effect of virtual ciling quantum mechanics and special these confined particles can play a role gluons that is responsible for antiscreen- relativity. Something would have to give. as building blocks. If we have, nearby to ing, which does not occur in QED. Either quantum mechanics or special a source particle, its antiparticle (for It became evident to us very early on relativity might ultimately fail, or else example, quark and antiquark), then the that one particular asymptotically free essentially new methods would have to catastrophic growth of the antiscreening theory was uniquely suited as a candi- be invented, beyond quantum field the- thundercloud is no longer inevitable. date to provide the theory of the strong ory, to reconcile them. Landau was not For where they overlap, the cloud of the interaction. On phenomenological displeased with this conclusion, because source can be canceled by the anticloud grounds, we wanted to have the possibil- in practice quantum field theory had not of the antisource. Quarks and anti- ity to accommodate baryons, based on been very helpful in understanding the quarks, bound together, can be accom- three quarks, as well as mesons, based strong interaction, even though a lot of modated with finite energy, although on quark and antiquark. In light of the effort had been put into it. But neither either in isolation would cause an infi- preceding discussion, this requires that he, nor anyone else, proposed a useful nite disturbance. the color charges of three different alternative. Because it was closely tied to detailed, quarks can cancel, when you add them So we had the paradox, that combin- quantitative experiments, the sharpest up. This can occur if the three colors ing quantum mechanics and special problem we needed to address was the exhaust all possibilities; so we arrived at relativity seemed to lead inevitably to paradoxical failure of quarks to radiate the gauge SU(3), with three col- quantum field theory; but quantum field when Friedman, Kendall, and Taylor ors and eight gluons. To be fair, several

Wilzcek PNAS ͉ June 14, 2005 ͉ vol. 102 ͉ no. 24 ͉ 8405 Downloaded by guest on September 23, 2021 physicists had, with various motivations, were helpful in organizing the hadrons, simple entities, whose properties are suggested the existence of a three-valued and even though Friedman, Kendall, fully defined by mathematically precise internal color label for quarks years be- and Taylor had ‘‘observed’’ them! The algorithms. fore.‡ It did not require a great leap of experimental facts wouldn’t go away, of You can even see them! Here’s a pic- imagination to see how we could adapt course, but their ultimate significance ture, which I’ll now explain. those ideas to our tight requirements. remained doubtful. Were quarks basic Asymptotic freedom is a great boon By using elaborate technical machin- particles, with simple properties that for , because it ery of quantum field theory (including could be used to in formulating a pro- leads to the beautiful phenomenon of the , operator found theory—or just a curious inter- jets. As I remarked before, an important product expansions, and appropriate mediate device that would need to be part of the atmosphere of mystery sur- dispersion relations), we were able to be replaced by deeper conceptions? rounding quarks arose from the fact that much more specific and quantitative Now we know how the story played they could not be isolated. But if we about the implications our theory than out, and it requires an act of imagina- change our focus, to follow flows of en- my loose pictorial language suggests. In tion to conceive how it might have been ergy and momentum rather than indi- particular, the strong interaction does different. But Nature is imaginative, and vidual hadrons, then quarks and gluons not simply turn off abruptly, and there so are theoretical physicists, and so it’s come into view, as I’ll now explain. is a nonzero probability that quarks will not impossible to fantasize alternative There is a contrast between two dif- radiate when poked. It is only asymptot- histories. For example, the quasiparticles ferent kinds of radiation, which expresses ically, as energies involved go to infinity, of the fractional quantum Hall effect, the essence of asymptotic freedom. Hard that the probability for radiation van- which are not basic but rather emerge as radiation, capable of significantly redi- ishes. We could calculate in great detail collective excitations involving ordinary recting the flow of energy and momen- the observable effects of the radiation at electrons, also cannot exist in isolation, tum, is rare. But soft radiation, which finite energy, and make experimental and they have fractional charge and produces additional particles moving in predictions based on these calculations. anomalous statistics! Related things the same direction without deflecting At the time, and for several years later, happen in the Skyrme model, where nu- the overall flow, is common. Indeed, the data were not accurate enough to cleons emerge as collective excitations soft radiation is associated with the test these particular predictions, but by of pions. One might have fantasized that build-up of the clouds I discussed be- the end of the 1970s they began to look good, and by now they’re beautiful. quarks would follow a similar script, fore, as it occurs in time. Let’s consider Our discovery of asymptotic freedom, emerging somehow as collective excita- what it means for experiments, say to be and its essentially unique realization in tions of hadrons, or of more fundamen- concrete the sort of experiment done at quantum field theory, led us to a new tal , or of strings. the Large Electron collider attitude toward the problem of the Together with the new attitude to- (LEP) at CERN during the 1990s, and strong interaction. In place of the broad ward the strong interaction problem, contemplated for the International Lin- research programs and fragmentary in- that I just mentioned, came a new atti- ear Collider (ILC) in the future. At sights that had characterized earlier tude toward quarks and gluons. These these facilities, one studies what emerges work, we now had a single, specific can- words were no longer just names at- from the annihilation of electrons and didate theory—a theory that could be tached to empirical patterns, or to no- that collide at high energies. tested, and perhaps falsified, but could tional building blocks within rough By well understood processes that be- not be fudged. Even now, when I reread phenomenological models. Quarks and long to QED or electroweak theory, the our declaration (2) (especially) gluons had become ideally annihilation proceeds through a virtual ‘‘Finally let us recall that the pro- posed theories appear to be uniquely singled out by nature, if one takes both the SLAC results and the renor- malization-group approach to quan- tum field theory at face value.’’ I relive the mixture of exhilaration and anxiety that I felt at the time. A Foursome of Paradigms Our resolution of the paradoxes that drove us had ramifications in unantici- pated directions, and extending far be- yond their initial scope.

Paradigm 1: The Hard Reality of Quarks and Gluons. Because, to fit the facts, you had to ascribe several bizarre properties to quarks—paradoxical dynamics, peculiar charge, and anomalous statistics—their ‘‘reality’’ was, in 1972, still very much in question. This despite the fact that they

Fig. 1. A photograph from the L3 collaboration, showing three jets emerging from electron–positron ‡An especially clear and insightful early paper, in which a annihilation at high energy. These jets are the materialization of a quark, antiquark, and . dynamical role for color was proposed, is Y. Nambu (6). (Reprinted with permission of the L3 Collaboration.)

8406 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0501642102 Wilzcek Downloaded by guest on September 23, 2021 each experiment, which may involve hundreds of independent measurements, be fit consistently, but one can then check whether the values of the cou- pling change with the energy scale in the way we predicted. As you can see, it does. A remarkable tribute to the suc- cess of the theory, which I’ve been amused to watch evolve, is that a lot of the same activity that used to be called testing QCD is now called calculating backgrounds. As a result of all this success, a new paradigm has emerged for the operational meaning of the concept of a fundamental particle. Physicists designing and interpret- ing high-energy experiments now routinely Fig. 2. These Feynman graphs are schematic representations of the fundamental processes in electron– describe their results in terms of produc- positron annihilation, as they take place in space and time. They show the origin of two-jet and three-jet ing and detecting quarks and gluons: what events. they mean, of course, is the corresponding jets. photon or Z into a quark and an many different kinds of experiments. In antiquark. Conservation and energy and most of these applications, including the Paradigm 2: Mass Comes from Energy. My momentum dictate that the quark and original one to deep inelastic scattering, friend and mentor Sam Treiman liked to antiquark will be moving at high speed the analysis necessary to separate out relate his experience of how, during in opposite directions. If there is no hard and soft radiation is much more World War II, the U.S. Army responded hard radiation, then the effect of soft involved and harder to visualize than in to the challenge of training a large num- radiation will be to convert the quark the case of electron-positron annihila- ber of radio engineers starting with very into a spray of hadrons moving in a tion. A lot of ingenuity has gone, and different levels of preparation, ranging common direction: a jet. Similarly, the continues to go, into this subject, known down to near zero. They designed a antiquark becomes a jet moving in the as perturbative QCD. The results have crash course for it, which Sam took. In opposite direction. The observed result been quite successful and gratifying. Fig. the training manual, the first chapter is then a 2-jet . Occasionally 3 shows one aspect of the success. Many was devoted to Ohm’s three laws. Ohm’s Ϸ ( 10% of the time, at LEP) there will different kinds of experiments, per- first law is V ϭ IR. Ohm’s second law is be hard radiation, with the quark (or formed at many different energies, have I ϭ V͞R. I’ll leave it to you to recon- antiquark) emitting a gluon in a signifi- been successfully described by QCD struct Ohm’s third law. cantly new direction. From that point on predictions, each in terms of the one Similarly, as a companion to Ein- the same logic applies, and we have a relevant parameter of the theory, the stein’s famous equation E ϭ mc2,we three-jet event, like the one shown in overall strength. Not only must have his second law, m ϭ E͞c2. Here, of Fig. 1. The theory of the underlying space-time process is depicted in Fig. 2. And Ϸ1% of the time, four jets will oc- cur, and so forth. The relative probabil- ity of different numbers of jets, how it varies with the overall energy, the rela- tive frequency of different angles at which the jets emerge and the total en- ergy in each—all these detailed aspects of the ‘‘antenna pattern’’ can be pre- dicted quantitatively. These predictions reflect the basic couplings among quarks and gluons, which define QCD, quite directly. The predictions agree well with very comprehensive experimental measure- ments. So we can conclude with confi- dence that QCD is right, and that what you are seeing, in Fig. 1, is a quark, an antiquark, and a gluon—although, since the predictions are statistical, we can’t say for sure which is which! By exploiting the idea that hard radia- tion processes, reflecting fundamental quark and gluon interactions, control Fig. 3. Many quite different experiments, performed at different energies, have been successfully the overall flow of energy and momen- analyzed by using QCD. Each fits a large quantity of data to a single parameter, the strong coupling ␣s.By tum in high-energy processes, one can comparing the values they report, we obtain direct confirmation that the coupling evolves as predicted. analyze and predict the behavior of (Figure courtesy S. Bethke, ref. 8.)

Wilzcek PNAS ͉ June 14, 2005 ͉ vol. 102 ͉ no. 24 ͉ 8407 Downloaded by guest on September 23, 2021 course, E denotes the energy of a body at rest, and m its mass. All this isn’t quite as silly as it may seem, because different forms of the same equation can suggest very different things. The usual way of writing the equation, E ϭ mc2, suggests the possibil- ity of obtaining large amounts of energy by converting small amounts of mass. It brings to mind the possibilities of nu- clear reactors, or bombs. Stated as m ϭ E͞c2, Einstein’s law suggests the possi- bility of explaining mass in terms of en- ergy. That is a good thing to do, be- cause in energy is a more basic concept than mass. Actually, Einstein’s original paper does not con- 2 tain the equation E ϭ mc , but rather Fig. 4. Comparison of observed masses to the energy spectrum predicted by QCD, upon direct m ϭ E͞c2. In fact, the title is a question: numerical integration of the equations, exploiting immense computer power. The small remaining ‘‘Does the Inertia of a Body Depend discrepancies are consistent with what is expected given the approximations that were necessary to make Upon its Energy Content?’’ From the the calculation practical. (Figure reprinted with permission from the Center for Computational Physics, beginning, Einstein was thinking about University of Tsukuba, Tsukuba, Japan.) the origin of mass, not about making bombs. and localization energy. This gives rise Y The calculated spectrum does not Modern QCD answers Einstein’s to mass, according to m ϭ E͞c2, even if question with a resounding ‘‘Yes!’’ In- contain anything with the charges or deed, the mass of ordinary matter de- the gluons and quarks started out with- other quantum numbers of quarks; rives almost entirely from energy—the out any nonzero mass of their own. So nor of course does it contain massless energy of massless gluons and nearly the different stable compromises will be gluons. The observed particles do not massless quarks, which are the ingredi- associated with particles that we can map in a straightforward way to the ents from which protons, neutrons, and observe, with different masses; and primary fields from which they ulti- atomic nuclei are made. metastable compromises will be associ- mately arise. Y The runaway build-up of antiscreen- ated with observable particles that have Lattice discretization of the quantum ing clouds, which I described before, finite lifetimes. field theory provides a cutoff proce- cannot continue indefinitely. The result- To determine the stable compromises dure that is independent of any ex- ing color fields would carry infinite en- concretely, and so to predict the masses pansion in the number of virtual ergy, which is not available. The color of mesons and baryons, is hard work. It particle loops. The renormalization charge that threatens to induce this run- requires difficult calculations that con- procedure must be, and is, carried out away must be cancelled. The color tinue to push the frontiers of massively without reference to perturbation the- charge of a quark can be cancelled ei- parallel processing. I find it quite ironic ory, as one takes the lattice spacing to ther with an antiquark of the opposite that, if we want to compute the mass of zero. Asymptotic freedom is crucial color (making a ), or with two a proton, we need to deploy something for this, as I discussed—it saves us 30 quarks of the complementary colors like 10 protons and neutrons, doing from Landau’s catastrophe. (making a ). In either case, per- trillions of multiplications per second, working for months, to do what one By fitting some fine details of the pat- fect cancellation would occur only if the Ϫ particles doing the canceling were lo- proton does in 10 24 seconds, namely tern of masses, one can get an estimate cated right on top of the original figure out its mass. Maybe it qualifies as of what the quark masses are and how quark—then there would be no uncan- a paradox. At the least, it suggests that much their masses are contributing to celled source of color charge anywhere there may be much more efficient ways the mass of the proton and . It in space, and hence no color field. to calculate than the ones we’re using. turns out that what I call QCD Lite— Quantum mechanics does not permit In any case, the results that emerge the version in which you put the u and d this perfect cancellation, however. The from these calculations are very gratifying. quark masses to zero, and ignore the quarks and antiquarks are described by They are displayed in Fig. 4. The ob- other quarks entirely—provides a re- wave functions, and spatial gradients in served masses of prominent mesons and markably good approximation to reality. these wave function cost energy, and so baryons are reproduced quite well, stating Since QCD Lite is a theory whose basic there is a high price to pay for localizing from an extremely tight and rigid theory. building blocks have zero mass, this re- the wave function within a small region Now is the time to notice also that one of sult quantifies and makes precise the of space. Thus, in seeking to minimize the data points in Fig. 3, the one labeled idea that most of the mass of ordinary the energy, there are two conflicting ‘‘Lattice,’’ is of a quite different character matter—90% or more—arises from pure considerations: to minimize the field from the others. It is based not on the energy, via m ϭ E͞c2. energy, you want to cancel the sources perturbative physics of hard radiation, but The calculations make beautiful im- accurately; but to minimize the wave- rather on the comparison of a direct inte- ages, if we work to put them in eye- function localization energy, you want to gration of the full equations of QCD with friendly form. Derek Leinweber has keep the sources fuzzy. The stable con- experiment, using the techniques of lattice done made some striking animations of figurations will be based on different . QCD fields as they fluctuate in empty ways of compromising between these The success of these calculations rep- space. Fig. 5 is a snapshot from one of two considerations. In each such config- resents the ultimate triumph over our his animations. Fig. 6 from Greg Kilcup, uration, there will be both field energy two paradoxes: displays the (average) color fields, over

8408 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0501642102 Wilzcek Downloaded by guest on September 23, 2021 description of the properties of ultra- high temperature matter that are rele- vant to . We can even, over an extremely limited volume of space and time, reproduce condi- tions in terrestrial laboratories. When heavy ions are caused to collide at high energy, they produce a fireball that briefly attains temperatures as high as 200 MeV. ‘‘Simple’’ may not be the word that occurs to you in describing the explosive outcome of this event, as displayed in Fig. 7, but in fact, detailed study does permit us to reconstruct as- pects of the initial fireball, and to check that it was a plasma of quarks and gluons.

Paradigm 4: Rules. Over the course of the , symmetry has been immensely fruitful as a source of insight into Nature’s basic operating principles. QCD, in particular, is con- Fig. 5. A snapshot of spontaneous quantum fluctuations in the gluon fields. For experts: what is shown is the topological charge density in a typical contribution to the functional integral, with high-frequency structed as the unique embodiment of a modes filtered out. (Image courtesy of Derek B. Leinweber, CSSM, University of Adelaide, Adelaide, huge symmetry group, local SU(3) color Australia; www.physics.adelaide.edu.au͞theory͞staff͞leinweber͞VisualQCD͞Nobel.) gauge symmetry (working together with special relativity, in the context of quan- tum field theory). As we try to discover and above the fluctuations, that are as- tion based on hadrons to a description new laws, that improve on what we sociated with a very simple hadron, the based on quark and gluon variables, and know, it seems good strategy to con- , moving through space–time. Inser- focus on quantities that, like total en- tinue to use symmetry as our guide. tion of a quark–antiquark pair, which we ergy, are not sensitive to soft radiation, This strategy has led physicists to sev- subsequently remove, produces this dis- then the treatment of the strong interac- eral compelling suggestions, which I’m turbance in the fields. tion, which was the great difficulty, be- sure you’ll be hearing more about in These pictures make it clear and tan- comes simple. We can calculate to a future years! QCD plays an important gible that the quantum vacuum is a dy- first approximate by pretending that the role in all of them—either directly, as namic medium, whose properties and quarks, antiquarks and gluons behave as their inspiration, or as an essential tool responses largely determine the behav- free particles, then add in the effects of in devising strategies for experimental ior of matter. In quantum mechanics, rare hard interactions. This makes it exploration. energies are associated with frequencies, quite practical to formulate a precise I will discuss one of these suggestions according to the Planck relation E ϭ h␯. The masses of hadrons, then, are uniquely associated to tones emitted by the dynamic medium of space when it disturbed in various ways, according to ␯ ϭ mc2͞h. [1]

We thereby discover, in the reality of masses, an algorithmic, precise Music of the Void. It is a modern embodiment of the ancients’ elusive, mystical ‘‘Music of the Spheres.’’

Paradigm 3: The Early Universe Was Simple. In 1972, the early universe seemed hope- lessly opaque. In conditions of ultra-high temperatures, as occurred close to the Big Bang singularity, one would have lots of hadrons and antihadrons, each one an extended entity that interacts strongly and in complicated ways with its neigh- bors. They’d start to overlap with one another, and thereby produce a theoreti- cally intractable mess. Fig. 6. The calculated net distribution of field energy caused by injecting and removing a quark– But asymptotic freedom renders ultra- antiquark pair. By calculating the energy in these fields and the energy in analogous fields produced by high temperatures friendly to theorists. other disturbances, we predict the masses of hadrons. In a profound sense, these fields are the hadrons. It says that if we switch from a descrip- (Figure courtesy of G. Kilcup.)

Wilzcek PNAS ͉ June 14, 2005 ͉ vol. 102 ͉ no. 24 ͉ 8409 Downloaded by guest on September 23, 2021 extended symmetry. Fortunately, asymp- totic freedom informs us that the ob- served interaction strengths at a large distance can be different from the basic strengths of the seed couplings viewed at short distance. To see whether the basic theory might have the full symme- try, we have to look inside the clouds of virtual particles, and to track the evolu- tion of the couplings. We can do this by using the same sort of calculations that underlie Fig. 3, extended to include the electroweak interactions, and extrapo- lated to much shorter distances (or equivalently, larger energy scales). It is convenient to display inverse couplings and work on a logarthmic scale, for then the evolution is (approximately) linear. When we do the calculation using only the virtual particles for which we have convincing evidence, we find that the couplings do approach each other in a promising way, although ultimately they Fig. 7. A picture of particle tracks emerging from the collision of two gold ions at high energy. The resulting fireball and its subsequent expansion recreate, on a small scale and briefly, physical conditions that last don’t quite meet. This is shown in Fig. occurred during the Big Bang. (Figure courtesy of Brookhaven National Laboratory– Collaboration.) 10 Upper. Interpreting things optimistically, we might surmise from this near-success schematically, and mention three others three different interaction symmetries, that the general idea of is on telegraphically. and five disconnected sets of particles the right track, as is our continued reli- Unified field theories. Both QCD and the (actually 15 sets, taking into account the ance on quantum field theory to calcu- standard electroweak standard model threefold repetition of families). late the evolution of couplings. After all, are founded on gauge symmetries. This We can do much better by having it is hardly shocking that extrapolation combination of theories gives a wonder- more symmetry, implemented by addi- of the equations for evolution of the fully economical and powerful account tional gluons that also change strong couplings beyond their observational of an astonishing range of phenomena. into weak colors. Then clicks foundation by many orders of magni- Just because it is so concrete and so into place quite beautifully, as displayed tude is missing some quantitatively sig- successful, this rendering of Nature can in Fig. 9. nificant ingredient. In a moment I’ll and should be closely scrutinized for its There seems to be a problem, how- mention an attractive hypothesis for aesthetic flaws and possibilities. Indeed, ever. The different interactions, as ob- what’s missing. the structure of the gauge system gives served, do not have the same overall A very general consequence of this line of thought is that an enormously powerful suggestions for its further strength, as would be required by the 15 fruitful development. Its product struc- large energy scale, of order 10 GeV or ture SU(3) ϫ SU(2) ϫ U(1), the reduc- more, emerges naturally as the scale of ibility of the representation unification. This is a profound and wel- come result. It is profound, because the (that is, the fact that the symmetry does large energy scale—which is far beyond not make connections linking all of the any energy we can access directly— fermions), and the peculiar values of the emerges from careful consideration of assigned experimental at energies more to the known particles all suggest the than 10 orders of magnitude smaller! desirability of a larger symmetry. The underlying logic that gives us this The devil is in the details, and it is leverage is a synergy of unification and not at all automatic that the superfi- asymptotic freedom, as follows. If evolu- cially complex and messy observed pat- tion of couplings is to be responsible for tern of matter will fit neatly into a sim- their observed gross inequality then, ple mathematical structure. But, to a since this evolution is only logarithmic remarkable extent, it does. in energy, it must act over a very wide Most of what we know about the range. strong, electromagnetic, and weak interac- Fig. 8. A schematic representation of the symme- The emergence of a large mass scale tions is summarized (rather schemati- try structure of the standard model. There are for unification is welcome, first, because cally!) in Fig. 8. QCD connects particles three independent symmetry transformations, un- many effects we might expect to be as- horizontally in groups of three [SU(3)], der which the known fermions fall into five inde- sociated with unification are observed to the connects particles pendent units (or fifteen, after threefold family be highly suppressed. Symmetries that vertically in groups of two [SU(2)], in the repetition). The color gauge group SU(3) of QCD ϫ ϫ acts horizontally, the weak interaction gauge unify SU(3) SU(2) U(1) will almost horizontal direction and hypercharge group SU(2) acts vertically, and the hypercharge inevitably involve wide possibilities for [U(1)] senses the little subscript numbers. U(1) acts with the relative strengths indicated by transformation among quarks, , Neither the different interactions nor the the subscripts. Right-handed do not par- and their antiparticles. These extended different particles are unified. There are ticipate in any of these symmetries. possibilities of transformation, mediated

8410 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0501642102 Wilzcek Downloaded by guest on September 23, 2021 Fig. 9. The hypothetical enlarged symmetry SO(10) [unification based on SO(10) symmetry was Fig. 10. We can test the hypothesis that the disparate coupling strengths of the different gauge first outlined in ref. 7] accommodates all of the interactions derive a common value at short distances by doing calculations to take into account the effect symmetries of the standard model, and more, into of clouds (9). These are the same sort of calculations that go into Fig. 3, but extrapolated a unified mathematical structure. The fermions, to much higher energies, or equivalently shorter distances. (Upper) Extrapolated by using known virtual including a right-handed that plays an particles. (Lower) Including also the virtual particles required by low-energy (10). important role in understanding observed neu- trino phenomena, now form an irreducible unit (neglecting family repetition). The allowed color decay. The emergence of a large mass ger when viewed with high-energy charges, both strong and weak, form a perfect scale for unification is welcome, sec- probes. In moving from the small en- match to what is observed. The phenomenologi- ondly, because it opens up possibilities ergies where we ordinarily measure to cally required hypercharges, which appear so pe- for making quantitative connections to unification energy scales, the ratio culiar in the standard model, are now theoretically the remaining GE2͞␣ ascends to values that are no determined by the color and weak charges, accord- ing to the formula displayed. in Nature: . It is notorious that longer absurdly small. gravity is absurdly feebler than the other Y If gravity is the primary force, and interactions, when they are compared special relativity and quantum me- by the corresponding gauge bosons, un- acting between fundamental particles at chanics frame the discussion, then dermine conservation laws including accessible energies. The gravitational Planck’s system of physical units, and conservation. force between proton and electron, at based on Newton’s constant G, the Violation of lepton number is closely any macroscopic distance, is about speed of light c, and Planck’s quan- 2 Ϫ40 associated with neutrino oscillations. Gmemp͞e Ϸ 10 of the electric force. tum of action h, is privileged. Dimen- Violation of baryon number is closely On the face of it, this fact poses a se- sional analysis then suggests that the associated with proton instability. In vere challenge to the idea that these value of naturally defined quantities, recent years, neutrino oscillations have forces are different manifestations of a measured in these units, should be of been observed; they correspond to mi- common source—and an even more se- order unity. But when we measure the niscule neutrino masses, indicating a vere challenge to the idea that gravity, proton mass in , we dis- Ϸ Ϫ18͌ ͞ very feeble violation of lepton number. because of its deep connection to cover mp 10 hc G. On this hy- Proton instability has not yet been ob- space–time dynamics, is the primary pothesis, it makes no sense to ask served, despite heroic efforts to do so. force. ‘‘Why is gravity so feeble?’’ Gravity, To keep these processes sufficiently By extending our consideration of the as the primary force, just is what it is. small, so as to be consistent with obser- evolution of couplings to include gravity, The right question is the one we con- vation, a high scale for unification, we can begin to meet these challenges. front here: ‘‘Why is the proton so which suppresses the occurrence of the light?’’ Given our new, profound un- transformative gauge bosons as virtual Y Whereas the evolution of gauge the- derstanding of the origin of the pro- particles, is most welcome. In fact, the ory couplings with energy is a subtle ton’s mass, which I’ve sketched for unification scale we infer from the evo- quantum-mechanical effect, the gravi- you today, we can formulate a tenta- lution of couplings is broadly consistent tational coupling evolves even classi- tive answer. The proton’s mass is set with the observed value of neutrino cally, and much more rapidly. For by the scale at which the strong cou- masses and encourages further vigorous gravity responds directly to energy pling, evolved down from its primary pursuit of the quest to observe proton momentum, and so it appears stron- value at the Planck energy, comes to

Wilzcek PNAS ͉ June 14, 2005 ͉ vol. 102 ͉ no. 24 ͉ 8411 Downloaded by guest on September 23, 2021 be of order unity. It is then that it pursuit of unified field theories, which in ence clearly distinguishes ‘‘up and becomes worthwhile to cancel off the past (and many present) incarnations has down’’ from ‘‘sideways’’ directions in our growing color fields of quarks, ab- been vague and not fruitful of testable local environment. Newton, of course, sorbing the cost of quantum localiza- consequences, has in the circle of ideas traced this asymmetry to the influence tion energy. In this way, we find, I’ve been describing here attained entirely of Earth’s gravity. In the framework of quantitatively, that the tiny value of new levels of concreteness and fecundity. electroweak theory, modern physicists the proton mass in Planck units arises .¶ As I have emphasized repeat- similarly postulate that the physical from the fact that the basic unit of edly, QCD is in a profound and literal world is described by a solution wherein color coupling strength, g2, is of order sense constructed as the embodiment of all space, throughout the currently ob- 1͞2 at the Planck scale! Thus, dimen- symmetry. There is an almost perfect served Universe, is permeated by one or sional reasoning is no longer mocked. match between the observed properties more (quantum) fields that spoil the full The apparent feebleness of gravity of quarks and gluons and the most gen- symmetry of the primary equations. results from our partiality toward the eral properties allowed by color gauge Fortunately, this hypothesis, which perspective supplied by matter made symmetry, in the framework of special might at first hearing sound quite ex- from protons and neutrons. relativity and quantum mechanics. The travagant, has testable implications. The exception is that the established symme- symmetry-breaking fields, when suitably Supersymmetry. As I mentioned a moment tries of QCD fail to forbid one sort of excited, must bring forth characteristic ago, the approach of couplings to a uni- behavior that is not observed to occur. particles: their quanta. Using the most fied value is suggested, but not accurately The established symmetries permit a economical implementation of the re- realized, if we infer their evolution by in- sort of interaction among gluons—the quired , one predicts cluding the effect of known virtual parti- so-called ␪ term—that violates the in- the existence of a remarkable new parti- cles. There is one particular proposal to variance of the equations of QCD under cle, the so-called Higgs particle. More expand the world of virtual particles, a change in the direction of time. Ex- ambitious speculations suggest that there which is well motivated on several inde- periments provide extremely severe lim- should be not just a single Higgs parti- pendent grounds. It is known as low- its on the strength of this interaction, cle, but rather a complex of related par- energy supersymmetry.§ As the name much more severe than might be ex- ticles. Low-energy supersymmetry, for suggests, supersymmetry involves expand- pected to arise accidentally. example, requires at least five Higgs ing the symmetry of the basic equations of By postulating a new symmetry, we particles. physics. This proposed expansion of sym- can explain the absence of the undesired Elucidation of the Higgs complex will metry goes in a different direction from interaction. The required symmetry is be another major task for the LHC. In the enlargement of gauge symmetry. Su- called Peccei–Quinn symmetry after the planning this endeavor, QCD and as- persymmetry makes transformations be- physicists who first proposed it. If it is ymptotic freedom play a vital supporting tween particles having the same color present, this symmetry has remarkable role. The strong interaction will be re- charges and different spins, whereas ex- consequences. It leads us to predict the sponsible for most of what occurs in col- panded gauge symmetry changes the color existence of new very light, very weakly lisions at the LHC. To discern the new charges while leaving spin untouched. Su- interacting particles, axions. (I named effects, which will be manifest only in a persymmetry expands the space–time them after a laundry detergent, since small proportion of the events, we must symmetry of special relativity. they clean up a problem with an axial understand the dominant backgrounds To implement low-energy supersym- current.) In principle, axions might be very well. Also, the production and de- metry, we must postulate the existence observed in a variety of ways, although cay of the Higgs particles themselves of a whole new world of heavy particles, none is easy. They have interesting im- usually involves quarks and gluons. To none of which has yet been observed plications for cosmology, and they are a anticipate their signatures, and eventu- directly. There is, however, a most in- leading candidate to provide cosmologi- ally to interpret the observations, we triguing indirect hint that this idea may cal . must use our understanding of how pro- be on the right track: If we include the In search of symmetry lost.ʈ It has been al- tons—the projectiles at LHC—are as- particles needed for low-energy super- most four decades since our current, sembled from quarks and gluons, and symmetry, in their virtual form, in the wonderfully successful theory of the how quarks and gluons show themselves calculation of how couplings evolve with was formulated. as jets. energy, then accurate unification is Central to that theory is the concept of achieved! This is shown in Fig. 10 spontaneously broken gauge symmetry. The Greatest Lesson Lower. According to this concept, the funda- Evidently asymptotic freedom, besides By ascending a tower of speculation, mental equations of physics have more resolving the paradoxes that originally involving now both extended gauge sym- symmetry than the actual physical world concerned us, provides a conceptual foun- metry and extended space-time symmetry, does. Although its specific use in elec- dation for several major insights into Na- we seem to break although the clouds, troweak theory involves exotic hypothet- ture’s fundamental workings, and a versa- into clarity and breathtaking vision. Is it ical substances and some sophisticated tile instrument for further investigation. an illusion, or reality? This question cre- , the underlying theme of The greatest lesson, however, is a ates a most exciting situation for the broken symmetry is quite old. It goes moral and philosophical one. It is truly (LHC), due to back at least to the dawn of modern awesome to discover, by example, that begin operating at CERN in 2007, for this physics, when Newton postulated that we humans can come to comprehend great accelerator will achieve the energies the basic laws of mechanics exhibit full Nature’s deepest principles, even when necessary to access the new world of symmetry in three of space they are hidden in remote and alien heavy particles, if it exists. How the story despite the fact that everyday experi- realms. Our minds were not created for will play out, only time will tell. But in this task, nor were appropriate tools any case, I think it is fair to say that the ready at hand. Understanding was ¶A standard review is J. Kim (12). I also recommend F. achieved through a vast international Wilczek (13). effort involving thousands of people §A standard review is H. P. Nilles (11). ʈI treat this topic more amply in ref. 14. working hard for decades, competing in

8412 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0501642102 Wilzcek Downloaded by guest on September 23, 2021 the small but cooperating in the large, love with her. Also Robert Shrock and Bill work was particularly important in leading abiding by rules of openness and hon- Caswell, my fellow graduate students, from to ours, and who have not (yet?) received a esty. Using these methods—which do whom I learned a lot, and who made our ex- Nobel Prize for it. These are Yoichiro not come to us effortlessly, but require tremely intense lifestyle seem natural and Nambu, Stephen Adler, and James Bjorken. Those heroes advanced the cause of trying to nurture and vigilance—we can accom- even fun. On the scientific side, I must of course thank David Gross above all. He understand hadronic physics by taking the plish wonders. swept me up in his drive to know and to cal- concepts of quantum field theory seriously, culate, and through both his generous guid- and embodying them in specific mechanistic Thanks. Before concluding I’d like to distrib- ance and his personal example started and models, when doing so was difficult and un- ute thanks. inspired my whole career in physics. The en- fashionable. I’d like to thank Murray Gell- First, I’d like to thank my parents, who vironment for in Princeton Mann and Gerard ’t Hooft for not quite cared for my human needs and encouraged in the 1970s was superb. There was an atmo- inventing everything, and so leaving us some- my curiosity from the beginning. They were sphere of passion for understanding, intellec- thing to do. And finally I’d like to thank children of immigrants from Poland and It- tual toughness, and inner confidence whose Mother Nature for her extraordinarily good aly, and grew up in difficult circumstances creation was a great achievement. Murph taste, which gave us such a beautiful and during the Great Depression, but managed to Goldberger, Sam Treiman, and Curt Callan powerful theory to discover. This work is supported in part by funds emerge as generous souls with an inspiring especially deserve enormous credit for this. provided by the U.S. Department of Energy admiration for science and learning. I’d like Also Sidney Coleman, who was visiting under cooperative research agreement DE- to thank the people of New York, for sup- Princeton at the time, was very actively inter- FC02-94ER40818. porting a public school system that served me ested in our work. Such interest from a physi- extremely well. I also got a superb under- cist I regarded as uniquely brilliant was A Note to Historians. graduate education at the University of Chi- I have not, here, given inspiring in itself; Sidney also asked many an extensive account of my personal experi- cago. In this connection, I’d especially like to challenging specific questions that helped us ences in discovery. In general, I don’t believe mention the inspiring influence of Peter come to grips with our results as they devel- that such accounts, composed well after the Freund, whose tremendous enthusiasm and oped. Ken Wilson had visited and lectured a fact, are reliable as history. I urge historians clarity in teaching a course on group theory little earlier, and his renormalization group of science instead to focus on the contempo- in physics was a major influence in nudging ideas were reverberating in our heads. rary documents, and especially the original me from pure mathematics toward physics. Fundamental understanding of the strong papers, which by definition accurately reflect Next I’d like to thank the people around interaction was the outcome of decades of the understanding that the authors had at the Princeton who contributed in crucial ways to research involving thousands of talented peo- time, as they could best articulate it. From the circumstances that made my development ple. I’d like to thank my fellow physicists this literature, it is I think not difficult to and major work in the 1970s possible. On the more generally. My theoretical efforts have identify where the watershed changes in atti- personal side, this includes especially my wife been inspired by, and of course informed by, tude I mentioned earlier occurred, and where Betsy Devine. I don’t think it’s any coinci- the ingenious persistence of my experimental the outstanding paradoxes of strong interac- dence that the beginning of my scientific colleagues. Thanks, and congratulations, to tion physics and quantum field theory were maturity, and a special surge of energy, hap- all. Beyond that generic thanks I’d like to resolved into modern paradigms for our un- pened at the same time as I was falling in mention specifically a trio of physicists whose derstanding of Nature.

1. Gross, D. & Wilczek, F. (1973) Phys. Rev. Lett. 30, 5. Yang, C.-N. & Mills, R. (1954) Phys. Rev. 96, 9. Georgi, H., Quinn, H. R. & Weinberg, S. (1974) 1343–1346. 191–195. Phys. Rev. Lett. 33, 451–454. 2. Gross, D. & Wilczek, F. (1973) Phys. Rev. D 8, 6. Nambu, Y. (1966) in Preludes in Theoretical Phys- 10. Dimopoulos, S., Raby, S. & Wilczek, F. (1981) 3633–3652. ics, eds. De-Shalit, A., Feshbach, H. & van Hove, Phys. Rev. D 24, 1681–1683. 3. Gross, D. & Wilczek, F. (1974) Phys. Rev. D 9, L. (North-Holland, Amsterdam), pp. 133–142. 11. Nilles, H. P. (1984) Phys. Rep. 110, 1–162. 980–993. 7. Georgi, H. (1975) in Particles and Fields 1974, ed. 12. Kim, J. (1987) Phys. Rep. 150, 1–177. 4. Landau, L. (1955) in Niels Bohr and the Develop- Carlson, C. (AIP, New York), pp. 575–582. 13. Wilczek, F. (2004) http:͞͞arxiv.org͞abs͞hep-ph͞ ment of Physics, ed. Pauli, W. (McGraw-Hill, New 8. Bethke, S. (2000) J. Phys. G 26, R27; hep-ex͞ 0408167. York), pp. 52–69. 0004021. 14. Wilczek, F. (2005) Nature 433, 239–247.

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