Physics Today The conceptual origins of Maxwell’s equations and gauge theoryChen Ning Yang Citation: Physics Today 67(11), 45 (2014); doi: 10.1063/PT.3.2585 View online: http://dx.doi.org/10.1063/PT.3.2585 View Table of Contents: http://scitation.aip.org/content/aip/magazine/physicstoday/67/11?ver=pdfcov Published by the AIP Publishing

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I t is often said that after Charles Augustin deCoulomb, Carl Friedrich Gauss, André Marie Am-père, and Michael Faraday discovered the fourexperimental laws concerning electricity andmagnetism, James Clerk Maxwell added the dis-

placement current and thereby created the great setof Maxwell’s equations. That view is not entirelywrong, but it obscures the subtle interplay betweensophisticated geometrical and physical intuitionsthat led not only to the replacement of “action at adistance” by field theory in the 19th century butalso, in the 20th century, to the very successful stan-dard model of particle physics.

The 19th centuryIn 1820 Hans Christian Oersted (1777–1851) discov-ered that an electric current would always causemagnetic needles in its neighborhood to move. Thediscovery electrified the whole of Europe and led tothe successful mathematical theory of “action at adistance” by Ampère (1775–1836). In England, Fara-day (1791–1867) was also greatly excited by Oersted’s

discovery. But he lacked the mathematical trainingneeded to understand Ampère. In a letter to Ampèredated 3 September 1822, Faraday lamented, “I amunfortunate in a want of mathematical knowledgeand the power of entering with facility into abstractreasoning. I am obliged to feel my way by facts closelyplaced together.”1

Faraday’s “facts” were his experiments, bothpublished and unpublished. During a period of 23years, 1831–54, he compiled the results of those ex-periments into three volumes, called ExperimentalResearches in Electricity, which we shall refer to as ER(figure 1). A most remarkable thing is that there wasnot a single formula in this monumental compila-tion, which showed that Faraday was feeling his way,guided only by geometric intuition without any pre-cise algebraic formulation.

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C. N. Yang is the honorary director of the Institute for Advanced Study at Tsinghua University in Beijing and a Distinguished Professor-at-Large in the physics departmentat the Chinese University of Hong Kong.

Chen Ning Yang

Already in Faraday’s electrotonic state and Maxwell’s vector potential,gauge freedom was an unavoidable presence. Converting that presence

to the symmetry principle that underpins our successful standard model is a story worth telling.

Maxwell’s equationsThe conceptual origins of

and gauge theory

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Figure 2 shows a diagram from Faraday’s diary,dated 17 October 1831, the day he found that mov-ing a bar magnet either into or out of a solenoidwould generate electric currents in the solenoid.Thus he had discovered electric induction, which, aswe know, eventually led to making big and small gen-erators of electricity and thereby changed the techno-logical history of mankind.

Throughout the volumes of ER, Faraday ex-plored variations of his induction experiment: Hechanged the metal used for winding the solenoid,immersed the solenoid in various media, created in-duction between two coils, and so on. He was espe-cially impressed by two facts—namely, that themagnet must be moved to produce induction, andthat induction seemed to produce effects perpendic -ular to the cause.

Feeling his way toward an understanding of in-duction, he introduced two geometric concepts:magnetic lines of force and the electrotonic state.The former was easily visualized by sprinkling ironfilings around magnets and solenoids. Those linesof force are today designated by the symbol H, themagnetic field. The latter, the electrotonic state, re-mained indefinite and elusive throughout the entireER. It first appeared early in volume 1, section 60,without any precise definition. Later it was vari-ously called the peculiar state, state of tension, pe-culiar condition, and other things. For example, insection 66 he wrote, “All metals take on the pecu-liar state,” and in section 68, “The state appears to beinstantly assumed.” More extensively, we read insection 1114,

If we endeavour to consider electricityand magnetism as the results of twofaces of a physical agent, or a peculiarcondition of matter, exerted in determi-nate directions perpendicular to eachother, then, it appears to me, that wemust consider these two states or forcesas convertible into each other in agreater or smaller degree.

When Faraday ceased his compi-lation of ER in 1854 at age 63, his geo-metric intuition, the electrotonic state,remained undefined and elusive.

Enter MaxwellAlso in 1854, Maxwell (1831–79) gradu-ated from Trinity College. He was 23years old and full of youthful enthusi-asm. On February 20 he wrote to WilliamThomson,

Suppose a man to have a popularknowledge of electric show experi-ments and a little antipathy to Mur-phy’s Electricity, how ought he to pro-ceed in reading & working so as to geta little insight into the subject wh[sic]may be of use in further reading?

If he wished to read Ampere Faraday&c how should they be arranged, andat what stage & in what order might heread your articles in the CambridgeJournal?2

Thomson (later Lord Kelvin, 1824–1907) was aprodigy. At the time, he had already occupied theChair of Natural Philosophy at Glasgow Universityfor eight years. Maxwell had chosen well: Earlier in1851 Thomson had introduced what we now call thevector potential A to express the magnetic field Hthrough

H = ∇ × A , (1)

an equation that would be of crucial importance forMaxwell, as we shall see.

We do not know how Thomson responded toMaxwell’s inquiry. What we do know is that, amaz-ingly, only a little more than one year later, Maxwellwas able to use equation 1 to give meaning to Fara-day’s elusive electrotonic state and publish the firstof his three great papers, which revolutionizedphysics and forever changed human history. Thosethree papers together with others by Maxwell hadbeen edited by William Davidson Niven in 1890 intoa two-volume collection, Scientific Papers of JamesClerk Maxwell, which we shall refer to as JM.

Maxwell’s first paper, published in 1856, is fullof formulas and therefore easier to read than Fara-day’s ER. Its central ideas are contained in part 2,which has as its title “Faraday’s Electro-tonic State.”On page 204 in this part 2 we find an equation thatin today’s vector notation is

E = −A , (2)

where A is Faraday’s electrotonic intensity.Three pages later, on page 207 of JM, the result

is restated in words:

Law VI. The electro-motive force on anyelement of a conductor is measured bythe instantaneous rate of change of theelectro-tonic intensity on that element,whether in magnitude or direction.

The identification of Faraday’s elusive idea ofthe electrotonic state (or electrotonic intensity, or elec-

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Maxwell’s equations

Figure 1. Three volumes of Faraday’s ExperimentalResearches in Electricity, published separately in 1839, 1844, and 1855. On the right is the first page of the first volume.

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trotonic function) with Thomson’s vector potentialA defined in equation 1 above is, in my opinion, thefirst great conceptual breakthrough in Maxwell’sscientific research: Taking the curl of both sides ofequation 2 we obtain

∇ × E = −H , (3)

which is the modern form of Faraday’s law. Anothermodern form of it is

∫E ·dl = − ∬ H · dσ, (4)

where dl is a line element and dσ is an element ofarea. Maxwell did not write Faraday’s law in eitherof the two forms in equations 3 and 4 because hisaim was to give precise definition to Faraday’s elu-sive concept of the electrotonic state. Indeed, theconcept of the vector potential A remained centralin Maxwell’s thinking throughout his life.

Maxwell was aware of what we now call thegauge freedom in equations 1–3, namely, that thegradient of an arbitrary scalar function can be addedto A without changing the result. He discussed thatfreedom explicitly in theorem 5 on page 198 of JM. Sowhich gauge did he choose for A in equations 1–3?He did not touch on that question, but left it com-pletely indeterminate. My conclusion: Maxwell im-plied that there exists a gauge for A in which equa-tions 1–3 are satisfied.

Maxwell was also fully aware of the importanceof his identification of Faraday’s electrotonic inten-sity with Thomson’s A. He was afraid that Thomsonmight take offense concerning the priority question.He therefore concluded part 2 of his first paper withthe following remark:

With respect to the history of the pres-ent theory, I may state that the recogni-tion of certain mathematical functionsas expressing the “electro-tonic state”of Faraday, and the use of them in de-termining electro-dynamic potentialsand electro-motive forces is, as far as Iam aware, original; but the distinct con-ception of the possibility of the mathe-matical expressions arose in my mindfrom the perusal of Prof. W. Thomson’spapers. (JM, page 209)

Maxwell’s vorticesFive years after completing his first paper, Maxwellbegan publishing his second, which appeared infour parts during 1861 and 1862. In contrast to theearlier paper, the second is very difficult to read. Themain idea of the paper was to account for electro-magnetic phenomena “on the hypothesis of the mag-netic field being occupied with innumerable vor-tices of revolving matter, their axes coinciding withthe direction of the magnetic force at every point ofthe field,” as we read in JM on page 489.

Maxwell gave an explicit example of such an in-tricate group of vortices in a diagram reproducedhere as figure 3, which he explained in detail onpage 477 of JM in the following passage:

Let AB, Plate VIII., p. 488, fig. 2, representa current of electricity in the directionfrom A to B. Let the large spaces aboveand below AB represent the vortices,and let the small circles separating thevortices represent the layers of particlesplaced between them, which in our hy-pothesis represent electricity.

Now let an electric current from leftto right commence in AB. The row ofvortices gh above AB will be set in mo-tion in the opposite direction to that ofa watch. (We shall call this direction +,and that of a watch –.) We shall supposethe row of vortices kl still at rest, thenthe layer of particles between theserows will be acted on by the row gh ontheir lower sides, and will be at restabove. If they are free to move, they willrotate in the negative direction, and willat the same time move from right to left,or in the opposite direction from thecurrent, and so form an induced electriccurrent.

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in an etched portrait. Theinset shows the diagram Faraday drew in his diary on 17 October1831, the day he discovered induction.

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That detailed explanation of Maxwell’s modelappeared in part 2 of his second paper and was pub-lished in Philosophical Magazine, volume 21, April–May 1861. Maxwell evidently took his intricate net-work of vortices very seriously and devoted theremaining 11 pages of part 2 to detailed studies ofthe model.

Then in January and February 1862, Maxwellpublished part 3 of his second paper with the title,“The Theory of Molecular Vortices Applied to Stat-ical Electricity.” Seven pages of analysis led to hisproposition 14: “To correct the equations of electriccurrents for the effect due to the elasticity of themedium” (JM, page 496). The correction was to adda “displacement current” Ė to Ampère’s law, whichin modern notation then reads ∇ × H = 4πj + Ė.

I had made several attempts to read the last 11pages of part 2 and the first 7 pages of part 3, tryingto see how Maxwell was led to his correction. In par-ticular, I wanted to learn what he meant by “the ef-fect due to the elasticity of the medium.” All my at-tempts failed. It is noteworthy that in the last 11pages of part 2, the word “displacement” occurs onlyonce, on page 479, in an unimportant sentence,whereas in the beginning 7 pages of part 3 that wordbecomes the center of Maxwell’s focus. Thus it seemsthat in the eight months between the publication ofthe two parts, Maxwell had found new features ofhis network of vortices to explore, leading to the dis-placement current.

After proposition 14 Maxwell quickly con-cluded that there should be electromagnetic waves.He calculated their velocity, compared it with theknown velocity of light, and reached the momen-tous conclusion that “We can scarcely avoid the in-ference that light consists in the transverse undulations

of the same medium which is the cause of electric andmagnetic phenomena” (page 500; the italics areMaxwell’s).

Maxwell was a religious person. I wonderwhether after this momentous discovery he had inhis prayers asked for God’s forgiveness for revealingone of His greatest secrets.

The birth of field theoryMaxwell’s third paper, published in 1865, gave riseto what today we call Maxwell’s equations, of whichthere are four in vector notation. Maxwell used 20equations: He wrote them in component form andalso included equations for dielectrics and electriccurrents.

That third paper is historically the first to givea clear enunciation of the conceptual basis of fieldtheory—that energy resides in the field:

In speaking of the Energy of the field,however, I wish to be understood liter-ally. All energy is the same as mechanicalenergy, whether it exists in the form ofmotion or in that of elasticity, or in anyother form. The energy in electromag-netic phenomena is mechanical energy.The only question is, Where does it re-side? On the old theories it resides inthe electrified bodies, conducting cir-cuits, and magnets, in the form of an un-known quality called potential energy,or the power of producing certain ef-fects at a distance. On our theory it re-sides in the electromagnetic field, in thespace surrounding the electrified andmagnetic bodies, as well as in those bod-ies themselves, and is in two differentforms, which may be described withouthypothesis as magnetic polarizationand electric polarization, or, accordingto a very probable hypothesis, as themotion and the strain of one and thesame medium. (JM, page 564)

But, in conformity with the prevalent ideas of thetime, Maxwell also wrote,

We have therefore some reason to be-lieve, from the phenomena of light andheat, that there is an aethereal mediumfilling space and permeating bodies, ca-pable of being set in motion and oftransmitting that motion from one partto another, and of communicating thatmotion to gross matter so as to heat itand affect it in various ways. (JM, page528)

Maxwell realized the great importance of hisdiscovery of the displacement current and his con-clusion that light is electromagnetic waves. In histhird paper, he collected the formulas of the two ear-lier papers and listed them together. In the processhe must have reviewed the arguments that had ledto those formulas. So after his review, how did hefeel about the intricate network of vortices that hadled to the displacement current three years earlier?

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Maxwell’s equations

Figure 3. A figure of vortices from a plate in the 1890 collection Scientific Papers of James Clerk Maxwell, facing page 488. The directionsof the arrows in two of the hexagonal vortices in the second row fromthe bottom are incorrect, presumably mistakes of Maxwell’s draftsman.

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Maxwell did not discuss that point. But we noticethat the word “vortex” did not appear in any of the 71 pages of his third paper. It thus seems reason-able to assume that in 1865 Maxwell no longer considered as relevant the network of vortices of his second paper. But he still saw the necessity of“an aethereal medium filling space and permeatingbodies.”

In 1886 Heinrich Hertz (1857–94) experimentallyverified an important consequence of Maxwell’sequations: that electromagnetic waves can be gener-ated by one set of electric circuits and detected byanother.

Starting in the mid 1880s Oliver Heaviside(1850–1925) and Hertz independently discovered thatone can eliminate from Maxwell’s equations the vec-tor potential A. The simplified equations then havethe additional attractive feature of exhibiting a highdegree of symmetry between the electric and mag-netic fields. We now know that in quantum mechan-ics, the vector potential cannot be eliminated. It hasobservable effects as in the Aharonov–Bohm effect.

Into the 20th centuryA conceptual revolution in field theory came earlyin the 20th century following Albert Einstein’s 1905special theory of relativity, which asserted that thereis no other medium at all: The electromagnetic fieldis the medium. The vacuum is then the state of a re-gion of spacetime where there is no electromagneticradiation and no material particles. That solved thepuzzle posed by the 1887 Michelson–Morley exper-iment, which looked for the aethereal medium butfailed to find it. Most physicists today believe Einstein’s motivation in formulating the theory ofspecial relativity was not to solve the puzzle posed by the Michelson–Morley experiment but rather to recognize the correct meaning of the concept ofsimultaneity.

In the years 1930–32, with the experimental dis-covery of the positron, it became necessary to dras-tically modify one’s view of the vacuum and toadopt instead Paul Dirac’s theory of the infinite seaof negative-energy particles. That was another con-ceptual revolution in field theory, and it culminatedin the theory of quantum electrodynamics. QEDproved successful for low-order calculations in the1930s but was beset with infinity-related difficultiesin calculations carried out to higher orders.

In a series of brilliant and dramatic experimen-tal and theoretical breakthroughs in the 1947–50 pe-riod, QED became quantitatively successful throughthe method of renormalization—a recipe for calcu-lating high-order corrections. The latest report ofthe calculated value of the anomalous magnetic mo-ment of the electron,3 a = (g − 2)/2, is in agreementwith its experimental value to an incredible accu-racy of one part in 109 (see the Quick Study by Gerald Gabrielse, PHYSICS TODAY, December 2013,page 64).

With the success of the renormalization pro-gram of QED and with the experimental discoveryof many mesons and strange particles, efforts weremade to extend field theory to describe the interac-tions between all of the new particles. Papers and

books appeared on scalar meson theory with vectorinteraction, pseudoscalar meson theory with pseu-doscalar interaction, and other esoteric topics. Noneof those efforts produced fundamental advances inour conceptual understanding of interactions. Therewere also enthusiastic supporters of efforts to findalternatives to field theory, but again no real break-throughs.

Return to field theoryIn the 1970s physicists returned to field theory,specifically to non-abelian gauge theory, which wasan elegant generalization of Maxwell’s theory. Theterm non-abelian means that the order in which ro-tations or other operations take place matters. (I dis-cussed gauge theories and other things in “Ein-stein’s impact on theoretical physics,” PHYSICSTODAY, June 1980, page 42. For a more technical dis-cussion, see the article by Isidore Singer, PHYSICSTODAY, March 1982, page 41.) Gauge theory is todayrecognized as of fundamental conceptual impor-tance in the structure of interactions in the physicaluniverse. It started with three papers published in1918–19 by mathematician Hermann Weyl, who wasinfluenced by Einstein’s call for the geometrizationof electromagnetism.4

Weyl was motivated by the importance of par-allel displacement. He argued that “the fundamen-tal conception on which the development of Rie-mann’s geometry must be based if it is to be inagreement with nature, is that of the infinitesimalparallel displacement of a vector.” Weyl then saidthat if, in the infinitesimal displacement of a vector,its direction keeps changing, then “Warum nichtauch seine Länge?” (Why not also its length?) ThusWeyl proposed a nonintegrable “Streckenfacktor,”or “Proportionalitätsfacktor,” which he related tothe electromagnetic field through

exp(−∫eAμdxμ/γ) , (5)

in which Aμ is the four-dimensional vector potentialand the coefficient γ is real. Weyl attached such astretch factor to every charged object movingthrough spacetime. To the second of Weyl’s three pa-pers, Einstein appended a postscript that criticizedWeyl’s idea of length changes in displacements.Weyl was unable to effectively respond to this dev-astating criticism.

After the development of quantum mechanicsin 1925–26, Vladimir Fock and Fritz London inde-pendently pointed out that in the new quantumframework, it was necessary to replace (p – eA) by

www.physicstoday.org November 2014 Physics Today 49

Maxwell was a religious person.I wonder whether he had inhis prayers asked for God’s

forgiveness for revealing one of His greatest secrets.

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(6)

which suggested the replacement in equation 5 ofeAμdxμ/γ by ieAμdxμ/ħ, that is, of γ by −iħ.

Evidently, Weyl accepted the idea that γ shouldbe imaginary, and in 1929 he published an impor-tant paper in which he explicitly defined the conceptof gauge transformation in QED and showed thatunder such a transformation, Maxwell’s theory inquantum mechanics is invariant.

Under a gauge transformation, Weyl’s length-change factor is replaced by

(7)

which evidently should have been called a phase-change factor. The replacement also renders inoper-ative Einstein’s criticism of Weyl’s idea mentionedabove.

That Maxwell’s equations have a high degree ofsymmetry was first shown in 1905–07 by Einsteinand Hermann Minkowski, who separately discov-ered the equations’ Lorentz invariance. Weyl’s 1929discovery that Maxwell’s equations are invariantunder gauge transformations revealed anothersymmetry property of the equations. Today we real-ize that these symmetry properties make Maxwell’sequations a fundamental pillar in the structure ofthe physical universe.

Weyl’s gauge transformation involves, at everyspacetime point, a so-called U(1) rotation—essentiallya simple rotation in the complex plane. There is thusa striking similarity between Weyl’s gauge transfor-mation and Maxwell’s network of rotating vortices.The similarity is, of course, fortuitous.

Mathematically, the phase factors of formula 7form a Lie group U(1), and one of Weyl’s favorite research fields was Lie groups. Going one step fur-ther for the more technical reader, had fiber-bundletheory been developed before 1929, Weyl could cer-tainly have realized that electromagnetism was aU(1) bundle theory and would likely have general-ized it to non-abelian gauge theory as a naturalmathematical extension in 1929.

In the event, the extension was made in 1954,motivated not by mathematical considerations but bythe need to find a principle for interactions in the newfield of particle physics in which there were foundmany new “strange” particles. The physical motiva-tion was concisely stated in a short 1954 abstract:

The electric charge serves as a source ofelectromagnetic field; an important con-cept in this case is gauge invariancewhich is closely connected with (1) theequation of motion of the electromag-netic field, (2) the existence of a currentdensity, and (3) the possible interactionsbetween a charged field and the electro-magnetic field. We have tried to gener-alize this concept of gauge invariance toapply to isotopic spin conservation.5

That extension led to a non-abelian field theory thatwas very beautiful but was not embraced by the

physics community for many years because itseemed to require the existence of massless chargedparticles.

To give mass to the massless particles in a non-abelian field theory, the concept of spontaneous sym-metry breaking was introduced in the 1960s. Thatconcept in turn led to a series of major advances, andfinally to a U(1) × SU(2) × SU(3) gauge theory ofelectroweak interactions and strong interactionsthat we now call the standard model. In the fifty-some years since 1960, the international theoreticaland experimental research community working in“particles and fields” combined their individualand collective efforts to develop and verify the stan-dard model. Those efforts met with spectacular suc-cess, climaxing in the discovery of the Higgs bosonin 2012 by two large experimental groups at CERN,each consisting of several thousand physicists (seePHYSICS TODAY, September 2012, page 12).

Despite its impressive success, the standardmodel is not the final story. To start with, dozens ofconstants need to enter the model. More important,one of its chief ingredients, the symmetry-breakingmechanism, is a phenomenological construct that inmany respects is similar to Fermi’s proposed “four-ψ interaction” to explain beta decay.6 That 1934 the-ory was also successful for almost 40 years. But itwas finally replaced by the deeper U(1) × SU(2) elec-troweak theory.

Gauge freedom was explicitly known to Thom-son and Maxwell in the 1850s. It probably had alsobeen vaguely sensed by Faraday in his elusive for-mulation of the electrotonic state. The gauge free-dom was converted by Weyl in 1929 to a symmetry(or invariant) property of Maxwell’s equations inquantum mechanics. That symmetry property, nowcalled gauge symmetry, forms the structural back-bone of the standard model today.

Maxwell’s equations are linear. In non-abeliangauge theory, the equations are nonlinear. The non-linearity arises conceptually from the same origin as the nonlinearity of the equations of general rela-tivity. About the latter nonlinearity Einstein hadwritten,

We shall speak only of the equations ofthe pure gravitational field.

The peculiarity of these equationslies, on the one hand, in their compli-cated construction, especially their non-linear character as regards the field-variables and their derivatives, and, onthe other hand, in the almost compellingnecessity with which the transforma-tion-group determines this complicatedfield-law. (reference 7, page 75)

The true laws can not be linear norcan they be derived from such. (page 89)

Entirely independent of developments in physics,there emerged during the first half of the 20th cen-tury a mathematical theory called fiber-bundle the-ory, which had diverse conceptual origins, includingdifferential forms (mainly due to Élie Cartan), statistics (Harold Hotelling), topology (HasslerWhitney), global differential geometry (Shiing-Shen

eħ( (exp −I A dx ,

µ

µ∫

− − ,iħ ∂ Aμ μieħ( (

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Maxwell’s equations

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Chern), and connection theory (Charles Ehres-mann). The great diversity of its conceptual originindicates that the fiber bundle is a central mathe-matical construct.

It came as a great shock to both physicists andmathematicians in the 1970s that the mathematics ofgauge theory, both abelian and non-abelian, turnedout to be exactly the same as that of fiber-bundle the-ory.8 But it was a welcome shock because it servedto bring back the close relationship between the twodisciplines, a relationship that had been interruptedby the increasingly abstract nature of mathematicssince the middle of the 20th century.

In 1975, after learning the rudiments of fiber-bundle theory from my mathematician colleagueJames Simons, I showed him the 1931 paper by Diracon the magnetic monopole. He exclaimed, “Dirachad discovered trivial and nontrivial bundles beforemathematicians.”

It is perhaps not inappropriate to conclude thisbrief sketch of the conceptual origin of gauge theoryby quoting a few paragraphs from Maxwell’s tributeupon Faraday’s death in 1867:

The way in which Faraday made use ofhis idea of lines of force in co-ordinatingthe phenomena of magneto-electric in-duction shows him to have been in re-ality a mathematician of a very highorder—one from whom the mathemati-cians of the future may derive valuableand fertile methods. . . .

From the straight line of Euclid to thelines of force of Faraday this has beenthe character of the ideas by which science has been advanced, and by thefree use of dynamical as well as geo -metrical ideas we may hope for furtheradvance. . . .

We are probably ignorant even of thename of the science which will be devel-oped out of the materials we are now col-lecting, when the great philosopher nextafter Faraday makes his appearance.

References1. F. A. J. L. James, ed., The Correspondence of Michael Fara-

day, Vol. 1, Institution of Electrical Engineers (1991), p. 287.

2. J. Larmor, Proc. Cambridge Philos. Soc. 32, 695 (1936), p. 697.

3. T. Kinoshita, in Proceedings of the Conference in Honour ofthe 90th Birthday of Freeman Dyson, K. K. Phua et al.,eds., World Scientific (2014), p. 148.

4. For this and the ensuing history, see C. N. Yang, in Her-mann Weyl, 1885–1985: Centenary Lectures, K. Chan-drasekharan, ed., Springer (1986), p. 7; and A. C. T. Wu,C. N. Yang, Int. J. Mod. Phys. A 21, 3235 (2006).

5. C. N. Yang, R. Mills, Phys. Rev. 95, 631 (1954). 6. An English translation of Fermi’s original paper is

available in F. L. Wilson, Am. J. Phys. 36, 1150 (1968).7. P. A. Schilpp, ed., Albert Einstein: Philosopher-Scientist,

Open Court (1949). The two passages are English trans-lations of Einstein’s autobiographical notes written in1946 when Einstein was 67 years old.

8. T. T. Wu, C. N. Yang, Phys. Rev. D 12, 3845 (1975). ■

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