陕南绿茶:Julian Schwinger: Nuclear Physics, the Radiation Laboratory, Renormalized QED, Source Theory, and Be

Julian Schwinger:Nuclear Physics, the Radiation Laboratory, Renormalized QED,Source Theory, andBeyond

Kimball A. Milton Homer L. Dodge Department of Physicsand Astronomy University of Oklahoma, Norman

Julian Schwinger’sinfluence on twentieth century science is profound and pervasive. Of course, heis most famous for his renormalization theory of quantum electrodynamics, for whichhe shared the Nobel Prize with Richard Feynman and Sin-itiro Tomonaga. Butalthough this triumph was undoubtedly his most heroic work, his legacy lives onchiefly through subtle and elegant work in classical electrodynamics, quantumvariational principles, proper-time methods, quantum anomalies, dynamical massgeneration, partial symmetry, and more. Starting as just a boy, he rapidlybecame the pre-eminent nuclear

physicist in thelate 1930s, led the theoretical development of radar technology at MIT duringWorld War II, and then, soon after the war, conquered quantum electrodynamics, andbecame the leading quantum field theorist for two decades, before taking a moreiconoclastic route during his last quarter century. Keywords: Julian Schwinger,nuclear physics, waveguides, quantum electrodynamics, renormalization, quantumaction principle, source theory, axial-vector anomaly

∗K.A.Milton is Professor of Physics at the University ofOklahoma. He was a Ph.D. student

of JulianSchwinger from 1968–71, and his postdoc at UCLA for the rest of the 1970s.

He has written ascientific biography of Schwinger, edited two volumes of Schwinger’s selected works,and co-authored two textbooks based on Schwinger’s lectures.

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1 Introduction

Given JulianSchwinger’s commanding stature in theoretical physics for half a century, it mayseem puzzling why he is relatively unknown now to the educated public, even to manyyounger physicists, while Feynman is a cult figure with his photograph needingno more introduction than Einstein’s.† This relative obscurity is even moreremarkable, in view of the enormous number of eminent physicists, as well asother leaders in science and industry, who received their Ph.D.’s underSchwinger’s direction, while Feynman had

practically none.In part, the answer lies in Schwinger’s retiring nature and reserved demeanor.Science, research and teaching, were his life, and he detested the limelight.Generally, he was not close to his students, so few knew him well. He was agracious host and a good conversationalist, and had a broad knowledge of manysubjects, but he was never one to initiate a relationship or flaunt his erudition.His style of doing physics was also difficult to penetrate. Oppenheimer oncesaid that others gave talks to show others how to do the calculation, whileSchwinger gave talks to show that only he could do it. Although a commonlyshared view, this witticism is unkind and untrue. He was, in fact,

a superb teacher,and generations of physicists, students and faculty alike, learned physics athis feet. On the one hand he was a formalist, inventing formalisms so powerfulthat they could lead, at least in his hands, unerringly to the correct answer.He did not, therefore, display the intuitive visualizations, for example, thatFeynman commanded, which eventually took over the popular and scientificculture.

But, moreprofoundly, he was a phenomenologist. Ironically, even some of his own studentscriticized him in his later years for his phenomenological orientation, notrecognizing that he had, from his earliest experiences in nuclear physics,insisted in grounding theoretical physics in the phenomena and data ofexperiment. Isidor Rabi, who discovered Schwinger and brought him to ColumbiaUniversity, generally had a poor opinion of theoretical physicists. But Rabiwas always very impressed with Schwinger because in nearly every paper, he “gotthe numbers out” to compare with experiment. Even in †An example is the seriesof posters produced by the American Physical Society in which the impression isgiven that Feynman was the chief innovator in quantum electrodynamics. Incontradiction to this, Norman Ramsey has stated that “it is my impression that Schwingeroverwhelmingly deserved the greatest credit for QED. I don’t think Feynman hadan explanation of the anomalous hyperfine structure before the [1948 APS]meeting.”1

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his most elaboratefield-theoretic papers he was always concerned with making

contact with thereal world, be it quantum electrodynamics, or strongly interacting hadrons. Althoughhis first, unpublished, paper, written at the age of 16, was on the subject ofthe then poorly-understood quantum electrodynamics, Julian Schwinger was almostexclusively a nuclear physicist until he joined the Radiation Laboratory at MITin 1943. There, faced with critical deadlines and the difficulty ofcommunicating with electrical engineers, he perfected variational techniquesfor solving the classical electrodynamic problems of waveguides. As the Warwound down, he started thinking about radiation produced by electrons inbetatrons and synchrotrons, and in so doing recognized that the reactive andresistive portions of the electromagnetic mass of the electron are united in ainvariant structure. Recruited by Harvard, he started teaching there in 1946,and at first continued research in nuclear physics and in classical diffraction.The Shelter Island conference of 1947 changed all that. He and Weisskopfsuggested to Bethe that electrodynamic processes were responsible for the Lambshift, which had been known for some time as the Pasternack effect.Immediately, however, Schwinger saw that the most direct consequence of quantumelectrodynamics lay in the hyperfine anomaly

reported for thefirst time at Shelter Island. He anticipated that the effect was due to aninduced anomalous magnetic moment of the electron. The actual calculation hadto wait three months, while Schwinger took an extended honeymoon, but byDecember 1947 Schwinger had his famous result for the gyromagnetic ratio. Inthe process he invented renormalization of mass and charge, only dimlyprefigured by Kramers. This first formulation of QED was rather crude, beingnoncovariant; to obtain the correct Lamb

shift, arelativistic formulation was required, which followed the next year. A comedyof errors ensued: Both Feynman and Schwinger made an incorrect patch betweenhard and soft photon processes, and so obtained identical, but incorrect,predictions for the Lamb shift, and the weight of their reputations delayed thepublication of the correct, if pedestrian, calculation by French andWeisskopf.‡ By 1950 Schwinger had started his third reformulation of quantumelectrodynamics, in terms of the quantum action principle. At the same time hewrote his remarkable paper, “On Gauge Invariance and Vacuum

Polarization,” formulated in a completely gauge-covariantway, which ‡Schwinger later claimed that his first noncovariant approach hadyielded the correct result, except that he had not trusted it.

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anticipated manylater developments, including the axial vector anomaly. His strongphenomenological bent eventually led him away from the mainstream of physics.Although he had given the basis for what is now called the Standard Model ofelementary particles in 1957, he never could accept the existence of quarksbecause they had no independent existence outside of hadrons. (A secondaryconsideration was that quarks were invented by Murray Gell-Mann, with whom along-running feud had developed.) He came to appreciate the notion ofsupersymmetry, but he rejected notions of “Grand

Unification” andof “Superstrings” not because of their structure but because he saw them as preposterousspeculations, based on the notion that nothing new remains to be found in theenormous energy range from 1 TeV to 1019 GeV. He was sure that totally new,unexpected phenomena were waiting just around the corner. This seems areasonable view, but it resulted in a selfimposed isolation, in contrast,again, to Feynman, who contributed mightly to the theory of partons and quantumchromodynamics up to the end.

A completebiography of Julian Schwinger was published six years ago.2 The present paperdraws upon that book, as well as on later interviews and research by theauthor. Quotations of Schwinger not otherwise attributed are based on anextended interview conducted for that book by my co-author Jagdish Mehra in1988.

2 Early Years

Julian Schwinger wasborn in Manhattan, New York City, on February 12, 1918, to rather well-offmiddle-class parents. His father was a well-known designer of women’s clothes.He had a brother Harold ten years older than himself, whom Julian idolized aschild. Harold claimed that he taught Julian physics until he was 13. AlthoughJulian was recognized as intelligent in school, everyone thought Harold was thebright one. (Harold in fact eventually became a respected lawyer, and hismother always considered him as the

successful one,even after Julian received the Nobel Prize.) The Depression cost Julian’sfather his business, but he was sufficiently appreciated that he was offeredemployment by other designers; so the family survived, but not so comfortablyas before. The Depression did mean that Julian would have to rely on freeeducation, which New York well-provided in those days: A year or two atTownsend Harris High School, a public preparatory school feeding into CityCollege,

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where Julian matriculated in 1933. Julian had alreadydiscovered physics, first through Harold’s Encyclopedia Britannica at home, andthen through the remarkable institution of the New York Public Library. LarryCranberg was a student at Townsend Harris at the same time as JulianSchwinger.3 They had some classes together, and both graduated in

January 1933, witha diploma that stated that graduates were entitled to automatic entry to CCNY.He recalled that Julian was “very, very quiet. He never gave recitations. Hesat in the last row, unsmiling and unspeaking, and was a real loner. But thescuttlebutt was that he was our star. He very early showed promise,” butCranberg saw nothing overt. “Rumors were that he was not very good outside mathand physics, and that he was flunking German.” Among the teachers at TownsendHarris, Cranberg particularly remembers Alfred Bender,§ who was apparently noton the regular faculty. Eileen

Lebow, whorecently wrote a history of Townsend Harris High School,4 does not recallBender’s name. Cranberg said that Bender “fixed me on the course to be aphysicist. He was diligent, passionate, and meticulous in his recitations. Hewas a great guy, one of the best teachers at Townsend Harris.” It seems verylikely that it was Bender to whom Schwinger referred to as an anonymousinfluence: I took my first physics course in High School. That instructor showedunlimited patience in answering my endless questions

about atomicphysics, after the class period was over. Although I try, I cannot live up tothat lofty standard.5 At City College Julian was reading and digesting thelatest papers from Europe, and starting to write papers with instructors whowere, at the same time, graduate students at Columbia and NYU. Joseph Weinberg,who went on to become a well-known relativist, was his closest friend at CityCollege. Weinberg recalled his first meeting with Julian.6 Because of hisoutstanding laboratory reports, Weinberg had been granted the privilege ofentering the closed library stacks at City College. One day he was seeking amathematics book,7 which had been mentioned at the Math Club the day before,and while he reached for it, another youngster was trying to get it. They hadboth heard the talk, on functions which are continuous but nowheredifferentiable, and

§Bender was thefather of physicist Carl Bender. Carl’s uncle Abram Bader was the physicsteacher of Richard Feynman at Far Rockaway High School.

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so they shared thebook between them, balancing the heavy volume on one knee each. The otherfellow kept finishing the page beforeWeinberg, who was a very fast reader. Ofcourse, his impatient co-reader was Julian Schwinger. Both were 15. Weinbergmentioned that he usually spent his time, not in the mathematics section of thelibrary, but in the physics section, which turned out to be Julian’s base aswell. Weinberg complained that Dirac’s book on quantum mechanics8 was veryinteresting and exciting, but difficult to follow. Julian concurred, and saidit was because it was polished too highly; he said

that Dirac’soriginal papers were much more accessible. Weinberg had never conceived ofconsulting the original literature, so this opened a door for him. This adviceabout over-refinement Schwinger himself forgot to follow in later life. Julianno longer had the time to spend in the classroom attending lectures. In physicsand mathematics he was able to skim the texts and work out the problems fromfirst principles, frequently leaving the professors baffled with his original,unorthodox solutions, but it was not so simple in history, English, and German.City College had an enormous number of required courses then in all subjects.His marks were not good, and he would have flunked out if the College had notalso had a rather forgiving policy toward grades. JoeWeinberg recalled anothervivid incident. Among the required courses were two years of gymnasium. One hadto pass exams in hurdling, chinning, parallel bars, and swimming. BecauseWeinberg and Julian had nearby lockers, they often fell into physicsconversations half dressed, and failed the class for lack of attendance.Weinberg remembered seeing Julian’s hurdling exam. Julian ran up to the bar,but came to a standstill when he was supposed to

jump oversideways. The instructor reprimanded him, at which point Julian said, sottovoce, “there’s not enough time to solve the equations of motion.” EdwardGerjuoy was another of Julian’s classmates at City College.9 “My main claim tofame is that Julian and I took the same course in mechanics together, taught bya man named Shea, and I got an A and Julian a B,” because Julian did not do thework. “It took about a week before the people in the class realized we weredealing with somebody of a different order

of magnitude.” Ata time when knowledge of a bit of vector algebra was considered commendable,“Julian could make integrals vanish—he was very, very impressive. The onlyperson in the classroom who didn’t understand this about Julian was theinstructor himself.” “He was flunking out of City College in everything exceptmath and physics. He was a phenomenon. He

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didn’t lead theconventional life of a high school student before he came to City College”—unlikeGerjuoy and Sidney Borowitz he was not on the math team in high school so theyhad not known him earlier—“when he appeared he was just a phenomenon.” MortonHamermesh recalled another disastrous course.10 We were in a class calledModern Geometry. It was taught by an old dodderer named Fredrick B. Reynolds.He was head of the math department. He really knew absolutely nothing. It

was amazing. Buthe taught this course on Modern Geometry. It was a course in projectivegeometry from a miserable book by a man named Graustein from Princeton, andJulian was in the class, but it was very strange because he obviously never

could get toclass, at least not very often, and he didn’t own the book. That was clear. Andevery once in a while, he’d grab me before class and ask me to show him my copyof this book and he would skim through it fast and see what was going on. And

this fellowReynolds, although he was a dodderer, was a very mean character.¶ He used tosend people up to the board to do a problem and he was always sending Julian tothe board to do problems because he knew he’d never seen the course and Julian

would get up atthe board, and—of course, projective geometry is a very strange subject. Theproblems are trivial if you think about them pictorially, but Julian neverwould do them this way. He would insist on doing them algebraically and so he’dget up at

the board at thebeginning of the hour and he’d work through the whole hour and he’d finish thething and by that time the course was over and anyway, Reynolds didn’tunderstand the proof, and that would end it for the day.

Sidney Borowitz,another classmate of Julian’s, recalled that “we had the pleasure of seeingJulian attack a problem de novo, and this used to drive Reynolds crazy.”12

¶In addition, hewas also apparently a notorious antisemite. He used to discourage Jewishstudents from studying mathematics, which worked to the advantage of physics.11

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3 Paper NumberZero

Not only wasJulian already reading the literature at City College, he quickly started to dooriginal research. Julian had studied a paper by Christian Møller13 in which hehad calculated the two-electron scattering cross section by using a retardedinteraction potential. Of course, Schwinger read all of Dirac’s papers onquantum field theory, and was particularly impressed by the one on“Relativistic Quantum Mechanics,”14 “in which Dirac went through his attempt torecreate an electrodynamics in which the particles and light were treateddifferently.” In a paper of Dirac, Fock, and Podolsky,15

it was recognizedthat this was simply a unitary transformation of the Heisenberg-Pauli theory,16in which the unitary transformation was applied to the electromagnetic field.And I said to myself, ‘Why don’t we apply a similar unitary transformation tothe second-quantized electron field?’ I did that and worked out the lowestapproximation to the scattering amplitudes in unrelativistic notation. It was arelativistic theory but it was not

covariant. Thatwas in 1934, and I would use it later; [the notion, called the interactionrepresentation,] is always ascribed to Tomonaga, but I had done it muchearlier.

In deriving hisresult, Schwinger had to omit a term which “represents the infinite self-energyof the charges and must be discarded.” This he eventually came to see as amistake: “The last injunction merely parrots the wisdom of my elders, to belater rejected, that the theory was fatally flawed, as witnessed by suchinfinite terms, which at best, had to be discarded, or subtracted. Thus, the‘subtraction physics’ of the 1930s.”17

This paper wasnever submitted to a journal, but was recently published in a selection ofSchwinger’s works.18

4 ColumbiaUniversity

It was Lloyd Motz,one the instructors at City College, who had learned about Julian from Harold,and with whom Julian was writing two papers, who introduced him to Isidor I.Rabi. Then, in a conversation between Rabi and Motz over the famous Einstein,Podolsky, and Rosen paper,19 which had just appeared, Julian’s voice appearedwith the resolution of a difficulty through

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the completenessprinciple, and Schwinger’s career was assured. Rabi, not without somedifficulty, had Schwinger transferred to Columbia, and by 1937 he had 7 paperspublished, mostly on nuclear physics, which constituted his Ph.D. thesis, eventhough his bachelor’s degree had not yet been granted. Schwinger still wasderelict in attending classes, and ran into trouble in a chemistry coursetaught by Victor LaMer. It was a dull course with a dull exam. A question onthe final exam was “Prove that d? = dξ + dη,” where none of the variables ?, ξ,or η were defined. Rabi recalled,20 LaMer was, for a chemist, awfully good. Agreat part of his lifework was testing the Debye-H¨uckel theory21 ratherbrilliantly. But he was this rigid, reactionary type. He had this mean way abouthim. He said, ‘You have this Schwinger? He didn’t pass my final exam.’ I said,‘He didn’t? I’ll look into it.’ So I spoke to a number of people who’d takenthe same course. And they had been greatly assisted in that subject by Julian.So I said, I’ll fix that guy. We’ll see what character he has. ‘Now Vicky, whatsort of guy are you anyway, what are your principles? What’re you going to doabout this?’ Well, he did flunk Julian, and I think it’s quite a badge ofdistinction for him, and I for one am not sorry at this point, they have thisblack mark on Julian’s rather elevated record. But he did get Phi Beta Kappa asan undergraduate, something I never managed to do.The papers which Julian wroteat Columbia were on both theoretical and experimental physics, and Rabi prizedJulian’s ability to “get the numbers out” to compare with experiment. Theformal awarding of the Ph.D. had to wait till 1939 to satisfy a Universityregulation. In the meantime,Schwinger was busy writing papers (one, forexample, was fundamental for the theory of nuclear magnetic resonance,22) andspent a somewhat lonely,but productive winter (1937) in Wisconsin,k where helaid the groundwork for his prediction that the deuteron had an electricquadrupole moment,23 independently established by his experimental colleaguesat Columbia a year later.24 Wisconsin confirmed his predilection for working atnight, so as not to be “overwhelmed” by his hosts, Eugene Wigner and GregoryBreit.Rabi later amusingly summarized Schwinger’s year in Wisconsin.20 kIt wasa cold winter as well, for he failed to unpack the trunk in which his mother hadplaced a warm winter coat.

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I thought that hehad about had everything in Columbia that we could offer—by we, as theoreticalphysics is concerned, [I mean] me. So I got him this fellowship to go toWisconsin, with the general idea that there were Breit and Wigner and theycould

carry on. It was adisastrous idea in one respect, because, before then, Julian was a regular guy.Present in the daytime. So I’d ask Julian (I’d see him from time to time) ‘Howare you doing?’‘Oh, fine, fine.’ ‘Getting anything out of Breit and Wigner?’

‘Oh yes, they’re very good, very good.’ I asked them.They said, ‘We never see him.’ And this is my own theory—I’ve never checked itwith Julian—that—there’s one thing about Julian you all know—I think he’s aneven more quiet man than Dirac. He is

not a fighter inany way. And I imagine his ideas and Wigner’s and Breit’s or theirpersonalities did not agree. I don’t fault him for this, but he’s such a gentlesoul, he avoided the battle by working at night. He got this idea of workingnights—it’s pure

theory, it hasnothing to do with the truth. But the theory seems validated.

5 Two Years in Berkeley By 1939, Rabi felt Schwinger hadoutgrown Columbia, so with a NRC Fellowship,he was sent to J. RobertOppenheimer in Berkeley. This exposed

him to new fields:quantum electrodynamics (although as we recall, he had written an early,unpublished paper on the subject while just 16) and cosmicray physics, but hemostly continued to work on nuclear physics. He had a number of collaborations;the most remarkable was with William Rarita,who was on sabbatical from BrooklynCollege; Rarita was Schwinger’s “calculatingarm”∗∗ on a series ofpapers extending the notion of nuclear tensor forces which he had conceived inWisconsin over a year earlier. Rarita and Schwinger also wrote the prescientpaper on spin-3/2 particles,25 which was to be influential decades later withthe birth of supergravity. Ed Gerjuoy, who had been an undergraduate withSchwinger at City College in 1934, now was one of Oppenheimer’s graduatestudents. He recalled9

∗∗Left-leaning Joe Weinberg accused Julian of exploitingRarita, but Julian responded

that these papersestablished Rarita’s reputation.

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an amusingincident which happened one day while he, Schwinger, and Oppenheimer weretalking in Oppenheimer’s office in LeConte Hall. Two other students, ChaimRichman and Bernard Peters, came in seeking a suggestion for a research problemfrom Oppenheimer. Schwinger listened with interest while Oppenheimer proposedcalculating the cross section for the electron disintegration of the deuteron.That midnight, when Gerjuoy came to pick up Schwinger for the latter’sbreakfast before their all-night work session, he noted that Schwinger, whilewaiting for him in the lobby of the International

House, whereJulian was living, had filled the backs of several telegram blanks withcalculations on this problem. Schwinger stuffed the sheets in his pocket andthey went to work. Six months later, again Gerjuoy and Schwinger were inOppenheimer’s office when Richman and Peters returned beaming.They had solvedthe problem, and they covered the whole board with the elaborate solution.Oppenheimer looked at it, said it looked reasonable, and then asked, “Julie,didn’t you tell me you worked this cross section out?”

Schwinger pulledthe yellowed, crumpled blanks from his pocket, stared at them a moment, andthen pronounced the students’ solution okay apart from a factor of two.Oppenheimer told them to find their error, and they shuffled out, dispirited.Indeed, Schwinger was right, they found they had made a mistake, and theypublished the paper,26 but they were sufficiently crushed that both switched toexperimental physics.

At the time, Schwinger and Gerjuoy were collaborating ona paper27 following from Schwinger’s tensor theory of nuclear forces. The work involvedcalculating about 200 fairly complicated spin sums, which sums Julian and Iperformed independently and then compared.To have the privilege of working withJulian meant I had to accommodate

myself to hisworking habits, as follows. Except on days when Julian had to come into theuniversity during conventional hours to confer with Oppenheimer, I would meethim at

11:45 pm in the lobby of his residence, the BerkeleyInternational House. He would then drive us to some Berkeley all-night bistro wherehe had breakfast, after which we drove to LeConte Hall,the Berkeley physicsbuilding, where we worked until about 4:00 am, Julian’s lunchtime. After lunchit was back to LeConte Hall until about 8:30 am, when Julian was ready to thinkabout dinner and poor TA me would meet my 9:00 am recitation class. Since I hadjust gotten married, and still was young enough to badly

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need my sleep,these months of working with Julian were trying, to put it mildly. What made iteven more trying is the fact that when Julian and I carefully worked outtogether the 20 or so spin sums where our independent calculations disagreed,Julian proved to be right every time! I accepted the fact that Julian was amuch better theorist than I, but I felt I was at least pretty good, and was infuriatedby his apparent constitutional inability to make a single error in 200complicated spin sum calculations. This inability stood Schwinger well when heembarked on the calculations that earned him the Nobel Prize. . . . [Al]thoughJulian certainly realized how extraordinarily talented he was, he never gloatedabout his error free calculations or in any other way put me down.28

The year of theNRC Fellowship was followed by a second year at Berkeley as Oppenheimer’sassistant. They wrote an important paper together which would prove crucialnearly a decade later: Although Oppenheimer was happy to imagine newinteractions, Schwinger showed that an anomaly in fluorine decay could beexplained by the existence of vacuum polarization, that is, by the virtualcreation of electron-positron pairs.29 This gave Schwinger a head start overFeynman, who for years suspected that vacuum polarization did not exist.

6 The War and the Radiation Laboratory

After two years atBerkeley, Oppenheimer and Rabi arranged a real job for Schwinger: He becamefirst an instructor, then an Assistant Professor at Purdue University, whichhad acquired a number of bright young physicists under the leadership of KarlLark-Horowitz. But the war was impinging on everyone’s lives, and Schwinger wassoon recruited into the work on radar. The move to the MIT Radiation Laboratorytook place in 1943. There

Schwinger rapidlybecame the theoretical leader, even though he was seldom seen, going home inthe morning just as others were arriving. He developed powerful variational methodsfor dealing with complicated microwave circuits, expressing results in terms ofquantities the engineers could understand, such as impedance and admittance.

At first it seems strange that Schwinger, by 1943 theleading nuclear theorist, should not have gone to Los Alamos, where nearly allhis colleagues

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eventually settledfor the duration. There seem to be at least three reasons why Schwinger stayedat the Radiation Laboratory throughout the war.

• The reason hemost often cited later in life was one of moral repugnance.

When he realized the destructive power of what was beingconstructed at Los Alamos, he wanted no part of it. In contrast, the radiation labwas developing a primarily defensive technology, radar, which had already savedBritain.

• He believed thatthe problems to solve at the Radiation Laboratory were more interesting. Bothlaboratories were involved largely in engineering, yet although Maxwell’sequations were certainly well known, the process of applying them to waveguidesrequired the development of special techniques that would prove invaluable toSchwinger’s later

career.

• Another factorprobably was Schwinger’s fear of being overwhelmed. In Cambridge he could livehis own life, working at night when no one was around the lab. This privacywould have been much more difficult to maintain in the microworld of LosAlamos. Similarly, the working conditions at the Rad Lab were much freer thanthose at Los Alamos. Schwinger never was comfortable in a team setting, aswitness his later

aversion to theatmosphere at the Institute for Advanced Study. The methods and the discoverieshe made at the Rad Lab concerning the reality of the electromagnetic mass wouldbe invaluable for his work on quantum electrodynamics a few years later. As thewar wound down, physicists started thinking about new accelerators, since thepre-war cyclotrons had been defeated by relativity, and Schwinger became aleader in this development: He proposed a microtron,†† an accelerator based onacceleration

through microwavecavities, developed the theory of stability of synchrotron orbits, and mostimportantly, worked out in detail the theory of synchrotron radiation,‡‡ at atime when many thought that such radiation would be negligible because ofdestructive interference. Schwinger never properly wrote up the work heconducted in his one and one-half years at the Rad Lab, an omission that hasnow be rectified in part by publication of a volume based ††The microtron isusually attributed to Veksler. ‡‡This was first circulated as a preprint in1945. The paper30 published in 1949 was substantially different.

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on his lecturesthen and later, and including both published and unpublished papers.31 Althoughhe never really published his considerations on self-reaction, he viewed thatunderstanding as the most important part of his work on synchrotron radiation:

It was a usefulthing for me for what was to come later in electrodynamics, because thetechnique I used for calculating the electron’s classical radiation was one ofself-reaction, and I did it relativistically, and it was a situation in which Ihad to take seriously

the part of theself-reaction which was radiation, so why not take seriously the part of theself-reaction that is mass change? In other words, the ideas of massrenormalization and relativistically handling them were already present at thisclassical level.

Just after theTrinity atomic bomb test, Schwinger traveled to Los Alamos to speak about hiswork on waveguides, electromagnetic radiation, and his ideas about futureaccelerators. There he met Richard Feynman for the first

time. Feynmanrecalled that at the time Schwinger33 had already a great reputation because hehad done so much work . . . and I was very anxious to see what this man waslike. I’d always thought he was much older than I was [they were the

same age] becausehe had done so much more. At the time I hadn’t done anything.

7 QED

In 1945 Harvardoffered Schwinger an Associate Professorship,∗ which he promptlyaccepted, partly because in the meantime he had met his future wife ClariceCarrol. Counteroffers quickly appeared, from Columbia, Berkeley, and elsewhere,and Harvard shortly made Schwinger the youngest full professor on the facultyto that date. There Schwinger quickly established a pattern that was to persistfor many years—he taught brilliant courses on classical electrodynamics,nuclear physics, and quantum mechanics, surrounded himself with a devotedcoterie of graduate students and post-doctoral assistants, and conductedincisive research that set the tone for theoretical physics

throughout theworld.

∗He beat out Hans Bethe for the job.

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Work on classicaldiffraction theory, begun at the Radiation Lab, continued for several yearslargely due to the presence of Harold Levine, whom Schwinger had brought alongas an assistant. Variational methods, perfected in the electrodynamic waveguidecontext, were rapidly applied to problems in nuclear physics. But these wereold problems, and it was quantum electrodynamics that was to define Schwinger’scareer. But it took new experimental data to catalyze this development. That datawas presented at the famous Shelter Island meeting held in June 1947, a weekbefore Schwinger’s wedding. There he, Feynman, Victor Weisskopf, Hans Bethe,and the other participants learned the details of the new experiments of Lamband Retherford32 that confirmed the pre-war Pasternack

effect, showing asplitting between the 2S1/2 and 2P1/2 states of hydrogen, that should bedegenerate according to Dirac’s theory. In fact, on the way to the conference,Weisskopf and Schwinger speculated that quantum electrodynamics could explainthis effect, and outlined the idea to Bethe there, who worked out the details,nonrelativistically, on his famous train ride to Schenectady after themeeting.34

But the otherexperiment announced there was unexpected: This was the experiment by Rabi’sgroup and others35 of the hyperfine anomaly that would prove to mark the existenceof an anomalous magnetic moment of the electron, expressing the coupling of thespin of the electron to an applied magnetic field, deviating from the valueagain predicted by Dirac. Schwinger immediately saw this as the crucialcalculation to carry out first, because it was purely relativistic, and muchcleaner to understand theoretically, not involving the complication of boundstates. However, he was delayed three months

in beginning thecalculation because of an extended honeymoon through the West. He did return toit in October, and by December 1947 had obtained a result36 completelyconsistent with experiment. He also saw how to compute the relativistic Lambshift (although he did not have the details quite right), and found the errorin the pre-war Dancoff calculation of the radiative correction to electronscattering by a Coulomb field.37 In effect, he had solved all the fundamentalproblems that had plagued quantum electrodynamics in the 1930s: The infinitieswere entirely isolated in quantities which renormalized the mass and charge ofthe electron. Further progress, by himself and

others, was thus amatter of technique. Concerning Schwinger’s technique at the time, Schweberwrites38 The notes of Schwinger’s calculation [of the Lamb shift] are extant

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[and] give proof of his awesome computational powers. . ..These traces over photon polarizations and integrations over photon energies .. .were carried out fearlessly and seemingly effortlessly. . . .Often, involvedsteps were carried out mentally and the answer

was written down.And, most important, the lengthy calculations are error free!

8 Covariant Quantum Electrodynamics

During the next two years Schwinger developed two newapproaches to quantum electrodynamics. His original approach, which made use ofsuccessive canonical transformations, while sufficient for calculating theanomalous magnetic moment of the electron, was noncovariant, and as such, led toinconsistent results. In particular, the magnetic moment appeared also as partof the Lamb shift calculation, through the coupling with the electric fieldimplied by relativistic covariance; but the noncovariant scheme gave the wrongcoefficient. (If the coefficient were modified by hand to the correct

value, what turnedout to be the correct relativistic value for the Lamb shift emerged, but whatthat was was unknown in January 1948, when he announced his results at theAmerican Physical Society meeting.) Norman Ramsey added an amusing footnote tothe story about LaMer, the chemist who flunked Julian.39 In 1948 Schwinger hadto repeat his brilliant lecture on quantum electrodynamics three times at theAmerican Physical

Society meeting at Columbia, in successively largerrooms.† It was a superb lecture. We were impressed. And as we walked backtogether—Rabi and I were sitting together during the lecture —Rabi invited meto the Columbia Faculty Club for lunch. We

got in theelevator [in the Faculty Club] when who should happen to walk in the elevatorwith us but LaMer. And as soon as Rabi saw that, a mischievous gleam came intohis eye and he began by saying that was the most sensational thing that’s everhappened

in the AmericanPhysical Society. The first time there’s been †K. K. Darrow, secretary of thePhysical Society, who apparently had little appreciation of theory, alwaysscheduled the theoretical sessions in the smallest room. Schwinger’s secondlecture was given in the largest lecture hall in Pupin Lab, and the third inthe largest theatre on campus.

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this threerepeats—it’s a marvelous revolution that’s been done— LaMer got more and moreinterested and finally said, ‘Who did this marvelous thing?’ And Rabi said,‘Oh, you know him, you gave him an F, Julian Schwinger.’ So first at the PoconoConference in April 1948, then in the Michigan Summer School that year, andfinally in a series of three monumental papers, “Quantum Electrodynamics I, II,and III,”40 Julian set forth his covariant approach to QED. At about the sametime Feynman was formulating his covariant path-integral approach; and althoughhis presentation at Pocono was not well-received, Feynman and Schwingercompared notes and realized that they had climbed the same mountain bydifferent routes. Feynman’s systematic papers41 were published only after Dysonhad proved the equivalence of Schwinger’s and Feynman’s schemes.42 It is worthremarking that Schwinger’s approach was conservative. He

took field theoryat face value, and followed the conventional path of Pauli, Heisenberg, andDirac. His genius was to recognize that the well-known divergences of thetheory, which had stymied all pre-war progress, could be consistently isolatedin renormalization of charge and mass. This bore a superficial resemblance tothe ideas of Kramers advocated as early as 1938,43 but Kramers proceededclassically. He had insisted that first the classical theory had to be renderedfinite and then quantized. That idea was a blind alley. Renormalization ofquantum field theory is unquestionably the discovery

of Schwinger. Feynmanwas more interested in finding an alternative to field theory,

eliminatingentirely the photon field in favor of action at a distance. He was, by 1950,quite disappointed to realize that his approach was entirely equivalent to theconventional electrodynamics, in which electron and photon fields are treatedon the same footing.

As early asJanuary 1948, when Schwinger was expounding his noncovariant QED to overflow crowdsat the American Physical Society meeting at Columbia University, he learnedfrom Oppenheimer of the existence of the work of Tomonaga carried out in Tokyoduring the terrible conditions of wartime. Tomonaga had independently inventedthe “Interaction Representation” which Schwinger had used in his unpublished1934 paper, and had come up with a covariant version of the Schr¨odingerequation as had Schwinger, which upon its Western rediscovery was dubbed theTomonaga- Schwinger equation.44 Both Schwinger and Tomonaga independently wrote

17

the same equation,a generalization of the Schr¨odinger equation to an arbitrary spacelikesurface, using nearly the same notation. The formalism found by Tomonaga andhis school was essentially identical to that developed by Schwinger five yearslater; yet they at the time calculated nothing, nor did they discoverrenormalization. That was certainly no reflection on the ability of theJapanese; Schwinger could not have carried the formalism to its logical conclusionwithout the impetus of the postwar experiments, which overcame

prewar paralysisby showing that the quantum corrections “were neither infinite nor zero, butfinite and small, and demanded understanding.”17 However, at first Schwinger’s covariantcalculation of the Lamb shift contained another error, the same as Feynman’s.45By this time I had forgotten the number I had gotten by just artificiallychanging the wrong spin-orbit coupling. Because I was now thoroughly involvedwith the covariant calculation and it was the covariant calculation thatbetrayed me, because something went wrong there as well. That was a human errorof stupidity.

French andWeisskopf46 had gotten the right answer, because they put in the correct valueof the magnetic moment and used it all the way through. I, at an earlier stage,had done that, in effect, and also got the same answer. But now he and Feynman“fell into the same trap. We were connecting a relativistic calculation of highenergy effects with a nonrelativistic calculation of low energy effects, a laBethe.” Based on the result Schwinger had presented at the APS meeting inJanuary 1948, Schwinger claimed priority

for the Lamb shiftcalculation: I had the answer in December of 1947. If you look at those [other]papers you will find that on the critical issue of the spin-orbit coupling,they appeal to the magnetic moment. The deficiency in the calculation I did [in1947] was [that it was] a non-covariant calculation. French and Weisskopf werecertainly doing a noncovariant

calculation.Willis Lamb47 was doing a non-covariant calculation. They could not possiblyhave avoided these same problems. The error Feynman and Schwinger made had todo with the infrared problem that occurred in the relativistic calculation,which was handled by

giving the photona fictitious mass.

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Nobody thoughtthat if you give the photon a finite mass it will also affect the low energyproblem. There are no longer the two transverse degrees of freedom of amassless photon, there’s also a longitudinal degree of freedom. I suddenlyrealized this absolutely

stupid error, thata photon of finite mass is a spin one particle, not a helicity one particle.

Feynman was moreforthright and apologetic in acknowledging45 his error which substantiallydelayed the publication of the French and Weisskopf paper.

9 Quantum ActionPrinciple

Schwinger learnedfrom his competitors, particularly Feynman and Dyson. Just as Feynman hadborrowed the idea that henceforward would go by the name of Feynman parametersfrom Schwinger, Schwinger recognized that the systematic approach of Dyson andFeynman was superior in higher orders. So by 1949 he replaced theTomonaga-Schwinger approach by a much more powerful engine, the quantum actionprinciple. This was a logical outgrowth of the formulation of Dirac,48 as wasFeynman’s path integrals; the latter was an integral approach, Schwinger’s adifferential. The formal solution of Schwinger’s differential equations wasFeynman’s functional integral; yet

while the latterwas ill-defined, the former could be given a precise meaning, and for example,required the introduction of fermionic variables, which initially gave Feynmansome difficulty. It may be fair to say that while the path integral formulationto quantum

field theoryreceives all the press, the most precise exegesis of field theory is providedby the functional differential equations of Schwinger resulting from his actionprinciple. He began in the “Theory of Quantized Fields I”49 by introducing acomplete set of eigenvectors “specified by a spacelike surface . . . and theeigenvalues . . . of a complete set of commuting operators constructed fromfield quantities attached to that surface.” The question is how to compute thetransformation function from one spacelike surface to another. After remarkingthat this development, time-evolution, must be described by

a unitarytransformation, he assumed that any infinitesimal change in the transformationfunction must be given in terms of the infinitesimal change in a quantum actionoperator, or of a quantum Lagrange function. This is the quantum dynamicalprinciple, a generalization of the principle of least

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action, or ofHamilton’s principle in classical mechanics. If the parameters of the systemare not altered, the only changes arise from those of the initial and finalstates, from which Schwinger deduced the Principle of Stationary Action, fromwhich the field equations may be derived. A series of six papers followed withthe same title, and the most important “Green’s Functions of Quantized Fields”published in the Proceedings of the National Academy.50 Paul Martin presentedan entertaining account of the prehistory of their

work together.51 Duringthe late 1940s and early 1950s Harvard was the home

of a school ofphysics with a special outlook and a distinctive set of rituals. Somewhatbefore noon three times each week, the master would arrive in his blue chariotand, in forceful and beautiful lectures, reveal profound truths to hisCantabridgian

followers, Harvardand M.I.T. students and faculty.‡ Cast in a language more powerful and generalthan any of his listeners had ever encountered, these ceremonial gatherings hadsome sacrificial overtones—interruptions were discouraged and since the sermons

usually lastedpast the lunch hour, fasting was often required. Following a mid-afternoonbreak, private audiences with the master were permitted and, in uncertainanticipation, students would gather in long lines to seek counsel. During thisperiod the religion had its own golden rule—the action principle—and its owncryptic testament—‘On the Green’s

Functions of Quantized Fields.’50 Mastery of this paperconferred on followers a high priest status.§ The testament was couched in termsthat could not be questioned, in a language whose elements ‡In a laterrecollection,52 Martin elaborated: “Speaking eloquently, without notes, and writingwith both hands, he expressed what was already known in new, unified ways, incorporatingoriginal examples and results almost every day. Interrupting the flow with questionswas like interrupting a theatrical performance. The lectures continued through Harvard’sreading period and then the examination period. In one course we attended, he presentedthe last lecture—a novel calculation of the Lamb Shift—during Commencement Week.The audience continued coming and he continued lecturing.” §Schwinger evidentlywas aware of the mystique. In a later letter recommending Martin for apermanent appointment at Harvard he stated that Martin was “superior inintrinsic ability and performance. Quantum field theory is the new religion ofphysics, and Paul C. Martin is one of its high priests.”5 However, as the lastparagraph of the present essay demonstrates, Schwinger throughout his lifemaintained a tension between an elitist and a democratic view of science.

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were the values ofreal physical observables and their correlations.The language was enlightening,but the lectures were exciting because they were more than metaphysical. Alongwith structural insights, succinct and implicit self-consistent methods forgenerating

true statementswere revealed. Recently, a perceptive analysis of Schwinger’s Green’s functionspapers has been given by Schweber53. There he concludes that

Schwinger’sformulation of relativistic QFTs [quantum field theories] in terms of Green’sfunctions was a major advance in theoretical physics. It was a representationin terms of elements (the Green’s functions) that were intimately related toreal physical

observables andtheir correlation. It gave deep structural insights into QFTs; in particular,it allowed the investigation of the structure of the Green’s functions whentheir variables are analytically continued to complex values, thus establishingdeep connections

to statisticalmechanics. 10 “Gauge Invariance and Vacuum Polarization” The paper “On GaugeInvariance and Vacuum Polarization”54, submitted by Schwinger to the PhysicalReview near the end of December 1950, is nearly universally acclaimed as hisgreatest publication. As his lectures have rightfully been compared to theworks of Mozart, so this might be compared to a mighty construction ofBeethoven, the 3rd Symphony, the Eroica, perhaps. It is most remarkable becauseit stands in splendid isolation. It was written over a year after the last of hisseries of papers on his second, covariant, formulation of quantumelectrodynamics was completed: “Quantum Electrodynamics III. TheElectromagnetic Properties of the Electron—Radiative Corrections toScattering”40 was submitted in May 1949. And barely two months later, in March1951, Schwinger would submit the first of the series on his third reformulationof quantum field theory, that based on the

quantum actionprinciple, namely, “The Theory of Quantized Fields I.”49 But “Gauge Invarianceand Vacuum Polarization” stands on its own, and has endued the rapid changes intastes and developments in quantum field

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theory, while thepapers in the other series are mostly of historical interest now. As LowellBrown55 pointed out, “Gauge Invariance and Vacuum Polarization” still has overone hundred citations per year, and is far and away Schwinger’s most citedpaper.¶ Yet even such a masterpiece was not without its critics. Abraham Klein,who was finishing his thesis at the time under Schwinger’s direction, and wouldgo on to be one of Schwinger’s second set of “assistants” (with RobertKarplus), as, first, an instructor, and then a Junior Fellow, recalled thatSchwinger (and, independently, he and Karplus) ran afoul of a temporary editorat the Physical Review. That editor thought Schwinger’s original paper repeatedtoo many complicated expressions and that symbols should be introduced torepresent expressions that appeared more than once. Schwinger complied, but hadhis assistants do the dirty work. Harold Levine, who was still sharingSchwinger’s office, working on

thenever-to-be-completed waveguide book, typed the revised manuscript, while Kleinwrote in the many equations. Klein recalled that he took much more care inwriting those equations than he did in his own papers.57 Schwinger recalledlater that he viewed this paper, in part, as a reaction to the “invariantregularization” of Pauli and Villars.58

It was this paper, with its mathematical manipulation,without physical insight particularly about questions such as photon mass andso forth, which was the direct inspiration for ‘Gauge Invariance and VacuumPolarization.’ The whole point is that if you

have a propagationfunction, it has a certain singularity when the two points coincide. Supposeyou pretend that there are several particles of the same type with differentmasses and with coupling constants which can suddenly become negative insteadof

positive. Then, ofcourse, you can cancel them. It’s cancellation again, subtraction physics, donein a more sophisticated way, but still, things must be made to add up to zero.Who needs it? In this paper, Schwinger obtained a closed form for the electronpropagator

in an externalmagnetic field, by solving proper-time equations of motion, opening a fieldwhich would be fashionable nearly three decades later with the discovery ofpulsars; gave the definitive derivation of the Euler-Heisenberg ¶In the 2005Science Citation Index, it had 105 citations, out of a total of 458 citations toall of Schwinger’s work.56 These numbers have remained remarkably constant overthe years.

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Lagrangiandescribing the scattering of light by light, a phenomenon still not observeddirectly; and gave the precise connection between axial-vector and pseudoscalarmeson theories, what became known as the axial-vector anomaly when it wasrediscovered nearly two decades later by Adler, Bell, and Jackiw.59 (We willdiscuss this the anomaly later in Sec. 17.) The paper is not only a thing ofgreat beauty, but a powerful storehouse of practical technique for solvinggauge-theory problems in a gauge-invariant way.

11 Harvard and Schwinger So it was no surprise that inthe late 1940s and early 1950s Harvard was the center of the world, as far astheoretical physics was concerned. Everyone, students and professors alike,flocked to Schwinger’s lectures. Everything

was revealed, longbefore publication; and not infrequently others received the credit because ofSchwinger’s reluctance to publish before the subject was ripe. A case in pointis the so-called Bethe-Salpeter equation,60 which as Gell-Mann and Low noted,61“first appeared in Schwinger’s lectures at Harvard.” At any one time, Schwingerhad ten or twelve Ph.D. students, who typically saw him but rarely. In part,this was because he was available to see his large flock but one afternoon aweek, but most saw him only when absolutely necessary, because they recognizedthat his time was too valuable to be wasted on trivial matters. A student mayhave seen him only a handful of times in his graduate career, but that was allthe student required. When admitted to his sanctum, students were never rushed,were listened to with respect, treated with kindness, and given inspiration andpractical advice. One must remember that the student’s problems were typicallyquite unrelated to what Schwinger himself was working on at the time; yet in a fewmoments, he could come up with amazing insights that would keep the

student going forweeks, if not months. A few students got to know Schwinger fairly well, andwere invited to the Schwingers’ house occasionally; but most saw Schwinger primarilyas a virtuoso in the lecture hall, and now and then in his office. A fewfaculty members were a bit more intimate, but essentially Schwinger was a veryprivate person.

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12 Custodian ofField Theory

Feynman left thefield of quantum electrodynamics in 1950, regarding it as essentially complete.Schwinger never did. During the next fifteen years, he continued to explorequantum field theory, trying to make it reveal the secrets of the weak andstrong interactions. And he accomplished much. In studying the relativisticstructure of the theory, he recognized that all the physically significantrepresentations of the Lorentz group were those that could be derived from the“attached” four-dimensional Euclidean group,

which is obtainedby letting the time coordinate become imaginary.62 This idea was originallyridiculed by Pauli, but it was to prove a most fruitful suggestion. Related tothis was the CPT theorem, first given a proof for interacting systems bySchwinger in his “Quantized Field” papers of the early 1950s, and elaboratedlater in the decade.63

By the end of the1950s, Schwinger, with his former student Paul Martin, was applying fieldtheory methods of many-body systems, which led to a revolution in that field.64Methods suitable for describing systems out of equilibrium, usually associatedwith the name of Keldysh,65 were obtained some four years earlier bySchwinger.66 Along the way, in what he considered rather modest papers, hediscovered Schwinger terms,67 anomalies in the commutation relations betweenfield operators, and the Schwinger model,68 still the only known example ofdynamical mass generation. The beginnings

of a quantum fieldtheory for non-Abelian fields was made;69 the original example of a non-Abelianfield being that of the gravitational field, he laid the groundwork for latercanonical formulations of gravity.70

13 Measurement Algebra

In 1950 or so, aswe mentioned, Schwinger developed his action principle, which applies to anyquantum system, including nonrelativistic quantum mechanics. Two years later,he reformulated quantum kinematics, introducing symbols that abstracted theessential elements of realistic measurements. This was measurement algebra,which yielded conventional Dirac quantum mechanics. But although the result wasas expected, Schwinger saw the approach as of great value pedagogically, and asproviding a interpretation of quantum mechanics that was self-consistent. Hetaught quantum mechanics this way for many years, starting in 1952 at the LesHouches summer school;

24 but only in 1959 did he start writing a series ofpapers expounding the method

to the world. Healways intended to write a definitive textbook on the subject, but only anincomplete version based on the Les Houches lectures ever appeared during hislifetime.71 Englert has now put his later undergraduate UCLA lectures togetherin a lovely book published by Springer.72 One cannot conclude a retrospectiveof Schwinger’s work without mentioning two other magnificent achievements inthe quantum mechanical domain. He presented a definitive development of angularmomentum theory derived in terms of oscillator variables in “On AngularMomentum,” which was never properly published;73k and he developed a“time-cycle” method of calculating matrix elements without having to find allthe wavefunctionsin his beautiful “Brownian Motion of a Quantum Oscillator,”66which as we mentioned above anticipated the work of Keldysh.65 We should alsomention the famous Lippman-Schwinger paper,75 which is chiefly remembered forwhat Schwinger considered a standard exposition of quantum scattering

theory, not forthe variational methods expounded there.

14 Electroweak Synthesis

In spite of hisawesome ability to make formalism work for him, Schwingerwas at heart aphenomenologist. He was active in the search for higher symmetry; while he cameup with W3, Gell-Mann found the correct approximate symmetry of hadronicstates, SU(3). Schwinger’s greatest success in this period was contained in hismasterpiece, his 1957 paper “A Theory of the Fundamental Interactions”.76 Alongwith many other insights, such as the existence of two neutrinos and the V − Astructure of weak interactions, Schwinger there laid the groundwork for theelectroweak unification. He introduced two charged intermediate vector bosonsas partners to the photon, which couple to charged weak currents. A few yearslater, his former student, Sheldon Glashow, as an outgrowth

of his thesis,would introduce a neutral heavy boson to close the system to the modernSU(2)×U(1) symmetry group;77 Steven Weinberg78 would complete the picture bygenerating the masses for the heavy bosons by spontaneous symmetry breaking.Schwinger did not have the details right in 1957, in particular becauseexperiment then seemed to disfavor the V − A theory his kThis and other ofSchwinger’s most important papers were reprinted in two selections of hiswork.18, 74

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approach implied,but there is no doubt that Schwinger must be counted as the grandfather of theStandard Model on the basis on this paper. 15 The Nobel Prize and Reaction Recognitionof Schwinger’s enormous contributions had come early. He received the CharlesL. Meyer Nature of Light Award in 1949 on the basis of the partly completedmanuscripts of his “Quantum Electrodynamics” papers. The first Einstein prizewas awarded to him, along with Kurt G¨odel, in 1951. The National Medal ofScience was presented to him by President Johnson in 1964, and, of course, theNobel Prize was received by him, Tomonaga, and Feynman from the King of Swedenin 1965.

But by that pointhis extraordinary command of the machinery of quantum field theory hadconvinced him that it was too elaborate to describe the real world, at leastdirectly. In his Nobel Lecture, he appealed for a phenomenological field theorythat would make immediate contact with the particles experiencing the stronginteraction. Within a year, he developed such a theory, Source Theory.

16 Source Theory and UCLA

It surely was thedifficulty of incorporating strong interactions into field theory that led to“Particles and Sources,” received by the Physical Review

barely six monthsafter his Nobel lecture, in July 1966,79 based on lectures Schwinger gave inTokyo that summer. One must appreciate the milieu in which Schwinger worked in1966. For more than a decade he and his students had been nearly the onlyexponents of field theory, as the community sought to understand weak andstrong interactions, and the proliferation of “elementary particles,” throughdispersion relations, Regge poles, current algebra, and the like, mostambitiously through the S-matrix bootstrap hypothesis of Geoffrey Chew andStanley Mandelstam.80–83 What work in field theory did exist then was largelyaxiomatic, an attempt to turn the structure of the theory into a branch ofmathematics, starting with Arthur Wightman,84 and carried on by many others,including Arthur Jaffe at Harvard.85 (The name changed from axiomatic fieldtheory to constructive field theory along the way.) Schwinger looked on all ofthis with considerable distaste; not that

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he did notappreciate many of the contributions these techniques offered in specificcontexts, but he could not see how they could form the basis of a theory. Thenew source theory was supposed to supersede field theory, much as Schwinger’ssuccessive covariant formulations of quantum electrodynamics had replaced hisearlier schemes. In fact, the revolution was to be more profound, because therewere no divergences, and no renormalization. The concept of renormalization issimply foreign to this phenomenological theory. In source theory, we begin byhypothesis

with thedescription of the actual particles, while renormalization is a field theoryconcept in which you begin with the more fundamental operators, which are thenmodified by dynamics. I emphasize that there never can be divergences in aphenomenological

theory. What onemeans by that is that one is recognizing that all further phenomena areconsequences of one phenomenological constant, namely the basic charge unit,which describes the probability of emitting a photon relative to the emissionof an electron.

When one says that there are no divergences one meansthat it is not necessary to introduce any new phenomenological constant. Allfurther processes as computed in terms of this primitive interaction automaticallyemerge to be finite, and in agreement with

those whichhistorically had evolved much earlier.86

16.1 EngineeringApproach to Particle Theory

In 1969 Schwingergave the Stanley H. Klosk lecture to the New York University School ofEngineering Science. Because that lecture captures his philosophy underpinningsource theory so well, at an early stage in the development of that approach, Iquote the transcription of it in full.87 It is a familiar situation in physicsthat when an experimental

domain is to becodified, even though a fundamental theory may be available, rarely is itbrought directly to bear upon the experimental material. The fundamental theoryis too complicated, generally too remote from the phenomena that you want

to describe.Instead, there is always an intermediate theory, a phenomenological theory,which is designed to deal directly with

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the phenomena, andtherefore makes use of the language of observation. On the other hand, it is agenuine theory, and employs abstract concepts that can make contact with thefundamental theory. The true role of fundamental theory is not to confront the

raw data, but toexplain the relatively few parameters of the phenomenological theory in termsof which the great mass of raw data has been organized. I learned this lesson25 years ago during World War II, when I became interested in the problems ofmicrowave systems, wave guides in particular. Being very naive, I started outsolving Maxwell’s equations. I soon learned better. Most of the information inMaxwell’s equations is really superfluous. As far as any particular problem isconcerned, one is only interested in

the propagation ofjust a few modes of the wave guide. A limited number of quantities that can bemeasured or calculated tell you how these few modes behave and exactly what thesystem is doing. You are led directly to a phenomenological theory of the kind

engineersinvariably use—a picture, say, in terms of equivalent transmission lines. Theonly role of Maxwell’s equations is to calculate the few parameters, theeffective lumped constants that characterize the equivalent circuits. Theengineer’s intermediate phenomenological theory looks in both directions. Itcan be connected to the fundamental theory at one end, and at the other it isapplied directly to the experimental data. This is an example of theengineering attitude. It is a pragmatic approach that is designed specificallyfor use. It is a nonspeculative procedure. Hypotheses that go beyond what is

relevant to theavailable data are avoided. Now, when we come to realm of high-energy physics,we are in a new situation. We do not know the underlying dynamics, theunderlying fundamental theory. That raises the question of finding the beststrategy. That is, what is the most effective way of confronting the data, oforganizing it, of learning lessons from results within a limited domain ofexperimental material?

I want to argue that we should adopt a pragmaticengineering approach. What we should not do is try to begin with some

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fundamental theoryand calculate. As we saw, this is not the best thing to do even when you have afundamental theory, and if you don’t have one, it’s certainly the wrong thingto do. Historically, relativistic quantum mechanics had proved very successfulin explaining atomic and nuclear physics until we got accelerators sufficientlyhigh in energy to create the strongly interacting particles, which includeparticles that are highly unstable

and decay throughvery strong forces. The ordinary methods that had evolved up to this point weresimply powerless in the face of this new situation. At the higher energies,particles can be—and are—created and destroyed with high probability.

In other words, the immutability of the particle—afoundation of ordinary physics—had disappeared. If the immutable particle hasceased to exist as the fundamental

concept in termsof which a situation can be described, what do we replace it with? There havebeen two different points of view about how to construct a fundamental theoryfor the strong interactions. The first—the point of view of conventionaloperator field

theory—proposes toreplace the particle with three-dimensional space itself. In other words, youthink of energy, momentum, electric charge, and other properties as distributedthroughout space, and of small volumes of three-dimensional space as the

things thatreplace particles. These volumes are the carriers of energy, momentum, and soon. People, including myself, have been actively developing the field idea formany years. I believe that this kind of theory may be the ultimate answer, butplease recognize that it is a speculation. It assumes that one is indeed ableto describe physical phenomena

down toarbitrarily small distance, and, of course, that goes far beyond anything weknow at the moment. All we are able to do experimentally as we go to higher andhigher energies is to plumb to smaller and smaller distances, but never to zerodistance.

The question is, should you, in discussing the phenomenathat are presently known, make use of a speculative hypothesis like operator fieldtheory? Can we not discuss particle phenomenology and handle the correlationsand organization of data without becoming

involved in aspeculative theory? In operator field theory

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you cannotseparate particle phenomena from speculations about the structure of particles.The operators of quantum-mechanical field theory conceptually mix these together.To be able to discuss anything from the operator-field-theory point of view,you must accept its fundamental hypothesis. You have to accept a speculationabout how particles are constructed before you can begin to discuss howparticles interact with each

other. Historically,this has proved to be a very difficult program to apply, and people have, ofcourse, been anxious to deal directly with the experimental data, and so therehas been a reaction. The extreme reaction to operator field theory is to insistthat there is

nothing morefundamental than particles and that, when you have a number of particlescolliding with each other and the number of particles ceases to be constant,all you can do is correlate what comes into a collision with what goes out, andcease to describe

in detail what ishappening during the course of the collision. This point of view is calledS-matrix theory. The quantitative description is in terms of a scatteringmatrix that connects the outgoing state with the incoming state. In this theorythe particles are basic and cannot be analyzed. Then, of course, the questioncomes up: what distinguishes the particular set of particles that do exist fromany other conceivable set? The only answer that has been suggested is that theobserved particles exist as a consequence of self-consistency. Given a certain setof particles, other particles can be formed as aggregates or composites ofthese. On the other hand, if particles are unanalyzable,

then this shouldnot be a new set of particles, but the very particles themselves.

That is the second idea, but I beg you to appreciate thatit is also a speculation. We do not know for a fact that our present inabilityto describe things in terms of something more

fundamental thanparticles reflects an intrinsic impossibility. So these are the two polarizedextremes in the search for a fundamental theory—the operator-field-theory pointof view and the S-matrix point of view. Now my reaction to all of this is to askagain why we must speculate, since the probability of falling on the rightspeculation is very small.

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Can we notseparate the theoretical problem of describing the properties of theseparticles from speculations about their inner structure? Can we not set asidethe speculation of whether particles are made from operator fields or are madefrom nothing but

themselves, andfind an intermediate theory, a phenomenological theory that directly confrontsthe data, but that is still a creative theory? This theory should besufficiently flexible so that it can make contact with a future, morefundamental theory of the structure

of particles, ifindeed any more fundamental theory ever appears. This is the line of reasoningthat led me to consider the theoretical problem for high-energy physics from anengineering point of view. Clearly I have some ideas in mind about how to carryout such a program, and I would like to give you an enormously simplifiedaccount of them. We want to eliminate speculation and take a pragmaticapproach. We are not going to say that particles are made out of fields, orthat particles sustain each other. We are simply going

to say thatparticles are what the experimentalists say they are. But we will construct atheory and not an experimenter’s manual in that we will look at realisticexperimental procedures and pick out their essence through idealizations.

There is one characteristic that the high-energyparticles have in common—they must be created. Through the act of creation, wecan define what we mean by a particle. How, in fact, do you create a particle?By a collision. The experimentalist arranges for a beam of particles to fall ona target. In the center-of-mass system, the target is just another beam, so twobeams of particles are colliding. Out of the collision, the particle that weare

interested in maybe produced. We say that it is a particle rather than a random bump on

an excitationcurve because its properties are reproducible. We still recognize the sameparticle event though we vary a number of experimental parameters, such asenergy, angles, and the kind of reaction. The properties of the particle inquestion remain the

same—it has thesame mass, the same spin, the same charge. These criteria can be applied to anobject that may last for only 10−24 sec—which decays even before it gets out ofthe nucleus

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in which it wascreated. Nevertheless, it is still useful to call this kind of object aparticle because it possesses essentially all of the characteristics that weassociate with the particle concept. What is significant is that within asomewhat controllable region of space and time, the properties characteristicof the particle have become transformed into the particle itself. The other particlesin the collision are there only to supply the net balance

of properties.They are idealized as the source of the particle. This is our new theoreticalconstruct. We introduce a quantitative description of the particle source interms of a source function, S(x), where x refers to the space and timecoordinates. This function indicates that, to some extent, we can control the regionthe particle comes from.

But we do not have to claim that we can make the sourcearbitrarily small as in operator field theory. We leave this question open, tobe tested by future experiment. A particular source may be more effective inmaking particles that go in one direction rather than another, so there must beanother degree of control expressed by a source function of

momentum, S(p).But from quantum mechanics we know that the dimension of the system and thedegree of directionality are closely related. The smaller the system, the lessdirectional it can be. And relativistic mechanics is incorporated from the very

beginning in thatthe energy and the momentum are related to its mass in the usual relativisticway. Now the experimenter’s job only begins with the production

of a beam. At theother end, he must detect the particles. What is detection? Unstable particleseventually decay, and the decay process is a detection device. More generally,any detection device can be regarded as a kind of collision that annihilatesthe particle

and hands itsproperties on in a more usable form. Thus the source concept can again beintroduced as an abstraction of an annihilation collision, with the sourceacting negatively, as a sink. We now have a complete theoretical picture of anyphysical situation, in which sources are used to create the initial particle ofinterest from the vacuum state, and sources are used to detect the finalparticles resulting from some interaction, thus returning to the vacuum state.

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[Schwinger thenwrote down an expression that describes the probability amplitude that thevacuum state before sources act remains the vacuum state after sources act, thevacuum persistence amplitude.] The basic things that appear in this expression

are the sourcefunctions and space-time functions that represent the state into which theparticle is emitted and from which it is absorbed, thus describing theintermediate propagation of the particle. This simple expression can begeneralized to apply to particles that have charge, spin, etc., and tosituations where more than one particle is present at a time. Interactionsbetween particles are described in terms containing more than two sources. Ourstarting point accepts particles as fundamental—we use sources to identify theparticles and to incorporate a simplified view of dynamics. From that we evolvea more complete dynamical theory in which we combine simple source arrangementslike building blocks to produce descriptions of situations that can in principlebe as complex as we want. A first test of this approach would be to see if wecan reproduce the results of some well-established theory such asquantumelectrodynamics. What is the starting point in this attack on electrodynamics?It is the photon, a particle that we know has certain striking properties suchas zero rest mass and helicity 1. So we must include all these aspects of thephoton in the picture, and describe how photons are emitted and absorbed. In

consequence, thesource must be a vector, and it must be divergenceless. This approach leads usto something resembling a vector potential, and when we ask what differentialequations it satisfies we find they are Maxwell’s equations. We start with theconcept

of the source asprimary and are led to Maxwell’s differential equations as derived concepts. Thedescription of interactions follows the tentative procedures

of life in thereal world. The theory is not stated once and for all. It begins with simplephenomena—for example, accelerated charges radiate. It then extrapolates thatinformation outside its domain, predicts more complicated phenomena, and awaitsthe test of experiment. We do not begin with a final de-

33

scription of, say,electron scattering. We extrapolate to it from more elementary situations, andthis is still not the final description. As the theory develops and becomesmore encompassing, we go back to refine the description of the scattering processand

obtain a morequantitative account of it. This is the concept of an interaction skeleton. Theprocess is there but it is not finally described to start with, its existenceis merely recognized. This simplified reconstruction of electrodynamics is completely

successful. Toindicate the wide sweep of the new approach, I mention that classicalgravitation theory (Einstein) can be reconstructed and simplified in a similarway by beginning with the quantum relativistic properties of the basicparticle, the graviton, although here indirect evidence for its properties mustbe adduced. But the real proving ground for source theory comes from the domainfor which it was invented, strong interactions. The starting point isexperimental information at low energies. The tentative

extrapolations aretoward higher energies. The method is quite elementary compared to othercurrent techniques. The successful correlations that have been obtainedemphasize the completely phenomenological nature of our present knowledge about

particles andrefute attempts to lend fundamental credence to this or that particle model.

A more fundamental theory may come into being one day,but it will be the outcome of continued experimental probing to higher energies,and will doubtless involve theoretical concepts that are now only dimly seen.But that day will be greatly speeded if the

flood ofexperimental results is organized and analyzed with the aid of a theory thatdoes not have built into it a preconception about the very question that isbeing attacked. This theory issource theory.

16.2 The Impact of Source Theory

Robert Finkelsteinhas offered a perceptive discussion of Schwinger’s source theory program:

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In comparingoperator field theory with source theory Julian revealed his politicalorientation when he described operator field theory as a trickle down theory(after a failed economic theory)— since it descends from implicit assumptionsabout unknown phenomena at inaccessible and very high energies to makepredictions at lower energies. Source theory on the other hand he described asanabatic (as in Xenophon’s Anabasis) by which he meant that it began with solidknowledge about known phenomena at accessible energies to make predictionsabout physical phenomena at higher energies. Although source theory was new, itdid not represent a complete break with the past but rather was a natural evolutionof Julian’s work with operator Green’s functions. His trilogy on source theoryis not only a stunning display of Julian’s power as an analyst but it is alsototally in the spirit of the modest scientific goals he had set in his QED workand which had

guided him earlieras a nuclear phenomenologist.88 But the new approach was not well received. Inpart this was because the times were changing; within a few years, ’t Hooft89would establish the renormalizability of the Glashow-Weinberg-Salam SU(2)×U(1)electroweak model, and field theory was seen by all to be viable again. Withthe discovery

of asymptoticfreedom in 1974,90 a non-Abelian gauge theory of strong interactions, quantumchromodynamics, which was proposed somewhat earlier,91 was promptly accepted bynearly everyone. An alternative to conventional field theory did not seem to berequired after all. Schwinger’s insistence on a clean break with the past, andhis rejection of “rules” as opposed to learning through serving as an“apprentice,” did not encourage conversions. Already before the source theoryrevolution, Schwinger felt a growing sense of unease with his colleagues atHarvard. But the chief reason Schwinger

left Harvard forUCLA was health related. Formerly overweight and inactive, he had become healthconscious upon the premature death of Wolfgang Pauli in 1958. (Ironically, bothdied of pancreatic cancer.) He had been fond of tennis from his youth, haddiscovered skiing in 1960, and now his doctor was recommending a daily swim forhis health. So he listened favorably to the entreaties of David Saxon, hisclosest colleague at the Radiation Lab during the war, who for years had beentrying to induce him to come to UCLA. Very much against his wife’s wishes, hemade the move in 1971. He brought along his three senior students at the time,Lester DeRaad, Jr., Wu-yang

35

Tsai, and thepresent author, who became long-term “assistants” at UCLA. He and Saxonexpected, as in the early days at Harvard, that students would flock to UCLA towork with him; but they did not. Schwinger was no longer the center oftheoretical physics. This is not to say that his little group at UCLA did notmake an heroic attempt to establish a source-theory presence. Schwingerremained a gifted innovator and an awesome calculator. He wrote 2-1/2 volumesof an exhaustive treatise on source theory, Particles, Sources, and Fields,92devoted primarily to the reconstruction of quantum electrodynamics in the newlanguage; unfortunately, he abandoned the project when it came time to deal withstrong interactions, in part because he became too busy writing papers on an“anti-parton” interpretation of the results of deep-inelastic scattering experiments.93He made some significant contributions to the theory of magnetic charge;particularly noteworthy was his introduction of dyons.94 He reinvigoratedproper-time methods of calculating processes in strong-field electrodynamics;95and he made some major contributions to the theory of the Casimir effect, whichare still having repercussions.96 But it was clear he was reacting, notleading, as witnessed by his quite pretty paper on the “Multispinor Basis ofFermi-Bose Transformation,”97 in which he kicked himself for not discoveringsupersymmetry, following a command private performance by Stanley Deser onsupergravity.

17 TheAxial-Vector Anomaly and Schwinger’s Departure from Particle Physics

In 1980 Schwingergave a seminar at MIT that marked his last scientific visit to the East Coast,∗∗ and caused him to abandon his attempt to influence thedevelopment of high-energy theory with his source theory revolution. The talkwas on a subject that he largely started in his famous “Gauge Invariance andVacuum Polarization” paper,54 the triangle or axial-vector anomaly. In itssimplest and basic manifestation, this “anomaly” describes how the neutral piondecays into two photons. The pion coupling could

be regarded asoccurring either through a pseudoscalar or an axial vector

∗∗This does not count a talk he gave at MIT in 1991 inhonor of birthdays of two of his students, where he gave a “progress report” onhis work on cold fusion and sonoluminescence, excerpts of which is given inRef. [2].

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coupling, whichformally appeared be equivalent, but calculations in the 1940s gave discrepantanswers. Schwinger resolved this issue in 1950 by showing that the two theorieswere indeed equivalent provided that proper care (gauge-invariance) was used,and that the formal result was modified by an additional term. Problem solved,and it was then forgotten for the next 18 years. In the late 1960s Adler, Bell,and Jackiw rediscovered this solution,59 but the language was a bit different.The extra term Schwinger had found was now called an anomaly, but the form ofthe equations, andthe prediction for the decay of the pion, were identical. Infact, at first it is apparent that Adler, Bell, and Jackiw were unaware ofSchwinger’s much earlier result, and it was the addition of Ken Johnson (one ofSchwinger’s many brilliant students) into the collaboration that corrected thehistorical

record.98 Shortlythereafter, Adler and Bardeen proved the “nonrenormalization” theorem,99 thatthe anomaly is exact, and is not corrected by higher-order quantum effects.This is in contrast to most physical phenomena, such as the anomalous magneticmoment of the electron, which is subject to corrections in all orders ofperturbation theory in the strength of the electromagnetic coupling, the finestructure constant. This seemed surprising to Schwinger, so he suggested to hispostdocs at UCLA that they work this out independently, and they did,publishing two papers in 1972,100 in which they showed,

using twoindependent methods, that there was indeed such a correction in higher order. However,Adler, who was the reviewer of these papers forced them to tone down theirconclusion, and to point out that the result depends on the physical point atwhich the renormalization is carried out. Nonrenormalization indeed can beachieved by renormalization at an unphysical point, which may be acceptable forthe use of the theorem in establishing renormalizability of gauge theories, itschief application, but it is nevertheless true that physical processes such asthe original process of pion decay receives higher-order corrections. As wastypical, Schwinger apparently took no notice of this dispute at the time. Buttoward the end of the 1970s, while he was writing the third

volume ofParticles, Sources, and Fields, he looked at the questions of radiative correctionsto neutral pion decay and found the same result as DeRaad, Milton, and Tsai. Hewrote an explicitly confrontational paper on the subject, which was the basisfor the above-mentioned talk at MIT. The paper was apparently definitivelyrejected, and the talk was harshly criticized, and on the basis of theseclosed-minded attacks, Schwinger left the field. For-

37

tunately for therecord, Schwinger’s paper exists as a chapter in the finally published thirdvolume of Particles, Sources, and Fields. However, the controversy lives on. In2004 Steve Adler wrote a historical perspective on his work on the axial-vectoranomaly.101 He devotes five pages of his retrospective to attack the work ofSchwinger and his group. He

even denies thatSchwinger was the first to calculate the anomaly, in blatant disregard of thehistorical record. Of course, physical understanding had increased in thenearly two decades between Schwinger’s and Adler’s papers, but to deny thatSchwinger was the first person to offer the basis for the connection betweenthe axial-vector and pseudoscalar currents, and the origin of the photonicdecay of the neutral pion, is preposterous.

18 Thomas-FermiAtom, Cold Fusion, and Sonoluminescence

When the last ofhis Harvard postdocs left UCLA in 1979, and the flap over the axial-vectoranomaly ensued, Schwinger abandoned high-energy physics altogether. In 1980,after teaching a quantum mechanics course (a notunusual sequence of events),Schwinger began a series of papers on the Thomas-Fermi model of atoms.102 Hesoon hired Berthold-Georg Englert, replacing Milton as a postdoc, to help withthe elaborate calculations. This endeavor lasted until 1985. It is interestingthat this work not only is regarded as important in its own right by atomicphysicists, but has led to some significant results in mathematics. A longseries of substantial papers by C. Fefferman and L. Seco103 has been devoted toproving his conjecture

about the atomicnumber dependence of the ground state energy of large atoms. As Seth Puttermanhas remarked, it is likely that, of all the work that Schwinger accomplished atUCLA, his work on the statistical atom will prove the most important.104

Following the Thomas-Fermi work, Schwinger continued tocollaborate with Englert, and with Marlan Scully, on the question of spincoherence. If an atomic beam is separated into sub-beams by a Stern-Gerlachapparatus, is it possible to reunite the beams? Scully had argued that it mightbe possible, but Julian was skeptical; the result was three joint papers,entitled “Is Spin Coherence Like Humpty Dumpty?”, which bore out Julian’sintuition of the

impossibility ofbeating the effects of quantum entanglement.105

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In March 1989began one of the most curious episodes in physical science in the last century,one that initially attracted great interest among the scientific as well as thelay community, but which rapidly appeared to be a characteristic example of“pathological science.”†† The effect to which we refer was the announcement byB. S. Pons and M. Fleischmann107 of the discovery of cold fusion. That is, theyclaimed that nuclear energy, in the form of heat, was released in a table-topexperiment, involving a palladium

cathodeelectrolyzing heavy water. So it was a shock to most physicists‡‡ whenSchwinger began speaking and writing about cold fusion, suggesting that theexperiments of Pons and Fleischmann were valid, and that the palladium latticeplayed a crucial role. In

one of his laterlectures on the subject in Salt Lake City, Schwinger recalled, “Apart from abrief period of apostasy, when I echoed the conventional wisdom that atomic andnuclear energy scales are much too disparate, I have retained my belief in theimportance of the lattice.”5 His first publication on the subject was submittedto Physical Review Letters, but was roundly rejected, in a manner thatSchwinger considered deeply insulting. In protest, he resigned as a member (hewas, of course, a fellow) of the American Physical

Society, of whichPhysical Review Letters is the most prestigious journal. (At first he intendedmerely to withdraw the paper from PRL, and his fellowship, but then he feltcompelled to respond to the referees’ comments: One comment was something tothe effect that no nuclear physicist could believe such an effect, to whichJulian angrily retorted, “I am a nuclear physicist!”5) In this letter to theeditor (G. Wells) in which he withdrew the paper and resigned from the AmericanPhysical Society, he also called for the removal of the source theory indexcategory the APS journals used: “Incidentally,

††This term wascoined in 1953 by Irving Langmuir, who gave a celebrated lecture at GeneralElectric’s Knolls Atomic Power Laboratory (transcribed from a disc recording byRobert Hall) on the phenomenon wherein reputable scientists are led to believethat an effect, just at the edge of visibility, is real, even though, asprecision increases, the effect remains marginal. The scientist becomesself-deluded, going to great lengths to convince one and all that theremarkable effect is there just on the margins of what can be measured. Greataccuracy is claimed nevertheless, and fantastic, ad hoc, theories are inventedto explain the effect. Examples include N-rays, the Allison effect, flyingsaucers, and ESP. It was not a coincidence that Physics Today published thearticle, without

comment, in thefall of 1989.106 ‡‡However, a few other eminent physicists spoke favorably ofthe possibility of cold fusion, notably Edward Teller and Willis Lamb, whopublished three articles in the Proceedings of the U.S. National Academy ofSciences on the subject.

39

the PACS entry(1987) 11.10.mn can be deleted. There will be no further occasion to useit.’;5, 108 A rather striking act of hubris: If he couldn’t publish source theory,neither could anybody else. But the Physical Review obliged. (Unfortunately,Schwinger failed to realize that the PACS index system has become thepredominant system for physics journals worldwide, a reflection of the premierstatus of the APS journals. So he largely spited his own contributions.) Notwishing to use any other APS venue, he turned to his friend and colleague,Berthold Englert, who arranged that “Cold Fusion: A Hypothesis” be published inthe Zeitschrift f¨ur Naturforschung, where it appeared in October of thatyear.109 Schwinger then went on to write three substantial papers, entitled“Nuclear Energy in an Atomic Lattice I, II, III,” to flesh out these ideas.5,110 The first was published in the Zeitschrift f¨ur Physik D,111 where it wasaccepted in spite of negative reviews,5 but directly preceded by an editorialnote, disclaiming any responsibility for the the paper

on the part of thejournal. They subsequently refused to publish the remaining papers.

Schwinger’s lastphysics endeavor marked a return to the Casimir effect, of which he had beenenamored nearly two decades earlier. It was sparked by the remarkable discoveryof single-bubble sonoluminescence, in which a small bubble of air in water,driven by a strong acoustic standing wave, undergoes a stable cycle of collapseand re-expansion; at minimum radius an intense flash of light, consisting of amillion optical photons, is emitted. It was not coincidental that the leadinglaboratory investigating this phenomenon was,

and is, at UCLA,led by erstwhile theorist Seth Putterman, long a friend and confidant.Putterman and Schwinger shared many interests in common, including appreciationof fine wines, and they shared a similar iconoclastic view of the decline ofphysics. So, of course, Schwinger heard about this remarkable phenomenon fromthe horse’s mouth, and was greatly intrigued.∗ Schwingerimmediately had the idea that a dynamical version of the Casimir effect mightplay a key role. He saw parallels between cold fusion and

sonoluminescencein that both deal with seemingly incommensurate energy scales, and both dependsignificantly on nonlinear effects. Since by the early 1990s, cold fusion waslargely discredited, he put his efforts to understanding sonoluminescence,which undoubtedly does exist. Unfortunately neither Schwinger, nor anyonesubsequently, was able to get very far with dynam-

∗For a review of the phenomena, and a detailed evaluationof various theoretical explanations, see Ref. [112].

40

ical zero-pointphenomena; he largely contented himself with an adiabatic approximation basedon static Casimir energies; and was able to obtain sufficient energy onlybecause he retained the “bulk energy,” which most now believe is unobservable,being subsumed in a renormalization of bulk material properties. His work onthe subject appeared as a series of short papers in the PNAS, the last appearing113shortly after his death in June 1994.

19 Conclusion

It is impossibleto do justice in a few words to the impact of Julian Schwinger on physicalthought in the 20th Century. He revolutionized fields from nuclear physics tomany body theory, first successfully formulated renormalized quantumelectrodynamics, developed the most powerful functional formulation of quantumfield theory, and proposed new ways of looking at quantum mechanics, angularmomentum theory, and quantum fluctuations.

His legacy includes“theoretical tools” such as the proper-time method, the quantum actionprinciple, and effective action techniques. Not only is he responsible forformulations bearing his name: the Rarita-Schwinger equation, theLippmann-Schwinger equation, the Tomonaga-Schwinger equation, theDyson-Schwinger equation, the Schwinger mechanism, and so forth, but someattributed to others, or known anonymously: Feynman parameters, theBethe-Salpeter equation, coherent states, Euclidean field theory; the list goeson and on. His legacy of nearly 80 Ph.D. students, including four Nobel laureates,lives on. It is impossible to imagine what physics would be like in the 21stcentury without the contributions of Julian Schwinger, a very private yetwonderful human being. It is most gratifying that a dozen years after hisdeath, recognition of his manifold influences is growing, and research projectshe initiated are still underway.

It is fitting toclose this retrospective with Schwinger’s own words, delivered some six monthsbefore his final illness, when he received an honorary degree from theUniversity of Nottingham.114† The Degree Ceremony is a modern version of amedieval rite

that seemed toconfer a kind of priesthood upon its recipients, thereby excluding all othersfrom its inner circle. But that will †This brief acceptance speech was followedby a brilliant lecture on the influence of George Green on Schwinger’s work.114

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not do for today.Science, with its offshoot of Technology, has an overwhelming impact upon allof us. The recent events at Wimbledon invite me to a somewhat outrageousanalogy. Very few of us, indeed, are qualified to step onto centre court. Yet thousandsof spectators gain great pleasure from watching these talented specialistsperform. Something similar should be, but generally is not, true for therelationship between the practitioners of Science and the general public. Thisis much more serious

than not knowingthe difference between 30 all and deuce. Science, on a big scale, is inevitablyintertwined with politics. And politicians have little practice indistinguishing between, say common law and Newton’s law. It is a suitablyeducated public that

must step into thebreach. This has been underlined lately by Minister Waldegrave’s cry forsomeone to educate him about the properties of the Higgs boson, to be rewardedwith a bottle of champagne. Any member of the educated public could have told himthat the cited particle is an artifact of a particular theoretical speculation,and the real challenge is to enter uncharted waters to see what is there. Thefailure to do this will inevitably put an

end to Science. Asociety that turn in on itself has sown the seeds of its own demise. Early inthe 16th century, powerful China had sea-going vessels exploring to the west.Then a signal came fromnew masters to return and destroy the ships. It was inthose years

that Portuguesesailors entered the Indian Ocean. The outcome was 400 years of dominance of theEast by the West. There are other threats to Science. A recent bestseller inEngland, Understanding the present, has the subtitle Science and the

soul of ModernMan. I shall only touch on the writer’s views toward quantum mechanics, surelythe greatest intellectual discovery of the 20th century. First, he complainsthat the new physics of quantum mechanics tosses classical physics in the trashbin.

This I would dismiss as mere technical ignorance; themanner in which classical and quantum mechanics blend into each other has longbeen established. Second, the author is upset that its theories can’t beunderstood by anyone not mathematically sophisticated

and so must beaccepted by most people on faith. He is, in short, saying that there is apriesthood. Against this I pose my own experience in presenting the basicconcepts of quantum

42

mechanics to aclass of American high school students. They understood it; they loved it. AndI used no more than a bit of algebra, a bit of geometry. So: catch them young;educate them properly; and there are no mysteries, no priests. It all comes

down to a properlyeducated public.

Acknowledgements

I am greatful tomany colleagues for the interviews and conversations granted me in writingabout Julian Schwinger. I am particularly grateful to Robert Finkelstein andEdward Gerjuoy for conversations in the past few months. Again I must thankCharlotte Brown, Curator of Special Collections at UCLA, for making theSchwinger archives available to me on many occasions. My research over theyear, not primarily historical, has been funded by grants from the USDepartment of Energy and the US National Science Foundation. I dedicate thismemoir to Julian’s widow, Clarice.

 

References[1] Norman Ramsey, interview by K. A. Milton, June 8, 1999.[2] Jagdish Mehra and Kimball A. Milton, Climbing the Mountain: TheScientific Biography of Julian Schwinger, (Oxford: Oxford UniversityPress, 2000).[3] Larry Cranberg, telephone interview by K. A. Milton, 2001.[4] Eileen F. Lebow, The Bright Boys: A History of Townsend Harris HighSchool (Westport CT: Greenwood Press, 2000).[5] Julian Schwinger Papers (Collection 371), Department of Special Collections,University Research Library, University of California, Los Angeles.[6] Joseph Weinberg, telephone interview by K. A. Milton, July 12, 1999.[7] E. J. Townsend, Functions of Real Variables (New York: Holt, 1928).43[8] P. A. M. Dirac, Principles of Quantum Mechanics (Oxford: OxfordUniversity Press, 1930).[9] Edward Gerjuoy, telephone interview by K. A. Milton, June 25, 1999.[10] M. Hamermesh, “Recollections” at Julian Schwinger’s 60th birthdaycelebration, UCLA, 1978 (AIP Archive).[11] Edward Gerjuoy, talk given at the University of Pittsburgh and atGeorgia Tech, 1994, private communication.[12] Sidney Borowitz, telephone interview by K. A. Milton, June 25, 1999.[13] C. Møller, “¨ Uber den Stoß zweier Teilchen unter Ber¨ucksichtigung derRetardation der Kr¨afte” Zeit. f¨ur Phys. 70 (1931), 786–795.[14] P. A. M. Dirac, “Relativistic Quantum Mechanics,” Proc. Roy. Soc.London A136 (1932), 453–464.[15] P. A. M. Dirac, V. A. Fock, and B. Podolsky, “On Quantum Electrodynamics,”Phys. Zeit. Sowjetunion 2 (1932), 468–479.[16] W. Heisenberg andW. Pauli, “Zur Quantendynamik der Wellenfelder,”Zeit. f¨ur Phys. 56 (1929), 1–61; ibid. “Zur Quantentheorie der Wellenfelder.II,” 59 (1930), 168–190.[17] J. Schwinger, “Quantum Electrodynamics—An Individual View,” J.Physique 43, Colloque C-8, supplement au no. 12 (1982), 409–421;“Renormalization Theory of Quantum Electrodynamics: An IndividualView,” in The Birth of Particle Physics, eds. L. M. Brown and L.Hoddeson (Cambridge University Press, 1983), p. 329–353.[18] K. A. Milton, ed., A Quantum Legacy: Seminal Papers of JulianSchwinger (Singapore: World Scientific, 2000).[19] A. Einstein, B. Podolsky and N. Rosen, “Can Quantum-MechanicalDescription of Physical Reality Be Considered Complete?” Phys. Rev.47 (1935), 777–780.[20] I. I. Rabi, talk at J. Schwinger’s 60th Birthday Celebration, February1978 (AIP Archive).44[21] P. Debye and E. H¨uckel, “Zum Theorie der Elektrolyte,” Phys. Z. 24(1923), 185–206; P. Debye, “Kinetische Theorie der Gesetze des OsmotischenDrucks bei starken Elektrolyten,” ibid. 24 (1923), 334–338;P. Debye, “Osmotische Zustandsgleichung und Aktivit¨at verd¨unnterstarker Elektrolyte,” ibid. 25 (1924), 97–107.[22] J. Schwinger, “On Nonadiabatic Processes in Inhomogeneous Fields,”Phys. Rev. 51 (1937), 648–561.[23] J. Schwinger, “On the Neutron-Proton Interaction,” Phys. Rev. 55(1939), 235.[24] J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey, and J. R. Zacharias, “AnElectrical Quadrupole Moment of the Deuteron,” Phys. Rev. 55 (1939),318–319.[25] W. Rarita and J. Schwinger, “On a Theory of Particles with Half-Integral Spin,” Phys. Rev. 60 (1941), 61.[26] B. Peters and C. Richman, “Deuteron Disintegration by Electrons,”Phys. Rev. 59 (1941), 804–807.[27] E. Gerjuoy and J. Schwinger, “On Tensor Forces and the Theory ofLight Nuclei,” Phys. Rev. 61 (1942), 138–146.[28] E. Gerjuoy, Newsletter of the Forum on the History of Physics,http://www.aps.org/units/fhp/FHPnews/news-fall05.cfm.[29] J. R. Oppenheimer and J. Schwinger, “On Pair Emission in the ProtonBombardment in Fluorine,” Phys. Rev. 56 (1939), 1066–1067.[30] J. Schwinger, “On the Classical Radiation of Accelerated Electrons,”Phys. Rev. 75 (1949), 1912–1925.[31] K. A. Milton and J. Schwinger, Electromagnetic Radiation: VariationalPrinciples, Waveguides, and Accelerators (Berlin: Springer, 2006).[32] W. E. Lamb, Jr., and R. C. Retherford, “Fine Structure of the HydrogenAtom by Microwave Method,” Phys. Rev. 72 (1947), 241–243.45[33] R. P. Feynman, talk given at Schwinger’s 60th birthday celebration,printed in Themes in Contemporary Physics II, ed. S. Deser and R. J.Finkelstein (Singapore: World Scientific, 1989), pp. 91–93.[34] H. A. Bethe, “The Electromagnetic Shift of Energy Levels,” Phys. Rev.72 (1947), 339–341.[35] J. E. Nafe, E. B. Nelson, and I. I. Rabi, “The Hyperfine Structure ofAtomic Hydrogen and Deuterium,” Phys. Rev. 71 (1947), 914–915; P.Kusch and H. M. Foley, “Precision Measurement of the Ratio of theAtomic ‘g Values’ in the 2P3/2 and 2P1/2 States of Gallium,” Phys. Rev.72 (1947), 1256–1257.[36] J. Schwinger, “On Quantum-Electrodynamics and the Magnetic Momentof the Electron,” Phys. Rev. 73 (1948), 416–417.[37] S. M. Dancoff, “On Radiative Corrections for Electron Scattering,”Phys. Rev. 55 (1939), 959–963.[38] Silvan S. Schweber, QED and the Men Who Made it: Dyson, Feynman,Schwinger, and Tomonaga (Princeton: Princeton University Press,1994).[39] Norman Ramsey, Reminiscences of the Thirties, videotaped at BrandeisUniversity, March 29, 1984, in Ref. [5].[40] J. Schwinger, “Quantum Electrodynamics I. A Covariant Formulation,”Phys. Rev. 74 (1948), 1439–1461; “Quantum ElectrodynamicsII. Vacuum Polarization and Self Energy,” Phys. Rev. 75 (1949), 651–679; “Quantum Electrodynamics III. The Electromagnetic Propertiesof the Electron—Radiative Corrections to Scattering,” Phys. Rev. 76(1949), 790–817.[41] R. P. Feynman, “The Theory of Positrons,” Phys. Rev. 76 (1949),749–759; “Space-Time Approach to Quantum Electrodynamics,” ibid.,769–789.[42] F. J. Dyson, “The Radiation Theories of Tomonaga, Schwinger, andFeynman,” Phys. Rev. 75 (1949), 486–502; “The S Matrix in QuantumElectrodynamics,” ibid. 1736–1755.46[43] H. A. Kramers, “Nonrelativistic Quantum Electrodynamics and CorrespondencePrinciple,” Rapports et discussions du 8e Conseil dePhysique Solvay 1948 (Bruxelles: Stoop, 1950), p. 241; M. Dresden, H.A. Kramers: Between Transition and Revolution (New York: Springer-Verlag, 1987); H. A. Kramers, Die Grundlagen den Quantentheorie—Quantentheorie des Elektrons und der Strahlung [Hand- und Jahrbuchder Chemischen Physik I, Abschnitt 1–2] (Leipzig: AkademischeVerlagsgesellschaft, 1938); “Die Wechselwirkung zwischen geladenenTeilchen und Strahlungsfeld,” Nuovo Cim. 15 (1938), 108–114.[44] S. Tomonaga, “On the Relativistically Invariant Formulation of theQuantum Theory of Wave Fields,” Prog. Theor. Phys. 1, 27–42 (1946);“On Infinite Field Reactions in Quantum Field Theory,” Phys. Rev. 74(1948), 224–225.[45] R. P. Feynman, “Relativistic Cut-Off for Quantum Electrodynamics,”Phys. Rev. 74 (1948), 1430–1438.[46] J. B. French and V. F. Weisskopf, “On the Electromagnetic Shift ofEnergy Levels,” Phys. Rev. 75 (1949), 338; “The Electromagnetic Shiftof Energy Levels,” 75 (1949), 1240–1248.[47] N. M. Kroll and W. E. Lamb, Jr., “On the Self-Energy of a BoundElectron,” Phys. Rev. 75 (1949), 388–389.[48] P. A. M. Dirac, “The Lagrangian in Quantum Mechanics,” Phys. Zeit.Sowjetunion 3 (1933), 64–72.[49] J. Schwinger, “The Theory of Quantized Fields. I,” Phys. Rev. 82(1951), 914–927.[50] J. Schwinger, “On the Green’s Functions of Quantized Fields. I, II,”Proc. Natl. Acad. Sci. USA, 37 (1951), 452–455, 455–459.[51] P. C. Martin, “Schwinger and Statistical Physics: A Spin-Off SuccessStory and Some Challenging Sequels,” in Themes in ContemporaryPhysics, eds. S. Deser, H. Feshbach, R. J. Finkelstein, K. A. Johnson,and P. C. Martin (Amsterdam: North-Holland, 1979) Physica 96A(1979), 70–88.47[52] P. C. Martin, “Julian Schwinger—Personal Recollections,” in JulianSchwinger: the Physicist, the Teacher, and the Man, ed. Y. J. Ng(Singapore: World Scientific, 1996), p. 83–89.[53] S. S. Schweber, “The Sources of Schwinger’s Green’s Functions,” Proc.Natl. Acad. Sci. USA, 102 (2005), 7783–7788.[54] J. Schwinger, “On Gauge Invariance and Vacuum Polarization,” Phys.Rev. 82 (1951), 664–679.[55] Lowell S. Brown, “An Important Schwinger Legacy: TheoreticalTools,” talk given at Schwinger Memorial Session at the April 1995meeting of the APS/AAPT. Published in Julian Schwinger: The Physicist,the Teacher, the Man, ed. Y. Jack Ng (Singapore: World Scientific,1996), p. 131–154.[56] Science Citation Index (Philadelphia: Institute for Scientific Information,2005) [ISI Web of Science].[57] Abraham Klein, telephone interview by K. A. Milton, December 14,1998.[58] W. Pauli and F. Villars, “On the Invariant Regularization in RelativisticQuantum Theory,” Rev. Mod. Phys. 21 (1949), 434–444.[59] J. S. Bell and R. Jackiw, “A PCAC Puzzle: π0 ! γγ in the σ-Model,”Nuovo Cimento 60A, Series 10 (1969), 47–61; S. L. Adler, “Axial-Vector Vertex in Spinor Electrodynamics,” Phys. Rev. 177 (1969),2426–2438.[60] E. Salpeter and H. Bethe, “A Relativistic Equation for Bound-StateProblems,” Phys. Rev. 84 (1951), 1232–1242.[61] M. Gell-Mann and F. Low, “Bound States in Quantum Field Theory,”Phys. Rev. 84 (1951), 350–354.[62] J. Schwinger, “On the Euclidean Structure of Relativistic Field Theory,”Proc. Natl. Acad. Sci. USA 44 (1958), 956–965; “Euclidean QuantumElectrodynamics,” Phys. Rev. 115 (1959), 721–731.48[63] J. Schwinger, “Spin, Statistics, and the TCP Theorem,” Proc. Natl.Acad. Sci. USA 44 (1958), 223–228; “Addendum to Spin, Statistics,and the TCP Theorem,” ibid. 44 (1958), 617–619.[64] P. C. Martin and J. Schwinger, “Theory of Many-Particle Systems,”Phys. Rev. 115 (1959), 1342–1373.[65] L. V. Keldysh, “Diagram Technique for Nonequilibrium Processes,”Soviet Physics JETP 20 (1965), 1018–1026 [Zh. Eksp. Teor. Fiz. 47(1964), 1515–1527].[66] J. Schwinger, “Brownian Motion of a Quantum Oscillator,” J. Math.Phys. 2 (1961), 407–432.[67] J. Schwinger, “Field Theory Commutators,” Phys. Rev. Lett. 3 (1959),296–297.[68] J. Schwinger, “Gauge Invariance and Mass,” Phys. Rev. 125 (1962),397–398; “Gauge Invariance and Mass II,” ibid. 128 (1962), 2425-2429.[69] J. Schwinger, “Non-Abelian Gauge Fields. Commutation Relations,”Phys. Rev. 125 (1962), 1043–1048; “Non-Abelian Gauge Fields. RelativisticInvariance,” ibid. 127 (1962), 324–330; “Non-Abelian GaugeFields. Lorentz Gauge Formulation,” ibid. 130 (1963), 402–405.[70] R. Arnowitt, S. Deser, and C. W. Misner, “Canonical Variables forGeneral Relativity,” Phys. Rev. 117 (1960), 1595–1602.[71] J. Schwinger, Quantum Kinematics and Dynamics (New York: Benjamin,1970).[72] J. Schwinger, Quantum Mechanics: Symbolism of Atomic Measurement,ed. B. Englert (Berlin: Springer, 2001).[73] J. Schwinger, “On Angular Momentum,” 1952, later published inQuantum Theory of Angular Momentum, eds. L. S. Biedenharn andH. Van Dam (New York: Academic Press, 1965), p. 229–279.[74] M. Flato, C. Fronsdal, and K. A. Milton, eds., Selected Papers (1937–1976) of Julian Schwinger (Dordrecht: Reidel, 1979).49[75] B. Lippmann and J. Schwinger, “Variational Principles for ScatteringProcesses. I” Phys. Rev. 79 (1950), 469–480.[76] J. Schwinger, “A Theory of Fundamental Interactions,” Ann. Phys.(N.Y.) 2 (1957), 407–434.[77] S. Glashow, “Partial-Symmetries of Weak Interactions,” Nucl. Phys.22 (1961), 579–588.[78] S. Weinberg, “A Model of Leptons,” Phys. Rev. Lett. 19 (1967), 1264–1266.[79] J. Schwinger, “Particles and Sources,” Phys. Rev. 152 (1966), 1219–1226.[80] For a contemporary account of S-matrix theory, see R. J. Eden, P. V.Landshoff, D. I. Olive, and J. C. Polkinghorne, The Analytic S-Matrix(Cambridge University Press, 1966).[81] For Regge poles, see S. C. Frautschi, Regge Poles and S-Matrix Theory(New York: Benjamin, 1963).[82] For current algebra, see S. L. Adler and R. F. Dashen, Current Algebrasand Applications to Particle Physics (New York: Benjamin, 1968).[83] Bootstrap calculations were introduced in G. F. Chew and S. Mandelstam,“Theory of the Low-Energy Pion-Pion Interaction–II,” NuovoCimento 19, Series 10 (1961), 752–776. A survey of S-matrix theoryjust before the bootstrap hypothesis may be found in G. F. Chew,S-Matrix Theory of Strong Interactions (New York: Benjamin, 1961).[84] An accessible early exposition of this approach is found in R. F. Streaterand A. S. Wightman, PCT, Spin and Statistics, and All That (NewYork: Benjamin, 1964).[85] For a modern exposition of some of these ideas, see J. Glimm and A.Jaffe, Quantum Physics: A Functional Integral Point of View (NewYork: Springer-Verlag, 1981).[86] J. Schwinger, “Back to the Source,” Proceedings of the 1967 InternationalConference on Particles and Fields, eds. C. R. Hagen, G. Guralnik,and V. A. Mathur (New York: Interscience, 1967), pp. 128–156.50[87] J. Schwinger, “Julian Schwinger’s Approach to Particle Theory,” StanleyH. Klosk lecture at NYU School of Engineering Science, publishedin Scientific Research, August 18, 1969, pp. 19–24.[88] R. Finkelstein, “Julian Schwinger: The QED Period at Michigan andthe Source Theory Period at UCLA” in Julian Schwinger: The Physicist,the Teacher, and the Man, ed. Y. J. Ng (Singapore: World Scientific,1996), p. 105–109.[89] G. ’t Hooft, “Renormalization of Massless Yang-Mills Fields,” Nucl.Phys. B33 (1971), 173–199; “Renormalizable Lagrangians for MassiveYang-Mills Fields,”B35 (1971), 167–188.[90] D. J. Gross and F. Wilczek, “Ultraviolet Behavior of Non-AbelianGauge Theories,”Phys. Rev. Lett. 30 (1973), 1343–1346; H. D. Politzer,“Reliable Perturbative Results for Strong Interactions?” Phys. Rev.Lett. 30 (1974), 1346–1349; “Asymptotic Freedom: An Approach toStrong Interactions,” Phys. Rep. 14C (1974), 129-180.[91] M. Gell-Mann, Acta Phys. Austriaca Suppl. IV (1972), 733; H. Fritzschand M. Gell-Mann, “Current Algebra: Quarks and What Else?” inProc. XVI Int. Conf. on High Energy Physics, ed. J. D. Jackson and A.Roberts (Batavia, IL: National Accelerator Laboratory, 1972) pp. 135–165; W. A. Bardeen, H. Fritzsch, and M. Gell-Mann, “Light-Cone,Current Algebra, π0 Decay, and e+e− Annihilation,” in Scale and ConformalSymmetry in Hadron Physics, ed. R. Gatto (New York: Wiley,1973), p. 139–151.[92] J. Schwinger, Particles, Sources, and Fields, Vol. I–III (Reading MA:Addison-Wesley, 1970, 1973, 1989).[93] J. Schwinger, “Source Theory Viewpoints in Deep Inelastic Scattering,”Proc. Natl. Acad. Sci. USA 72 (1975), 1–5; “Deep Inelastic Scatteringof Leptons,” ibid. 73 (1976), 3351–3354; “Deep Inelastic Scattering ofCharged Leptons,” ibid. 73 (1976), 3816–3819; “Deep Inelastic NeutrinoScattering and Pion-Nucleon Cross Sections,” Phys. Lett. 67B(1977), 89–90; “Adler’s Sum Rule in Source Theory,” Phys. Rev. D 15(1977), 910–912; “Deep Inelastic Sum Rules in Source Theory,” Nucl.Phys. B 123 (1977), 223–239.51[94] J. Schwinger, “Sources and Magnetic Charge,” Phys. Rev. 173 (1968),1536–1544; “A Magnetic Model of Matter,” Science 165 (1969), 757–767; “Magnetic Charge and the Charge Quantization Condition,” Phys.Rev. D 12 (1975), 3105–3111; J. Schwinger, K. A. Milton, W.-Y. Tsai,L. L. DeRaad, Jr., “Non-relativistic Dyon-Dyon Scattering,” Ann.Phys. (N.Y.) 101 (1975), 451–495.[95] J. Schwinger, “Classical Radiation of Accelerated Electrons II. A QuantumViewpoint,” Phys. Rev. D 7 (1973), 1696–1701; J. Schwinger,W.-y. Tsai, and T. Erber, “Classical and Quantum Theory of SynergicSynchrotron-ˇ Cerenkov Radiation,” Ann. Phys. (N.Y.) 96 (1976),303–331; J. Schwinger and W.-y. Tsai, “New Approach to QuantumCorrection in Synchrotron Radiation,” ibid. 110 (1978), 63–84.[96] J. Schwinger, “Casimir Effect in Source Theory,” Lett. Math. Phys.1 (1975), 43–47; J. Schwinger, L. L. DeRaad, Jr., and K. A. Milton,“Casimir Effect in Dielectrics,” Ann. Phys. (N.Y.) 115 (1978), 1–23;K. A. Milton, L. L. DeRaad, Jr., and J. Schwinger, “Casimir Self-Stresson a Perfectly Conducting Spherical Shell,” ibid. 115 (1978), 388–403.[97] J. Schwinger, “Multispinor Basis of Fermi-Bose Transformation,” Ann.Phys. (N.Y.) 119 (1979), 192–237.[98] R. Jackiw and K. Johnson, “Anomalies of the Axial-Vector Current,”Phys. Rev. 182 (1969), 1459–1469.[99] S. L. Adler and W. A. Bardeen, “Absence of Higher-Order Correctionsin the Anomalous Axial-Vector Divergence Equation,” Phys. Rev. 182(1969), 1517–1536.[100] L. L. DeRaad, Jr., K. A. Milton, and W.-y. Tsai, “Second-Order RadiativeCorrections to the Triangle Anomaly. I,” Phys. Rev. D 6 (1972),1766–1780; K. A. Milton, W.-y. Tsai, and L. L. DeRaad, Jr., “Second-Order Radiative Corrections to the Triangle Anomaly. II,” Phys. Rev.D 6 (1972), 3491–3500.[101] S. L. Adler, “Anomalies to All Orders,” in Fifty Years of Yang-MillsTheory, ed. G. ’t Hooft (Singapore: World Scientific, 2005) pp. 187–228[arXiv:hep-th/0405040]52[102] J. Schwinger, “Thomas-Fermi Model: The Leading Correction,” Phys.Rev. A 22 (1980), 1827–1832; “Thomas-Fermi Model: The SecondCorrection,” ibid. 24 (1982), 2353–2361; J. Schwinger and L. L. DeRaad,Jr., “New Thomas-Fermi Theory: A Test,” ibid. 25 (1982), 2399–2401; B.-G. Englert and J. Schwinger, “Thomas-Fermi Revisited: TheOuter Regions of the Atom,” ibid. 26 (1982), 2322–2329; “StatisticalAtom: Handling the Strongly Bound Electrons,” ibid. 29 (1984),2331–2338; “Statistical Atom: Some Quantum Improvements,” ibid.29 (1984), 2339–2352; “New Statistical Atom: A Numerical Study,”ibid. 29 (1984), 2353–2363; “Semiclassical Atom,” ibid. 32 (1985), 26–35; “Linear Degeneracy in the Semiclassical Atom,” ibid. 32 (1985),36–46; “Atomic-Binding-Energy Oscillations,” ibid. 32 (1985), 47–63.[103] C. Fefferman and L. Seco, “On the Energy of Large Atoms,” Bull.Am. Math. Soc. 23 (1990), 525–530, continuing through “The EigenvalueSum for a Three-Dimensional Radial Potential,” Adv. Math. 119(1996), 26–116. See also A. Cordoba, C. Fefferman, and L. Seco, “ANumber-Theoretic Estimate for the Thomas-Fermi Density,” Comm.Part. Diff. Eqn. 21 (1996), 1087–1102.[104] Seth Putterman, conversation with K. Milton, in Los Angeles, July 28,1997.[105] B.-G. Englert, J. Schwinger, and M. O. Scully, “Is Spin Coherence LikeHumpty Dumpty? I. Simplified Treatment,” Found. Phys. 18 (1988),1045–1056; “Is Spin Coherence Like Humpty Dumpty? II. GeneralTheory,” Z. Phys. D 10 (1988), 135–144; “Spin Coherence and HumptyDumpty. III. The Effects of Observation,” Phys. Rev. A 40 (1989),1775–1784.[106] I. Langmuir, Physics Today, October 1989, p. 36.[107] New York Times (National Edition), March 24, 1989, p. A16; M. Fleischmann,S. Pons, and M. Hawkins, “Electrochemically Induced NuclearFusion of Deuterium,” J. Electroanal. Chem. 261 (1989), 301–308; errata:263 (1989), 187–188.[108] Berthold Englert, correspondence to K. Milton, February 8, 1998.[109] J. Schwinger, “Cold Fusion: A Hypothesis,” Z. Naturforsch. A 45(1990), 756.53[110] Berthold-Georg Englert, telephone interview by K. Milton, March 16,1997.[111] J. Schwinger, “Nuclear Energy in an Atomic Lattice,” Z. Phys. D, 15(1990), 221–225.[112] M. P. Brenner, S. Hilgenfeldt, and D. Lohse, “Single-bubble sonoluminescence,”Rev. Mod. Phys. 74 (2002), 425–484.[113] J. Schwinger, “Casimir Light: A Glimpse,” Proc. Natl. Acad. Sci. USA90 (1993), 958–959; “Casimir Light: The Source,” ibid. 90 (1993),2105–2106; “Casimir Light: Photon Pairs,” ibid. 90 (1993), 4505–4507;“Casimir Light: Pieces of the Action,” ibid. 90 (1993), 7285–7287;“Casimir Light: Field Pressure,” ibid. 91 (1994), 6473–6475.[114] “Schwinger’s Response to an Honorary Degree at Nottingham,” in JulianSchwinger: The Physicist, the Teacher, and the Man, ed. Y. J. Ng(Singapore: World Scientific, 1996), p. 11–12.54