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The conceptual origins of Maxwell’s equations and 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 principle that underpins our successful is a story worth telling. t is often said that after Charles Augustin de discovery. But he lacked the mathematical training Coulomb, Carl Friedrich , André Marie Am- needed to understand Ampère. In a letter to Ampère père, and discovered the four dated 3 September 1822, Faraday lamented, “I am I experimental laws concerning electricity and unfortunate in a want of mathematical knowledge , added the dis- and the power of entering with facility into abstract placement current and thereby created the great set reasoning. I am obliged to feel my way by facts closely of Maxwell’s equations. That view is not entirely placed together.”1 wrong, but it obscures the subtle interplay between Faraday’s “facts” were his experiments, both sophisticated geometrical and physical intuitions published and unpublished. During a period of 23 that led not only to the replacement of “action at a years, 1831–54, he compiled the results of those ex- distance” by theory in the 19th century but periments into three volumes, called Experimental also, in the 20th century, to the very successful stan- Researches in Electricity, which we shall refer to as ER dard model of . (figure 1). A most remarkable thing is that there was not a single formula in this monumental compila- The 19th century tion, which showed that Faraday was feeling his way, In 1820 Hans Christian (1777–1851) discov- guided only by geometric intuition without any pre- ered that an electric current would always cause cise algebraic formulation. magnetic needles in its neighborhood to move. The discovery electrified the whole of Europe and led to C. N. Yang is the honorary director of the Institute for the successful mathematical theory of “action at a Advanced Study at Tsinghua University in Beijing and a distance” by Ampère (1775–1836). In , Fara- Distinguished Professor-at-Large in the physics department day (1791–1867) was also greatly excited by Oersted’s at the Chinese University of Hong Kong.

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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 Maxwell Also in 1854, Maxwell (1831–79) gradu- ated from Trinity College. He was 23 years old and full of youthful enthusi- asm. On February 20 he wrote to William Thomson, Suppose a man to have a popular knowledge of electric show experi- ments and a little antipathy to Mur- phy’sElectricity, how ought he to pro- ceed in reading & working so as to get a little insight into the subject wh[sic] Figure 1. Three volumes of Faraday’s Experimental may be of use in further reading? Researches in Electricity, published separately in 1839, 1844, If he wished to read Ampere Faraday and 1855. On the right is the first page of the first volume. &c how should they be arranged, and

STEFAN KABEN, LIBRARY AND ESVA at what stage & in what order might he read your articles in the Cambridge Figure 2 shows a diagram from Faraday’s diary, Journal?2 dated 17 October 1831, the day he found that mov- ing a bar magnet either into or out of a solenoid Thomson (later Lord , 1824–1907) was a would generate electric currents in the solenoid. prodigy. At the , he had already occupied the Thus he had discovered electric induction, which, as Chair of Natural Philosophy at Glasgow University we know, eventually led to making big and small gen- for eight years. Maxwell had chosen well: Earlier in erators of electricity and thereby changed the techno- 1851 Thomson had introduced what we now call the logical history of mankind. vector potential A to express the H Throughout the volumes of ER, Faraday ex- through plored variations of his induction experiment: He H = ∇ × A, (1) changed the metal used for winding the solenoid, immersed the solenoid in various media, created in- an equation that would be of crucial importance for duction between two coils, and so on. He was espe- Maxwell, as we shall see. cially impressed by two facts—namely, that the We do not know how Thomson responded to magnet must be moved to produce induction, and Maxwell’s inquiry. What we do know is that, amaz- that induction seemed to produce effects perpendic - ingly, only a little more than one year later, Maxwell ular to the cause. was able to use equation 1 to give meaning to Fara- Feeling his way toward an understanding of in- day’s elusive electrotonic state and publish the first duction, he introduced two geometric concepts: of his three great papers, which revolutionized magnetic lines of force and the electrotonic state. physics and forever changed human history. Those The former was easily visualized by sprinkling iron three papers together with others by Maxwell had filings around magnets and solenoids. Those lines been edited by William Davidson Niven in 1890 into of force are today designated by the symbol H, the a two-volume collection, Scientific Papers of James magnetic field. The latter, the electrotonic state, re- Clerk Maxwell, which we shall refer to as JM. mained indefinite and elusive throughout the entire Maxwell’s first paper, published in 1856, is full ER. It first appeared early in volume 1, section 60, of formulas and therefore easier to read than Fara- without any precise definition. Later it was vari- day’s ER. Its central ideas are contained in part 2, ously called the peculiar state, state of tension, pe- which has as its title “Faraday’s Electro-tonic State.” culiar condition, and other things. For example, in On page 204 in this part 2 we find an equation that section 66 he wrote, “All metals take on the pecu- in today’s vector notation is ˙ liar state,” and in section 68, “The state appears to be E =−A , (2) instantly assumed.” More extensively, we read in A section 1114, where is Faraday’s electrotonic intensity. Three pages later, on page 207 of JM, the result If we endeavour to consider electricity is restated in words: and magnetism as the results of two Law VI. The electro-motive force on any faces of a physical agent, or a peculiar element of a conductor is measured by condition of , exerted in determi- the instantaneous rate of change of the nate directions perpendicular to each electro-tonic intensity on that element, other, then, it appears to me, that we whether in magnitude or direction. must consider these two states or forces as convertible into each other in a The identification of Faraday’s elusive idea of greater or smaller degree. the electrotonic state (or electrotonic intensity, or elec-

This46 articleNovember is copyrighted 2014 as indicated Physics inToday the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. www.physicstoday.org Downloaded to IP: 129.2.19.102 On: Wed, 12 Nov 2014 16:06:41 trotonic function) with Thomson’s vector potential COLLECTION BARR SCOTT E. ARCHIVES, VISUAL SEGRÈ EMILIO AIP A defined in equation 1 above is, in my opinion, the first great conceptual breakthrough in Maxwell’s scientific research: Taking the curl of both sides of equation 2 we obtain ˙ ∇ × E =−H , (3) which is the modern form of Faraday’s law. Another modern form of it is

˙ ∫E ·dl =−∬ H · dσ, (4) where dl is a line element and dσ is an element of area. Maxwell did not write Faraday’s law in either of the two forms in equations 3 and 4 because his aim was to give precise definition to Faraday’s elu- sive concept of the electrotonic state. Indeed, the concept of the vector potential A remained central in Maxwell’s thinking throughout his life. Maxwell was aware of what we now call the gauge freedom in equations 1–3, namely, that the gradient of an arbitrary scalar function can be added to A without changing the result. He discussed that freedom explicitly in theorem 5 on page 198 of JM. So which 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 importance of his identification of Faraday’s electrotonic inten- sity with Thomson’s A. He was afraid that Thomson might take offense concerning the priority question. He therefore concluded part 2 of his first paper with Figure 2. Michael Faraday the following remark: in an etched portrait. The inset shows the diagram Faraday drew in his diary on 17 October With respect to the history of the pres- 1831, the day he discovered induction. ent theory, I may state that the recogni- tion of certain mathematical functions as expressing the “electro-tonic state” of Faraday, and the use of them in de- termining electro-dynamic potentials Let AB, Plate VIII., p. 488, fig. 2, represent and electro-motive forces is, as far as I a current of electricity in the direction am aware, original; but the distinct con- from A to B. Let the large above ception of the possibility of the mathe- and below AB represent the vortices, matical expressions arose in my and let the small circles separating the from the perusal of Prof. W. Thomson’s vortices represent the layers of papers. (JM, page 209) placed between them, which in our hy- pothesis represent electricity. Now let an electric current from left Maxwell’s vortices to right commence in AB. The row of Five years after completing his first paper, Maxwell vortices gh above AB will be set in mo- began publishing his , which appeared in tion in the opposite direction to that of four parts during 1861 and 1862. In contrast to the a watch. (We shall call this direction +, earlier paper, the second is very difficult to read. The and that of a watch –.) We shall suppose main idea of the paper was to account for electro- the row of vortices kl still at rest, then magnetic phenomena “on the hypothesis of the mag- the layer of particles between these netic field being occupied with innumerable vor- rows will be acted on by the row gh on tices of revolving matter, their axes coinciding with their lower sides, and will be at rest the direction of the magnetic force at every point of above. If they are free to move, they will the field,” as we read in JM on page 489. rotate in the negative direction, and will Maxwell gave an explicit example of such an in- at the same time move from right to left, tricate group of vortices in a diagram reproduced or in the opposite direction from the here as figure 3, which he explained in detail on current, and so form an induced electric page 477 of JM in the following passage: current.

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of the same medium which is the cause of electric and magnetic phenomena” (page 500; the italics are Maxwell’s). Maxwell was a religious person. I wonder whether after this momentous discovery he had in his prayers asked for God’s forgiveness for revealing one of His greatest secrets. The birth of field theory Maxwell’s third paper, published in 1865, gave rise to what today we call Maxwell’s equations, of which there are four in vector notation. Maxwell used 20 equations: He wrote them in component form and also included equations for dielectrics and electric currents. That third paper is historically the first to give a clear enunciation of the conceptual basis of field theory—that 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 mechanical energy, whether it exists in the form of Figure 3. A figure of vortices from a plate in the 1890 collection motion or in that of elasticity, or in any Scientific Papers of James Clerk Maxwell, facing page 488. The directions other form. The energy in electromag- of the arrows in two of the hexagonal vortices in the second row from netic phenomena is mechanical energy. the bottom are incorrect, presumably mistakes of Maxwell’s draftsman. The only question is, Where does it re- side? On the old theories it resides in the electrified bodies, conducting cir- cuits, and magnets, in the form of an un- That detailed explanation of Maxwell’s model known quality called potential energy, appeared in part 2 of his second paper and was pub- or the power of producing certain ef- lished in Philosophical Magazine, volume 21, April– fects at a distance. On our theory it re- May 1861. Maxwell evidently took his intricate net- sides in the electromagnetic field, in the of vortices very seriously and devoted the surrounding the electrified and remaining 11 pages of part 2 to detailed studies of magnetic bodies, as well as in those bod- the model. ies themselves, and is in two different Then in January and February 1862, Maxwell forms, which may be described without published part 3 of his second paper with the title, hypothesis as magnetic polarization “The Theory of Molecular Vortices Applied to Stat- and electric polarization, or, according ical Electricity.” Seven pages of analysis led to his to a very probable hypothesis, as the proposition 14: “To correct the equations of electric motion and the strain of one and the currents for the effect due to the elasticity of the same medium. (JM, page 564) medium” (JM, page 496). The correction was to add a “displacement current” Ė to Ampère’s law, which But, in conformity with the prevalent ideas of the in modern notation then reads ∇ × H =4πj + Ė. time, Maxwell also wrote, I had made several attempts to read the last 11 We have therefore some reason to be- pages of part 2 and the first 7 pages of part 3, trying lieve, from the phenomena of and to see how Maxwell was led to his correction. In par- heat, that there is an aethereal medium ticular, I wanted to learn what he meant by “the ef- filling space and permeating bodies, ca- fect due to the elasticity of the medium.” All my at- pable of being set in motion and of tempts failed. It is noteworthy that in the last 11 transmitting that motion from one part pages of part 2, the word “displacement” occurs only to another, and of communicating that once, on page 479, in an unimportant sentence, motion to gross matter so as to heat it whereas in the beginning 7 pages of part 3 that word and affect it in various ways. (JM, page becomes the center of Maxwell’s focus. Thus it seems 528) that in the eight months between the publication of the two parts, Maxwell had found new features of Maxwell realized the great importance of his his network of vortices to explore, leading to the dis- discovery of the displacement current and his con- placement current. clusion that light is electromagnetic . In his After proposition 14 Maxwell quickly con- third paper, he collected the formulas of the two ear- cluded that there should be electromagnetic waves. lier papers and listed them together. In the process He calculated their velocity, compared it with the he must have reviewed the arguments that had led known velocity of light, and reached the momen- to those formulas. So after his review, how did he tous conclusion that “We can scarcely avoid the in- feel about the intricate network of vortices that had ference that light consists in the transverse undulations led to the displacement current three years earlier?

This48 articleNovember is copyrighted 2014 as indicated Physics inToday the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. www.physicstoday.org Downloaded to IP: 129.2.19.102 On: Wed, 12 Nov 2014 16:06:41 Maxwell did not discuss that point. But we notice books appeared on scalar meson theory with vector that the word “vortex” did not appear in any of interaction, pseudoscalar meson theory with pseu- the 71 pages of his third paper. It thus seems reason- doscalar interaction, and other esoteric topics. None able to assume that in 1865 Maxwell no longer of those efforts produced fundamental advances in considered as relevant the network of vortices of our conceptual understanding of interactions. There his second paper. But he still saw the necessity of were also enthusiastic supporters of efforts to find “an aethereal medium filling space and permeating alternatives to field theory, but again no real break- bodies.” throughs. In 1886 (1857–94) experimentally verified an important consequence of Maxwell’s Return to field theory equations: that electromagnetic waves can be gener- In the 1970s returned to field theory, ated by one set of electric circuits and detected by specifically to non-abelian gauge theory, which was another. an elegant generalization of Maxwell’s theory. The Starting in the mid 1880s Oliver Heaviside term non-abelian means that the order in which ro- (1850–1925) and Hertz independently discovered that tations or other operations take place . (I dis- one can eliminate from Maxwell’s equations the vec- cussed gauge theories and other things in “Ein- tor potential A. The simplified equations then have stein’s impact on ,” PHYSICS the additional attractive feature of exhibiting a high TODAY, June 1980, page 42. For a more technical dis- degree of symmetry between the electric and mag- cussion, see the article by Isidore Singer, PHYSICS netic fields. We now know that in quantum mechan- TODAY, March 1982, page 41.) Gauge theory is today ics, the vector potential cannot be eliminated. It has recognized as of fundamental conceptual impor- observable effects as in the Aharonov–Bohm effect. tance in the structure of interactions in the physical universe. It started with three papers published in Into the 20th century 1918–19 by , who was A conceptual revolution in field theory came early influenced by Einstein’s call for the geometrization in the 20th century following ’s 1905 of .4 special theory of relativity, which asserted that there is no other medium at all: The electromagnetic field is the medium. The vacuum is then the state of a re- Maxwell was a religious person. gion of spacetime where there is no electromagnetic radiation and no material particles. That solved the I wonder whether he had in puzzle posed by the 1887 Michelson–Morley exper- iment, which looked for the aethereal medium but his prayers asked for God’s failed to find it. Most physicists today believe Einstein’s motivation in formulating the theory of forgiveness for revealing one was not to solve the puzzle posed of His greatest secrets. by the Michelson–Morley experiment but rather to recognize the correct meaning of the concept of simultaneity. Weyl was motivated by the importance of par- In the years 1930–32, with the experimental dis- allel displacement. He argued that “the fundamen- covery of the positron, it became necessary to dras- tal conception on which the development of Rie- tically modify one’s view of the vacuum and to mann’s geometry must be based if it is to be in adopt instead ’s theory of the infinite sea agreement with nature, is that of the infinitesimal of negative-energy particles. That was another con- parallel displacement of a vector.” Weyl then said ceptual revolution in field theory, and it culminated that if, in the infinitesimal displacement of a vector, in the theory of . QED its direction keeps changing, then “Warum nicht proved successful for low-order calculations in the auch seine Länge?” (Why not also its length?) Thus 1930s but was beset with infinity-related difficulties Weyl proposed a nonintegrable “Streckenfacktor,” in calculations carried out to higher orders. or “Proportionalitätsfacktor,” which he related to In a series of brilliant and dramatic experimen- the electromagnetic field through tal and theoretical breakthroughs in the 1947–50 pe- riod, QED became quantitatively successful through μ exp(− eAμdx /γ), (5) the method of —a recipe for calcu- ∫

lating high-order corrections. The latest report of in which Aμ is the four-dimensional vector potential the calculated value of the anomalous magnetic mo- and the coefficient γ is real. Weyl attached such a ment of the ,3 a =(g − 2)/2, is in agreement stretch factor to every charged object moving with its experimental value to an incredible accu- through spacetime. To the second of Weyl’s three pa- racy of one part in 109 (see the Quick Study by pers, Einstein appended a postscript that criticized Gerald Gabrielse, PHYSICS TODAY, December 2013, Weyl’s idea of length changes in displacements. page 64). Weyl was unable to effectively respond to this dev- With the success of the renormalization pro- astating criticism. of QED and with the experimental discovery After the development of of many mesons and strange particles, efforts were in 1925–26, and inde- made to extend field theory to describe the interac- pendently pointed out that in the new quantum tions between all of the new particles. Papers and framework, it was necessary to replace (p – eA) by

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ie physics community for many years because it −−iħ( ∂μμ A ( , (6) ħ seemed to require the existence of massless charged particles. which suggested the replacement in equation 5 of To give mass to the massless particles in a non- eA dxμ/γ by ieA dxμ/ħ, that is, of γ by −iħ. μ μ abelian field theory, the concept of spontaneous sym- Evidently, Weyl accepted the idea that γ should metry breaking was introduced in the 1960s. That be imaginary, and in 1929 he published an impor- concept in turn led to a series of major advances, and tant paper in which he explicitly defined the concept finally to a U(1) × SU(2) × SU(3) gauge theory of of gauge transformation in QED and showed that electroweak interactions and strong interactions under such a transformation, Maxwell’s theory in that we now call the standard model. In the fifty- quantum mechanics is invariant. some years since 1960, the international theoretical Under a gauge transformation, Weyl’s length- and experimental research community working in change factor is replaced by “particles and fields” combined their individual e µ and collective efforts to develop and verify the stan- exp( −IAdx,∫ µ ( (7) ħ dard model. Those efforts met with spectacular suc- cess, climaxing in the discovery of the Higgs which evidently should have been called a phase- in 2012 by two large experimental groups at CERN, change factor. The replacement also renders inoper- each consisting of several thousand physicists (see ative Einstein’s criticism of Weyl’s idea mentioned PHYSICS TODAY, September 2012, page 12). above. Despite its impressive success, the standard That Maxwell’s equations have a high degree of model is not the final story. To start with, dozens of symmetry was first shown in 1905–07 by Einstein constants need to enter the model. More important, and Hermann Minkowski, who separately discov- one of its chief ingredients, the symmetry-breaking ered the equations’ Lorentz invariance. Weyl’s 1929 mechanism, is a phenomenological construct that in discovery that Maxwell’s equations are invariant many respects is similar to Fermi’s proposed “four- under gauge transformations revealed another ψ interaction” to explain beta decay.6 That 1934 the- symmetry property of the equations. Today we real- ory was also successful for almost 40 years. But it ize that these symmetry properties make Maxwell’s was finally replaced by the deeper U(1) × SU(2) elec- equations a fundamental pillar in the structure of troweak theory. the physical universe. Gauge freedom was explicitly known to Thom- Weyl’s gauge transformation involves, at every son and Maxwell in the 1850s. It probably had also spacetime point, a so-called U(1) rotation—essentially been vaguely sensed by Faraday in his elusive for- a simple rotation in the complex plane. There is thus mulation of the electrotonic state. The gauge free- a striking similarity between Weyl’s gauge transfor- dom was converted by Weyl in 1929 to a symmetry mation and Maxwell’s network of rotating vortices. (or invariant) property of Maxwell’s equations in The similarity is, of course, fortuitous. quantum mechanics. That symmetry property, now Mathematically, the phase factors of formula 7 called gauge symmetry, forms the structural back- form a Lie group U(1), and one of Weyl’s favorite bone of the standard model today. research fields was Lie groups. Going one step fur- Maxwell’s equations are linear. In non-abelian ther for the more technical reader, had fiber-bundle gauge theory, the equations are nonlinear. The non- theory been developed before 1929, Weyl could cer- linearity arises conceptually from the same origin tainly have realized that electromagnetism was a as the nonlinearity of the equations of general rela- U(1) bundle theory and would likely have general- tivity. About the latter nonlinearity Einstein had ized it to non-abelian gauge theory as a natural written, mathematical extension in 1929. We shall speak only of the equations of In the event, the extension was made in 1954, the pure gravitational field. motivated not by mathematical considerations but by The peculiarity of these equations the need to find a principle for interactions in the new lies, on the one hand, in their compli- field of in which there were found cated construction, especially their non- many new “strange” particles. The physical motiva- linear character as regards the field- tion was concisely stated in a short 1954 abstract: variables and their derivatives, and, on The electric charge serves as a source of the other hand, in the almost compelling electromagnetic field; an important con- necessity with which the transforma- cept in this case is gauge invariance tion-group determines this complicated which is closely connected with (1) the field-law. (reference 7, page 75) equation of motion of the electromag- The true laws can not be linear nor netic field, (2) the existence of a current can they be derived from such. (page 89) density, and (3) the possible interactions Entirely independent of developments in physics, between a charged field and the electro- there emerged during the first half of the 20th cen- magnetic field. We have tried to gener- tury a mathematical theory called fiber-bundle the- alize this concept of gauge invariance to ory, which had diverse conceptual origins, including apply to isotopic spin conservation.5 differential forms (mainly due to Élie Cartan), That extension led to a non-abelian field theory that statistics (Harold Hotelling), topology (Hassler was very beautiful but was not embraced by the Whitney), global differential geometry (Shiing-Shen

This50 articleNovember is copyrighted 2014 as indicated Physics inToday the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. www.physicstoday.org Downloaded to IP: 129.2.19.102 On: Wed, 12 Nov 2014 16:06:41 Chern), and connection theory (Charles Ehres- From the straight line of Euclid to the mann). The great diversity of its conceptual origin lines of force of Faraday this has been indicates that the fiber bundle is a central mathe- the character of the ideas by which matical construct. has been advanced, and by the It came as a great shock to both physicists and free use of dynamical as well as geo - in the 1970s that the of metrical ideas we may hope for further gauge theory, both abelian and non-abelian, turned advance.... out to be exactly the same as that of fiber-bundle the- We are probably ignorant even of the ory.8 But it was a welcome shock because it served name of the science which will be devel- to bring back the close relationship between the two oped out of the materials we are now col- disciplines, a relationship that had been interrupted lecting, when the great philosopher next by the increasingly abstract nature of mathematics after Faraday makes his appearance. since the middle of the 20th century. In 1975, after learning the rudiments of fiber- References bundle theory from my mathematician colleague 1. F. A. J. L. James, ed., The Correspondence of Michael Fara- James Simons, I showed him the 1931 paper by Dirac day, Vol. 1, Institution of Electrical Engineers (1991), on the magnetic monopole. He exclaimed, “Dirac p. 287. 32 had discovered trivial and nontrivial bundles before 2. J. Larmor, Proc. Cambridge Philos. Soc. , 695 (1936), mathematicians.” p. 697. 3. T. Kinoshita, in Proceedings of the Conference in Honour of It is perhaps not inappropriate to conclude this the 90th Birthday of , K. K. Phua et al., brief sketch of the conceptual origin of gauge theory eds., World Scientific (2014), p. 148. by quoting a few paragraphs from Maxwell’s tribute 4. For this and the ensuing history, see C. N. Yang, in Her- upon Faraday’s death in 1867: mann Weyl, 1885–1985: Centenary Lectures, K. Chan- drasekharan, ed., Springer (1986), p. 7; and A. C. T. Wu, The way in which Faraday made use of C. N. Yang, Int. J. Mod. Phys. A 21, 3235 (2006). his idea of lines of force in co-ordinating 5. C. N. Yang, R. Mills, Phys. Rev. 95, 631 (1954). the phenomena of magneto-electric in- 6. An English translation of Fermi’s original paper is 36 duction shows him to have been in re- available in F. L. Wilson, Am. J. Phys. , 1150 (1968). ality a mathematician of a very high 7. P. A. Schilpp, ed., Albert Einstein: Philosopher-Scientist, Open Court (1949). The two passages are English trans- order—one from whom the mathemati- lations of Einstein’s autobiographical notes written in cians of the future may derive valuable 1946 when Einstein was 67 years old. and fertile methods.... 8. T. T. Wu, C. N. Yang, Phys. Rev. D 12, 3845 (1975). ■

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