A mathematician admired by the gods of mathematics, her theorems became the cornerstone of 20th century physics

Author's Note

In the summer of 1918, Emmy Noether published the theorem that now bears her name—establishing a profound two-way connection between symmetries and conservation laws. The impact of this insight is pervasive in physics; it underlies all our theories of fundamental interactions and gives conservation laws a deeper meaning that transcends the empirical rules that have worked well. Noether’s papers, lectures, and personal interactions with students and colleagues advanced the development of abstract algebra and established her place in the pantheon of twentieth-century mathematicians. This article traces her path from Erlangen to Göttingen to a brief but happy exile at Bryn Mawr College in Pennsylvania, illustrating the importance of Noether’s theorem for the way we think today.

By Chris Quigg

Translation| 1/137

Emmy Noether | Photo credit: Bryn Mawr College Special Collections

I

On July 26, 1918, Felix Klein gave a lecture at the Royal Academy of Sciences in Götingen[1]. The paper he read had been presented to him by a young colleague named Emmy Noether on the occasion of his Golden Doctorate, the fiftieth anniversary of his doctorate.[2] The paper contained two theorems that have had a profound impact on physics, including particle physics, and the centenary of the paper provided an occasion for this commemoration.

It was a busy week in Göttingen, especially for Klein. Not only was he celebrating his doctoral degree (Doktorjubilaum), he had also published a paper the week before [3] explaining how he and David Hilbert had come to agree on the idea of ​​conservation of energy in Einstein’s general theory of relativity. They had noticed that in general relativity the usual constraint of conservation of energy seemed to appear as an identity, and were puzzled by it. How could it constrain anything? This was the problem he turned to Emmy Noether for help. Hilbert was revered among mathematicians for his 23 problems [4] that he had posed in 1900 , and was well known to physicists for his large book on the methods of mathematical physics, co-authored with Richard Courant.

A few days later, on July 23, Emmy Noether summarized the contents of her two theorems at the German Mathematical Society. As a young person—and a woman—she was not qualified to speak at a meeting of the Royal Academy of Sciences. So Klein reported her results.

The title page of the paper he read (Figure 1) reveals Amy Noether’s interesting approach: she combines concepts from the calculus of variations (or, in more technical terms, the Euler–Lagrange equations) with group theory to explore what can be extracted from differential equations subject to symmetry constraints. Her main results can be stated in two theorems [5]:

What do these statements mean? Theorem I captures all the known theorems about first integrals in mechanics, including the well-known conservation laws shown in Table I [6]. Interestingly, examples of such relations in special cases were actually known before Noether's work. Theorem II, the differential identities, can be described as the most comprehensive generalization of "general relativity" in group theory.

Table I: Symmetries and conservation laws in classical mechanics

What’s so striking about these theorems is their radical generality. Instead of limiting yourself to a particular equation of motion, or stopping after a first-order derivative, you can, in principle, have any number of derivatives in the Lagrangian, and you can extend this generality beyond simple transformations to more complex ones.

To put these theorems into the language we physicists and students use, Theorem I relates conservation laws to every successive symmetry transformation under which the Lagrangian remains invariant in form. From our perspective, this is an amazing advance. Consider the example of the conservation of energy. Mechanics often develops step by step, by inspiration or misinspiration. Clever people guess about what quantities might be useful to measure and what might be constants of motion. Even something as basic as the law of conservation of energy is in some ways an empirical law. It didn’t just fall from the sky, but people found it to be a useful idea. After Noether’s Theorem I, we knew that the conservation of energy really came from a plausible idea: that the laws of nature should be independent of time. We can derive empirical laws that might be useful from symmetry principles [7].

Feza Gürsey, a distinguished group theorist who taught physics at Yale and the Middle East Technical University in Ankara, was ecstatic about the implications of the theory. Nathan Jacobson, in his introduction to Amy Noether’s collection,[8] quotes Gürsey:

Before Noether's theorem, the principle of conservation of energy was shrouded in mystery, leading to the obscure physical systems of Ernst Mach and Wilhelm Ostwald. Noether's simple yet profound mathematical form did much to illuminate physics.

Theorem II, in itself, contains the seeds of gauge theory (“symmetries determine interactions”) and shows the affinity between general relativity (generalized coordinate invariance) and gauge theory. We will talk more about the door to gauge theory at the end of this article. In the process, Noether’s analysis clarified the controversy between Klein and Hilbert about the conservation of energy in general relativity [9].

II

The person who gave us these theorems was Amalie Emmy Noether. She was called by her middle name (Emmy) because her mother and grandmother were both named Amalie. She was born on March 23, 1882, in Erlangen, a university town a little north of Nuremberg. At the time of her birth, Nuremberg had a population of about 15,000. The most famous child of Erlangen was Georg Simon Ohm, the discoverer of Ohm's law V=IR, who was not only born in Erlangen but also received his doctorate there. Emmy Noether's father, Max Noether[10], had been a professor of mathematics at the University of Erlangen since 1875. This undoubtedly influenced her upbringing. He worked in algebraic geometry, the study of curves on surfaces. Max was a scholar of considerable renown, having been elected to such organizations as Berlin, Göttingen, Munich, Budapest, Copenhagen, Turin, the Accademia dei Lincei, the Institut de France, and the London Mathematical Society.

Klein—who, as we have already mentioned, pronounced Noether’s theorem—worked in Erlangen for three years, making it famous in mathematical history. In his inaugural address (1872), Klein proposed a research program to study geometry from the perspective of group theory. Before this, the foundations of geometry had been based on rectilinear coordinate systems. Klein’s innovation was that, on the basis of Riemann’s theory, you should not be bound by a coordinate system, or Euclidean space, as we would say today. Instead, it should be the symmetries of your object of study—the group structure, not just the x, y, and z coordinates. Klein then moved on to a series of other positions, but he left his mark with the “Erlangen Program”[11], so that the university was seen as a place where mathematics was taken seriously.

Alfred Clebsch was Max Noether’s mentor and collaborator, and Max Noether later became a practitioner of Clebsch’s work. Clebsch also had a junior collaborator named Paul Gordan, who was a colleague of Max Noether. We know the Clebsch-Gordan coefficient for angular momentum coupling [12]. When Max Noether taught in Erlangen, Gordon was very influential in the mathematics department. He was portrayed as an eccentric guy who would wander around town with a cigar in his mouth, visit beer gardens, and think hard all the time. According to his colleagues, he could write a whole paper in one breath. It is said that he wrote a paper with twenty pages of continuous formulas without a single word between the lines. In the obituary written by Noether and his daughter, he was called an algorithmmiker, that is, a creator of algorithms.

What about Emmy Noether herself? How did she become a promising young mathematician?[13] Like many young women from her background – aspiring middle-class women with some intellectual tendencies – she attended the Municipal Gymnasium for Girls (Städtische Höheren Töchterschule) from 1889 to 1897. Nominally, this was a school to prepare women for life, if she had to pursue a career, which was nothing more than teaching other young women English and French. After completing the course, Emmy passed the Bavarian state examination for teachers in French and English in 1900. She could not enroll at the University of Erlangen, as women were not allowed to attend universities at the time.[14] However, she could apply for special permission to sit in on lectures. In Germany, such opportunities varied from one institute to another at different times. The rector of the University of Erlangen who approved this great reform was none other than Emmy’s father, Max.

While Emmy Noether was studying along the traditional ladies’ curriculum, she also took private lessons at the “Gymnasium Mathematics Course” in Stuttgart and Erlangen in preparation for university study.[15] She was able to present these qualifications when she applied to attend university lectures in October 1900 – allowing the university to recognize that she was indeed well prepared.

In 1903 she passed her university qualifying exams, but still wasn’t accepted to the University of Erlangen. (Perhaps because her father, the provost, didn’t move fast enough.) The University of Göttingen was more open-minded. She went there for a semester, during which she attended lectures by Karl Schwarzschild, Hermann Minkowski, Klein, and David Hilbert. I think if you do that in your first semester at university, you either change your ways or go down in history! Emmy Noether—she’ll make history. After a semester, Erlangen realized its mistake and began admitting women, with only two women in a class of about a thousand. She eventually got into Erlangen to study mathematics.

In 1907, Emmy Noether completed her thesis under the supervision of Gordon, whom she had known since childhood. She received a D. Phil. degree summa cum laude for "Formal System Construction of Ternary Quartic Forms" (Über die Bildung des Formensystems der ternären biquadratischen Form in German). This work involved meticulous calculations of some 331 quartic form invariants - a very "Gordanian" job. Later, she described her thesis title as "Mist" (German for feces), which was hardly an expression of pride, as she aspired to create, not just calculate. Noether appears to have been the second woman in Europe to receive a PhD in mathematics, after Sofia Kovalevskaya, who received her PhD in 1874 under Karl Weierstraß in Göttingen, became a full professor in Stockholm in 1889, and died in 1891 at the age of 41.

From 1908 to 1915, Dr. Emmy Noether remained in her hometown as an unpaid member of the Erlangen Institute for Mathematics. She gained extensive experience in teaching and research. When her father's health began to fail, Emmy took over his courses. Although she had neither salary nor status, she still disciplined herself as a teacher. In 1909, she became a member of the German Mathematical Society (Deutsche Mathematiker-Vereinigung) and in the same year became the first woman to speak at the annual meeting of the society. New positions were added to the mathematics department, and she came under the influence of Ernst Fischer, Gordon's successor[16], who introduced her to the world of abstract mathematics, not just calculation. It was in abstract mathematics that she showed great talent.

III

In 1915, Klein and Hilbert invited Emmy Noether to Göttingen. Göttingen was the Olympus of mathematics at the time, at least in Germany.[17] This was where Carl Friedrich Gauss lectured. If you look at the list of their heroes who made history, you will find many familiar names: Constantin Carathéodory, Kleibusch, Courant, Peter Gustav Dirichlet, Gustav Herglotz, Abraham Gotthelf Kästner, Minkowski, Carl Runge, and Weyl, to name a few. It was a great place for young people in mathematics.

Göttingen has a proud tradition in mathematics, including an unrivalled treasure trove of material on the early history of modern (18th and 19th century) mathematics. The mathematical library has a locked “poison cabinet” (Giftschrank), which contains treasures such as the seminar lecture notes of Klein, his colleagues, students, and distinguished visitors over a period of forty years[18] – 29 volumes and 8,000 pages in total!

Hilbert was very interested in Emmy Noether and worked hard to promote her career. In 1915, the Department of Mathematics and Science of the Faculty of Philosophy recommended her for the Habilitation Lectures, making her a Privatdozent in Göttingen, with unanimous approval from everyone, even some of the old guard. One of the votes came from the Göttingen mathematician Edmund Landau [19]:

My experience with female students had so far been unsatisfactory, and I had come to believe that the female brain was not suited to mathematics. Miss Knott seemed to be a rare exception.

However, in a special vote on November 19, 1915, against Emmy Noether's candidacy, the History-Linguistics Department blocked the move because of "fears that the sight of a female creature might distract the students".[20]

The university did not formally deny her a professorship; the administration did not take action at all. Thus, the professorship was not granted. But since Hilbert was her patron, Emmy Noether was allowed to lecture under his name. The courses were published as Hilbert's authorisation and Fräulein's assistance. Hilbert might appear at the first and last lectures, but Noether was responsible for everything else. She did not receive official remuneration from the university, but there are indications that some salary arrangements may have been made.

In 1917, mathematicians in Göttingen pressed the issue again, this time with new urgency: they feared that if Göttingen did not act quickly, Frankfurt would hire her. They asked the Ministry of Education for an exception to retain this indispensable talent, and the Ministry’s response[21] displayed impeccable bureaucratic logic.

Berlin, June 20, 1917

With regard to the admission of women to teaching posts, the rules of the University of Frankfurt are the same as those of all universities: it is not permitted to appoint women as supernumerary lecturers. It is simply impossible to make an exception at one university. Therefore, your fear that Miss Nott will go to Frankfurt and get a post there is completely unfounded: she will not be granted the right to teach there, just as she would not be granted a post at Göttingen or any other university. The Minister of Education has repeatedly stated and emphasized that he supports the instructions of his predecessors and therefore does not allow women to hold teaching posts at universities.

Therefore, you need not worry about losing Miss Nott as a supernumerary lecturer at the University of Frankfurt.

After Germany's defeat in World War I (1914-1918), the establishment of the Weimar Republic brought liberalization and many reforms: Women were no longer explicitly prohibited from teaching at universities. In 1919, Emmy Noether received her Habilitation based on her thesis on "Invariant Variational Problems." She is now a temporary faculty member and also has no documents proving her payment for her services.

IV

Symmetries might give rise to interactions? One of Emmy Noether’s colleagues, Hermann Weyl, was one of the pioneers in applying symmetries to modern physics. He frequently visited Göttingen and eventually accepted a teaching position there. In 1918, the year Noether’s theorem was born, Weyl had an interesting idea. He set out to build a unified theory for all the fundamental interactions known at the time—electromagnetism and gravity.[22] It occurred to him that this unified theory could be derived from the symmetry principle by building a theory that was invariant under scale transformations. Imagine a measuring stick with markings that changed as it moved, and requiring the theory to remain invariant under these changes. As a physical theory, this construction failed.[23] It did not lead to Maxwell’s equations, and in the case of gravity, Einstein himself objected to the idea that the time of a clock depended on the path taken from one point to another. So Weyl’s idea was wrong, but like many “wrong” ideas in physics, it had a brilliant point: interactions might arise from symmetries.[24]

No one noticed the connection between Weyl's intention and Noether's second theorem, which we now understand to indicate that such a construction is always possible. This was partly due to the lack of other specific conditions. After the birth of quantum mechanics, in the following decade, with the help of Einstein, V.A. Fock, and others, Weyl realized that it was indeed possible to derive electrodynamics by imposing certain symmetries on the wave function - an extremely important new feature of quantum mechanics. In the introductory course in quantum mechanics, we showed that the absolute phase of the quantum mechanical wave function is a convention with no observable consequences. If we go a step further and add the freedom to choose the phase convention independently at each point, then we can derive electrodynamics from the Schrödinger equation in the style of Noether's second theorem.

In 1931, in the paper that invented quantum electrodynamics and the magnetic monopole [25], the author spoke rather mysteriously about what he called a nonintegrable phase. We know that in classical electrodynamics, the potential contains an excess of information, and it was long thought that the electric and magnetic fields contained all the information needed. This turned out to be incorrect: in quantum mechanics, the fields contain too little information. There is an intermediate, path-dependent phase that is both nonlocal and topological and contains just the right amount of information, as explained by Yakir Aharonov and David Bohm in 1959 [26].

In late life (1955), Weyl wrote, in order to explain how he knew he was on the right track,[27]

The strongest argument for my theory seems to be that gauge invariance corresponds to the principle of conservation of electric charge, just as coordinate invariance corresponds to the laws of conservation of energy and momentum.

I interpret this as, to some extent, his understanding, either explicitly or vaguely, of the connection between Noether's theorem and symmetry and conservation laws.

V

A central feature of electrodynamics is that charge is conserved. The best current limit on charge conservation comes from the Borexino experiment [28], a sophisticated radiopure liquid scintillation detector deep underground at the Gran Sasso laboratory. They derived a new limit on the stability of the electron from the decay of an electron into a neutrino and a monoenergetic photon. This new limit is τ ≧ 6.6 × 1028 years at the 90% confidence level, which improves the previous limit by two orders of magnitude.

Where does the conservation of charge come from? Why is charge conserved? You might say it's implied by Maxwell's equations. But if you look back at how Maxwell developed his equations based on Faraday's observations, he adjusted the equations so that charge was conserved in all cases. This is where the displacement current comes from, as a supplement to Ampere's law in non-static situations. In other words, Maxwell's equations were used to explain experimental observations of charge conservation. So it's not a profound explanation to derive charge conservation from Maxwell's equations, even though it works well in most cases.

We can use the global phase invariance of Theorem I to mean that there is a conserved charge, which we identify as the electric charge. This is an important step in the derivation, but we can also think of the conserved charge as the baryon number. It seems to me that we can only be sure that the charge we defined is the electric charge when we apply the local phase invariance of Theorem II and show that the resulting theory is indeed electromagnetism. There is still a coupling constant, and you still have to determine its coupling to the charge, but you have derived the global form of Maxwell's equations, so it is not a big leap.

From this idea, we can choose phase conventions independently at points in space and time and derive quantum electrodynamics (the Lagrangian and equations of motion), which leads to charge conservation [29]. By analogy with the kinematic conservation laws, this pushes the origin of conservation laws back a step by showing that they can be derived from symmetry principles. They are not just empirical laws. Now, the exact degree of symmetry principles can still be challenged, and you can make unsuccessful choices about them, but Noether's theorem gives us a deeper understanding of why conservation laws should hold.

VI

Noether’s “invariant variational problem” caused a stir in general relativity circles, but otherwise received a lukewarm response. It is what our friends at inspirehep.net call “Sleeping Beauty.” Werner Heisenberg, a famous proponent of symmetry theory in fundamental physics (he was the inventor of isospin, after all), made this striking statement in his later years while discussing the meaning of everything with his disciples[30]:

“In the beginning there was symmetry” is obviously more correct than Democritus’s argument that “in the beginning there were particles.” Elementary particles embody symmetry and are the simplest representation of symmetry, but they are first and foremost the result of symmetry.

Although not conclusive, there is evidence from other interviews that Heisenberg never read Noether's paper: "[Noether's paper] did not go very far into quantum theory, so I did not realize the importance of the paper" [31]. I suspect that Heisenberg and his peers had so much else to do—inventing and applying quantum mechanics—that once they heard about the obvious consequences of Noether's theorem—the conservation laws of mechanics—they assumed that they already knew them and there was no need to pay attention. Another important point is that internal symmetries had not yet been invented. (From our point of view, we can apply these theorems to internal symmetries, thus forming gauge theories.) Internal symmetries such as isospin did not exist until 1932, when the neutron was discovered.

As a result, Emmy Noether was not immediately respected in the physics community. Some have speculated that the excitement of quantum physics and abstract algebra at Göttingen had so engrossed physicists and mathematicians that they failed to notice the relevance of their new developments. It can be argued that the "invariant variation problem" and Emmy Noether have had a place in physics and other sciences since the 1960s.[32]

You may have heard of Niels Bohr’s famous suggestion[33] that the continuous spectrum of beta decay could be explained by the hypothesis that the conservation of energy might be a statistical phenomenon rather than a strict law for microscopic phenomena.

At the present stage of atomic theory there is no reason either empirically or theoretically to adhere to the energy principle of beta-ray decay, and even the attempt to do so leads to complications and difficulties.

This was not the first time that Bohr had explored deviations from strict conservation of energy. In 1924, Bohr, Hendrik Anthony Kramers, and John Clarke Slater proposed in a paper[34] the possibility that energy conservation might hold in some statistical sense in radiation processes and at microscopic scales. Although many physicists objected[35], no one seemed to invoke Noether’s insight and say, “There is a theorem that says this is a dead end,” or at least, “It would be very costly.” The conjecture was buried within a year, thanks to precise measurements of final-state momentum in Compton scattering.

VII

In the hotbed of Göttingen, Noether's approach to mathematical problems changed. She stopped calculating and became interested in abstract algebra. Évariste Galois's famous pamphlet on the application of group theory to the solution of algebraic equations[36] became a source of inspiration. With Hilbert's support, Emmy Noether was appointed Außerordentlicher Professor (a real professorship, but again, the university did not pay a salary) in 1922. Hilbert could provide a small stipend, and she also had some family savings. She had close contacts with Soviet mathematicians and visited Moscow State University in 1928-1929. She did spend a short time in Frankfurt in 1930, but she was not hired there.

The honors began to come. In 1932, she and her colleague Emil Artin, another pioneer of algebraic equations,[37] received the Alfred Ackermann-Teubner Prize. In the same year, Noether became the first woman invited to give a plenary lecture at the International Congress of Mathematicians in Zurich. She was also a dedicated editor of the journal Mathematische Annalen.

Emmy Noether was, by many accounts, the hub of activity in Göttingen. She had a devoted group of students and young collaborators, mostly men, known as the “Noether boys” (die Noetherknaben). They were said to hang around in Göttingen in small groups, debating mathematical problems. They caused a minor scandal as they wandered around town in dishevelled attire, even though the only photo I stumbled across showed them in coats and ties. Many of them grew up to be distinguished and famous mathematicians. Weyl later confessed that the math boys in Göttingen called Emmy “Der Noether” (masculine Mr. Noether)—because she was as strong in mathematics as any man. Her groundbreaking work on rings and ideals earned her a less ambiguous title: the mother of modern algebra.

VIII

In 1933, the authorities issued a decree in the name of the notorious Minister of Culture Bernhard Rust, declaring that anyone with a Jewish background must be placed on compulsory leave from the university.[38] According to the Göttinger Tageblatt of 26 April,[39] Emmy Noether was one of the first six faculty members to be expelled. Other mathematicians and physicists were Felix Bernstein (one of the founders of biostatistics), Max Born (who won the 1954 Nobel Prize in Physics for his statistical interpretation of quantum mechanics), and Richard Courant. Courant had taken over the management of the Academy from Hilbert, who had passed the mandatory retirement age of 68. The two collaborated on the famous two-volume Courant-Hilbert work on mathematical physics,[40] which, incidentally, discussed Noether's theorem (Ch. IV, §12.8). Among the Nazi informers was Emmy Noether's doctoral student Werner Weber.

The ominous developments of the furloughs soon had a devastating effect. On May 10, 1933, German students burned tens of thousands of “un-German” books in Berlin’s Theaterplatz, Göttingen, and other university towns. The leaders of twenty-one American colleges and universities quickly formed an Emergency Committee in Aid of Displaced German Scholars. The Emergency Committee’s operating officer was Edward R. Murrow, who later became a legendary journalist.[41]

In September, the Prussian Ministry of Science, Art, and Education in Berlin sent a telegram[42] notifying her that Emmy Noether’s teaching license had been revoked in accordance with Section 3 of the Civil Service Reorganization Act of April 1933. The school was instructed that her current salary would be discontinued by the end of the month.

Sympathetic colleagues, including Hilbert, scrambled to find accommodations for Emmy Noether and many others—by the end of 1933, eighteen mathematicians had left or been driven out of the Institute in Göttingen alone. Born went to Cambridge, then Bangalore, India, and finally settled in Edinburgh as Tait Professor of Natural Philosophy.[43] Courant went via Cambridge to New York,[44] where he founded what is now the Courant Institute for Mathematical Sciences at New York University.

Amy Noether was hired as a visiting professor for two years at Bryn Mawr College in Pennsylvania. Founded in 1885, Bryn Mawr was one of the first women's colleges in the United States to open higher education to women. It offered rigorous intellectual training, including graduate study, and the opportunity to pursue original research in the tradition of European universities. In a brief announcement of her appointment, The New York Times, with the caution that is occasionally seen today, reported that "she and other members of the Göttingen faculty had been asked to resign last spring under the Nazi regime."[45]

The president of Bryn Mawr College and Emmy Noether's supporters in Germany realized that, despite her mastery of English—and the credentials to prove it—she might not be conceptually suited to undergraduate teaching. Bryn Mawr already had a small graduate program in mathematics, and she was an ideal candidate for it. To take advantage of this famous mathematician, the college expanded the circle of women in mathematics by establishing the Emmy Noether Fellowship and the Emmy Noether Scholars.[46] In addition, she was able to go to the Institute for Advanced Study every week, where she gave seminars and lecture courses. The Institute had become one of the great centers of mathematical research. This relationship brought her into contact with other outstanding immigrants, including colleagues she had known in Germany, such as Weyl. Einstein took note of her work, but it is unclear whether they ever had any substantial contact.

The Bryn Mawr ladies who followed "Miss Nott" on her brisk walks seemed equally engaged and enthusiastic, whether or not they were as carefree as the Nott boys in Göttingen. Inspired by her students and life at Princeton, Nott herself was excited, curious about the American way of life, and generally energetic. During spring break in 1935, she had a routine operation on her abdomen. She seemed to recover well, but suffered complications and died within a few days.

Einstein wrote a eulogy to The New York Times[47] in which he wrote that Amy Noether was

“…the most creative mathematical genius ever born since the beginning of higher education for women. In algebra, a field in which the most gifted mathematicians had been busy for centuries, she discovered methods which proved to be of great importance to the development of the younger generation of mathematicians of her day.”

At Bryn Mawr her ashes were interred in the aisle of the cloister, beneath a modest marker reading E.N. 1882-1935.

Weyl spoke at the Bryn Mawr (Knott) memorial service, in which he gave a long and very detailed, hymn-like tribute:[48]

I remember well that in the winter semester of 1926-1927 I was a visiting professor at Göttingen, giving a lecture on the representation theory of continuous groups. She was in the audience; for at that time she was very interested in hypercomplex number systems and their representations. I remember also that after the lectures, on the way home, I walked with her and von Neumann in the cold, dirty, rain-soaked streets, and we had many discussions. Von Neumann was then a Rockefeller Fellow at Göttingen. When I was offered a permanent appointment at Göttingen in 1930, I earnestly tried to obtain a better position for her from the Ministry, because I knew that as a mathematician she was in many ways superior to me, and I was ashamed to occupy such a superior position beside her.

Pavel Alexandrov of Moscow was one of Noether’s closest friends. Her fascination with the Russian school of mathematics came in part from her interactions with him. He wrote affectionately and respectfully about what a wonderful person she was and how generous she was to her students.[49] Apparently she would come up with ideas, elaborate plans for her students to implement, and ensure that her students wrote them down and received credit for them.

With the death of Amy Noether I have lost one of the most charming people I have ever known. Her extraordinary goodness of heart, so antithetical to affectation and hypocrisy; her cheerfulness and simplicity; her ability to overlook all the trivialities of life - created an atmosphere of warmth, peace and goodwill that will never be forgotten by those who were associated with her ... Although she was gentle and generous, she was also passionate, changeable and strong-willed by nature; she always spoke her mind and was not afraid of opposition. Her love for her students was touching, and they formed the content of her life and replaced the family she had not had. Her concern for their scientific and secular needs, her sentimentality and sympathy, were rare qualities. Her sense of humor made her very comfortable in public and her informal interactions particularly pleasant, and it also enabled her to deal with all the injustices and absurdities that encountered academic career with ease and without malice. In such cases, she did not mind, but laughed them off.

Bartel van der Waerden[50], synthesizing the stimulating lectures of Emile Aden and Emmy Noether, from which they created the axiomatic approach to abstract algebra, wrote[51]:

This completely non-graphic and non-computational way of thinking was probably one of the main reasons why it was so difficult for the average person to follow her lectures. She had no talent for didacticism, and she struggled to clarify statements and quickly added explanations before she had even finished them, which often had the opposite effect. Yet, what a profound impact her lectures had. Her few loyal listeners, usually composed of a few outstanding students and an equal number of professors and guests, had to work hard to keep up with her. However, those who succeeded gained far more than they would have from the most sophisticated lectures. She almost never presented complete theories; usually they were in development. Each of her lectures was a project plan. When the project was realized by her students, no one was happier than herself. She was completely not self-centered, not vain, and never sought anything for herself except to train her students. She always wrote the introduction to our papers...

Elsewhere, van der Waerden writes that when they walked in Göttingen, Noether, as she had with her students at Bryn Mawr, spoke so fast and so excitedly that he could not understand her at all, and that if he took her around the city several times, by the third lap she was a little out of breath and spoke slowly enough for him to understand.

postscript

More than twenty years after Emmy Noether’s death, physicists began to exploit the full power of Theorem II. The idea that internal symmetries can give rise to interactions was put into practice by Chen Ning Yang and R. Mills [52], who tried to derive a theory of the strong interactions between nucleons from isospin symmetry. They wondered whether it were not possible to choose the isospin convention independently at each point in spacetime, just as we set the phase convention of the quantum mechanical wave function locally to derive quantum electrodynamics. The mathematical construction was this: the symmetry implies a conserved isospin current, a massless vector field that interacts to transmit the forces between nucleons. This does not correspond to the real world. As with many ideas in physics, they do not work the first time they are applied, but the idea sticks. We have now found how to apply this idea successfully—in the theory of quantum chromodynamics (QCD) of the strong interactions between quarks and gluons, and in the electroweak theory, where gauge symmetries must remain hidden.

References and Notes

[1] Klein is well known in popular science culture for his concept of a Klein surface (Fläche), which has been mistranslated as a Klein bottle (Flasche).

[2] Emmy Noether. Invariante Variationsprobleme. Gott. Nachr., pages 235–257, 1918. http://bit.ly/2GQyfsm; 3–22. Springer, New York, 2011. doi: 10.1007/978-0-387-87868-3_1

[3] F. Klein. Differentialgesetze für die Erhaltung von Impuls und Energie in der Einsteinschen Gravitationstheorie. Königliche Gesellschaft der Wissenschaften zu Göttingen. Mathematischphysikalische Klasse. Nachrichten, pages 171–189, 1918. http://bit.ly/2VsEnKK

[4] David Hilbert. Mathematical problems. Bull. Amer. Math. Soc., 8:437–479, 1902. doi:10.1090/S0002-9904-1902-00923-3. Translated from Göttinger Nachrichten, 1900, pp. 253-297; Archiv der Mathernatik und Physik, 3d ser., vol. 1 (1901), pp. 44-63 and 213-237.

[5] Summary of Emmy Noether's report to the German Mathematics Club, Jahresbericht der Deutschen Mathematiker-Vereinigung Mitteilungen und Nachrichten vol 27, part 2, p.47 (1918).

[6] A skeletal but useful reference is EL integrals, including the familiar conservation laws 6 shown in Table 1. Hill. Hamilton's principle and the conservation theorems of mathematical physics. Rev. Mod. Phys., 23:253–260, 1951. doi: 10.1103/RevModPhys.23.253.

[7] For an example derivation, see Chapter 2 of Chris Quigg. Gauge Theories of the Strong, Weak, and Electromagnetic Interactions. Princeton University Press, Princeton, second edition, 2013.

[8] Emmy Noether. Gesammelte Abhandlungen=Collected papers. Nathan Jacobson, editor; Springer-Verlag, Berlin New York, 1983. See pages 23–25.

[9] General coordinate invariance gives rise to the Bianchi identities that cause the energy conservation law to seem trivial. Energy conservation arises from the symmetry, as explained in Katherine Brading. A Note on General Relativity, Energy Conservation, and Noether's Theorems. Einstein Stud., 11:125–135, 2005. doi: 10.1007/0-8176-4454-7_8. The canonical modern treatment is Richard L. Arnowitt, Stanley Deser, and Charles W. Misner. The Dynamics of General Relativity. Gen. Rel. Grav., 40:1997–2027, 2008. doi: 10.1007/s10714-008-0661-1, arXiv:gr-qc/0405109.

[10] Francis S. Macaulay. Life and work of the mathematician Max Noether (1844-1921). Proceedings of the London Mathematical Society. - 2. ser., 21:XXXVII–XLII, 1923. doi: 10.11588/heidok.00013182.

[11] Garrett Birkhoff and MK Bennett. Felix Klein and His “Erlanger Programm”. In William Aspray and Philip Kitcher, editors, History and Philosophy of Modern Mathematics: Volume XI, pages 145–176. University of Minnesota Press, 1988. https://www.jstor.org/stable/10.5749/j.cttttp0k.9

[12] The Particle Data Group's Table of Clebsch–Gordan coefficients, pdg.lbl.gov/2018/reviews/rpp2018-rev-clebsch-gordan-coefs.pdf.

[13] For a brief account of the early years, see Emmy Noether in Erlangen and Göttingen. In Bhama Srinivasan and Judith Sally, editors, Emmy Noether in Bryn Mawr: proceedings of a symposium, pages 133–137. Springer-Verlag, New York, 1983.

[14] In 1898, the Erlangen Academic Senate held that the “admission of women would overthrow all academic order.” See the Appendix for some examples of the integration of women into American universities.

[15] For a detailed account (in German), see Cordula Tollmien, “Das mathematische Pensum hat sie sich durch Privatunterricht angeeignet” — Emmy Noethers Zielstrebiger Weg an die Universität, in Mathematik und Gender 5, 1–12 (2016), Tagungsband zur Doppeltagung Frauen in der Mathematikgeschichte+ Herbsttreffen Arbeitskreis Frauen und Mathematik (edited by Andrea Blunck, Renate Motzer, Nicola Ostwald), Franzbecker-Verlag für Didaktik http://www.cordula-tollmien.de/pdf/tollmiennoether2016.pdf.

[16] JJ O'Connor and EF Robertson. Ernst Sigismund Fischer, MacTutor History of Mathematics. http://www-history.mcs.st-andrews.ac.uk/Biographies/Fischer.html, 2006.

[17] Benno Artmann, “Hochburg der Mathematik,” in Georgia Augusta (2008) http://bit.ly/2GQmQZL, pp. 14–23.

[18] Felix Klein, Seminar-Protokolle, http://www.claymath.org/publications/klein-protokolle. For a brief tour, see Eugene Chislenko and Yuri Tschinkel, “The Felix Klein Protocols,” Notices Amer. Math. Soc. 54, 961–970, (2007), http://www.ams.org/notices/200708/tx070800960p.pdf

[19] Norbert Schappacher. Edmund Göttingen mathematician Edmund Landau 19: Landau's Göttingen: From the Life and Death of a Great Mathematical Center. Math. Intelligencer, 13(4):12, 1991. http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1991b_Landau.pdf.

[20] For the full German text, see Cordula distracting to the students." Tollmien, "Weibliches Genie: Frau und Mathematiker: Emmy Noether," in Georgia Augusta (2008) http://bit.ly/2GQmQZL, pp. 38–44.

[21] Letter from the Ministry of Education, to Göttingen. The reply from the Ministry of Education 21 exhibits the Edelstein Collection, the National Library of Israel, http://bit.ly/2BFZHDs. English translation at https://blog.nli.org.il/en/noether/.

[22] H. Weyl. Gravitation und Elektrizität. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 1918:465. English translation in L. O'Raifeartaigh, The Dawning of Gauge Theory. Princeton University Press, Princeton, 1997, pp. 24–37.

[23] See §3.1 of HA Kastrup. On the Advancements of Conformal Transformations and their Associated Symmetries in Geometry and Theoretical Physics. Annalen Phys., 17:631–690, 2008. doi: 10.1002/andp.200810324, arXiv:0808.2730.

[24] See §3.1 of HA Kastrup. On the Advancements of Conformal Transformations and their Associated Symmetries in Geometry and Theoretical Physics. Annalen Phys., 17:631–690, 2008. doi: 10.1002/andp.200810324, arXiv:0808.2730.

[25] Paul Adrien Maurice Dirac. Quantised singularities in the electromagnetic field. Proc. Roy. Soc. Lond., A133(821):60–72, 1931. doi: 10.1098/rspa.1931.0130.

[26] Y. Aharonov and D. Bohm. Significance of electromagnetic potentials in the quantum theory. Phys. Rev., 115:485–491, 1959. doi: 10.1103/PhysRev.115.485.

[27] Quoted in Freeman J. Dyson, Birds and Frogs: Selected Papers of Freeman Dyson, 1990–2014, World Scientific, Singapore, 2015, p. 47.

[28] M. Agostini et al. A test of electric from the Borexino experiment 28, an exquisitely radiopure liquid charge conservation with Borexino. Phys. Rev. Lett., 115:231802, 2015. doi:10.1103/PhysRevLett.115.231802, arXiv:1509.01223

[29] For further discussion, see Katherine A. Brading. Which symmetry? Noether, Weyl, and conservation of electric charge. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 33(1):3 – 22, 2002. doi:10.1016/S1355-2198(01)00033-8.

[30] W. Heisenberg. Der Teil und das Ganze: Gespräche im Umkreis der Atomphysik. Piper, München, 2006. p. 280. »Am Anfang war die Symmetrie«, das ist sicher richtiger als die Demokritsche These »Am Anfang war das Teilchen«. verkörpern die Symmetrien, sie sind ihre einfachsten Darstellungen, aber sie sind erst eine Folge der Symmetrien.

[31] See pp. 85–86 of The Noether Theorems, Ref. 2.

[32] Crowned by a Google doodle: https://www.google.com/doodles/emmy-noethers-133rd-birthday.

[33] Niels Bohr. Chemistry and The Quantum Theory of Atomic Constitution. J. Chem. Soc., pages 349–384, 1932. doi:10.1039/JR9320000349. VIII. Faraday Lecture, May 8, 1930. See p. 383.

[34] Niels Bohr, Hendrik A. Kramers, John C. Slater. The Quantum Theory of Radiation. Phil. Mag., 47:785–802, 1924. http://bit.ly/2ETtID3.

[35] For a commentary, see Helge Kragh. Bohr–Kramers–Slater Theory. In Daniel Greenberger, Klaus Hentschel, and Friedel Weinert, editors, Compendium of Quantum Physics, pages 62–64. Springer, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-540-70626-7_19.

[36] Évariste Galois. OEuvres Mathématiques. Jacques Gabay, Sceaux, 1989. Éditions Jacques Gabay, Sceaux, 1989. Sophus Lie, Influence of Galois, a été publiée dans l'ouvrage Le Centenaire de l'École Normale 1795-1895; http://bit.ly/2QZy0jt, Hachette 1895.

[37] Emil Artin. Galois theory. Dover Publications, Mineola, NY, 1998. ISBN 978-0486623429. second edition; edited and supplemented with a Section on Applications by Arthur N. Milgram.

[38] For an account from the perspective of six decades, see Saunders Mac Lane. Mathematics at Göttingen under the Nazis. Notices Amer. Math. Soc., 42: 1134–1138, 1995. http://www.ams.org/notices/199510/maclane.pdf.

[39] http://www.tollmien.com/noethertelegrammapril1933.html.

[40] Richard Courant and David Hilbert. Methods of Mathematical Physics, 2 vols. John Wiley Interscience, New York, 1953 & 1962.

[41] Tufts University Digital Collections and Archives. The Life and Work of Edward R. Murrow: Murrow at the International Institute of Education (IIE), 1932–1935. https://dca.lib.tufts.edu/features/murrow/exhibit/iie.html.

[42] Several interesting documents from the Edelstein Collection in the National Library of Israel appear in Hadar Ben-Yehuda. Emmy Noether: The Jewish Mathematician Who Changed the World. https://blog.nli.org.il/en/noether/, 2018.

[43] Max Born. My life: recollections of a Nobel laureate. Scribner, New York, 1978. See Part 2, Chapter III: Arrival of the Nazis.

[44] An extensive discussion of the drama of 1933 appears in chapters 15 and 16 of Constance Reid. Courant in Göttingen and New York : the story of an improbable mathematician. Springer-Verlag, New York, 1976.

[45] To Join Bryn Mawr. New York Times, page 23, 4 Oct 1933. https://nyti.ms/2Riprj6.

[46] Four of her Bryn Mawr students and Emmy Noether Fellows have contributed admiring recollections: Grace S. Quinn, Ruth S. McKee, Marguerite Lehr, and Olga Taussky. Emmy Noether in Bryn Mawr. In Bhama Srinivasan and Judith Sally, editors, Emmy Noether in Bryn Mawr : proceedings of a symposium, pages 139–146. Springer-Verlag, New York, 1983. For additional information about Noether's association with Bryn Mawr, see Qinna Shen. A Refugee Scholar from Nazi Germany: Emmy Noether and Bryn Mawr College. The Mathematical Intelligencer, 2019. doi: 10.1007/s00283-018-9852-0. https://repository.brynmawr.edu/german_pubs/19/.

[47] A. Einstein. The Late Emmy Noether; Professor Einstein Writes in Appreciation of a Fellow-Mathematician. New York Times, page 12, 4 May 1935. https://nyti.ms/2GJc4o1.

[48] ​​Reprinted in Auguste Dick's Emmy Noether, 1882-1935, pp. 112–152.

[49] Reprinted in Auguste Dick's Emmy Noether, 1882-1935, pp. 153–179.

[50] BL van der Waerden. Algebra. Springer-Verlag, New York, 2003. Two volumes.

[51] Reprinted in Auguste Dick's Emmy Noether, 1882-1935, pp. 100–111.

[52] Chen-Ning Yang and Robert L. Mills. Conservation of Isotopic Spin and Isotopic Gauge Invariance. Phys. Rev., 96:191–195, 1954. doi: 10.1103/PhysRev.96.191.

[53] Margaret W. Rossiter. Doctorates for American Women, 1868-1907. History of Education Quarterly, 22(2):159–183, 1982. doi: 10.2307/367747; and Walter Crosby Eells. Earned doctorates for women in the nineteenth century. AAUP Bulletin, 42(4):644–651, 1956. doi:10.2307/40222081.

[54] Ruth H Howes and Caroline L Herzenberg. Women physicists in the women's colleges. In After the War: Women in Physics in the United States, pages 5–1 to 5–18. Morgan & Claypool Publishers, 2015. doi:10.1088/978-1-6817-4094-2ch5.

Additional Information

1. Auguste Dick. Emmy Noether, 1882-1935. Birkhäuser, Boston, 1981. https://archive.org/details/EmmyNoether1882-1935.

2. Martha K. Smith and James W. Brewer (editors). Emmy Noether: a tribute to her life and work. M. Dekker, New York, 1981.

3. Bhama Srinivasan and Judith Sally (Editors). Emmy Noether in Bryn Mawr: proceedings of a symposium. Springer-Verlag, New York, 1983.

4. HA Kastrup. The contribution of Emmy Noether, Felix Klein and Sophus Lie to the modern concept of symmetries in physical systems. In Manuel G. Doncel, Armin Hermann, Louis Michel, and Abraham Pais, editors, Symmetries in Physics (1600-1980), pages 115–163. Seminari d'Història de les Ciències, Universitat Autònoma de Barcelona, ​​Bellaterra (Barcelona) Spain, 1987. http://bit.ly/2LG7gyl.

5. Leon M. Lederman and Christopher T. Hill. Symmetry and the Beautiful Universe. Prometheus, Amherst, NY, 2008.

6. Celebrating Emmy Noether, a symposium at the Institute for Advanced Study. https://www.ias.edu/ideas/2016/emmy-noether, 2016; History Working Group. Emmy Noether's Paradise. The Institute Letter, Spring 2017. Institute for Advanced Study, http://bit.ly/2R2J0fU, page 8.

7. Olver, Peter J. Emmy Noether's Enduring Legacy in Symmetry. http://www-users.math.umn.edu/~olver/s_/noether.pdf, 2018.

8. Clark Kimberling. Emmy Noether, Greatest Woman Mathematician. The Mathematics Teacher, 75(3):246–249, 1982. http://www.jstor.org/stable/27962871.

9. Judy Green and Jeanne LaDuke. Pioneering Women in American Mathematics: The Pre-1940 PhD's. American Mathematical Society/London Mathematical Society, Providence & London, 2009. https://bookstore.ams.org/hmath-34; see also http://bit.ly/2VAR5qP.

10. Slides illustrating the colloquium on which this article is based are available at Chris Quigg. A Century of Noether's Theorem, August 2018. https://doi.org/10.5281/zenodo.1346275.

This article is part of the text of the special discussion given by the author at Fermilab on August 15, 2018. It is published in Fanpu with the author's authorization. The original title of the article is Colloquium: A Century of Noether's Theorem.
https://arxiv.org/abs/1902.01989v2.

# About the Author#

Chris Quigg (1944-) is a distinguished scientist emeritus at Fermilab National Accelerator Laboratory (FNAL). He has been a visiting scholar at CERN, École Normale Supérieure in Paris, Cornell University, and Princeton University, and has been the Erwin Schrödinger Professor at the University of Vienna. His research covers many topics in particle physics, from heavy quarks to cosmic neutrinos. His work on electroweak symmetry breaking and supercollider physics has earned him the 2011 American Physical Society J.J. Sakurai Prize for outstanding achievements in particle theory, which has pointed the way to exploration at Fermilab's Tevatron and CERN's Large Hadron Collider (LHC). His current research focuses on experiments at the Large Hadron Collider.

Quigley is a fellow of the American Association for the Advancement of Science and the American Physical Society. He has received the Alexander von Humboldt Senior Scientist Award. As chair of the APS Division of Particles and Fields, he led the 2001 SNOWMASS study on the future of particle physics. He has served as associate section editor of Physical Review Letters (1980-1983), associate editor of Reviews of Modern Physics (1981-1993), and editor of the Annual Review of Nuclear and Particle Science (1994-2004). He has been an advisor to CERN on plans for a future circular collider.

Quigley is also dedicated to science communication. He was the founding lecturer of Fermilab's Saturday Morning Physics Program and has presented workshops on the nature of science to high school students and teachers. He also writes and speaks regularly to the public. Outside of work, he enjoys hiking long-distance trails in Europe and cooking.

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