Emmy Noether and Her Theorems

Today Noether's principal theorem occupies a prominent place in theoretical physics, though for a long time its significance was largely overlooked. Even now, relatively few physicists realize that Emmy Noether's original paper from 1918 contains two fundamental theorems. Moreover, both theorems are essential for understanding her original motivation, namely to distinguish between proper and improper conservation laws in physics.


DOI: 10.1002/andp.202300479
Feza Gürsey, to write about her two papers connected with physical theories.One of these, "Invariante Variationsprobleme," [4] is today considered one of her most important works because of its relevance for particle physics (Figure 2). [5]oday, in fact, Noether's name is probably better known among younger physicists than it is to mathematicians.That alone tells us something about new trends and massive shifts in perspective that have taken place during the last 40 years.In 1980, when Herbert Goldstein published the second edition of his textbook on classical mechanics, [6] he added a final section on "Noether's Theorem," which afterward gradually became a standard topic in physics courses.This marks part of the growing recognition that conserved quantities are closely related with the symmetries associated with Lie algebras or Lie groups.Gürsey summarized the general viewpoint of modern theoretical physicists when he wrote: Noether's Theorem associates each element of the Lie algebra (generator of a one parameter transformation of the group) with a corresponding conserved quantity.…Before Noether's Theorem the principle of conservation of energy was shrouded in mystery, leading to the obscure physical systems of Mach and Ostwald.Noether's simple and profound mathematical formulation did much to demystify physics. [7]is bird's-eye view from afar sheds no light on what actually happened.For one thing, the ideas of Ernst Mach and Wilhelm Ostwald were long since forgotten when Noether wrote her paper.But more to the point, if physicists and mathematicians who knew her work-like Albert Einstein, Hermann Weyl, and Wolfgang Pauli-had read it this way, then surely it would have made quite a splash.The fact that it did not cause much attention, however, has been firmly established by Yvette Kosmann-Schwarzbach, author of a detailed examination of the paper and the very weak reception it received for decades afterward. [8]This raises a serious question: why was Noether's paper presenting "Noether's Theorem" largely neglected and overlooked by the mathematical and physical communities until long after her death?Was this a simple case of sexist discrimination?That may have been a factor, but other conceptual issues were also involved here, and so this question is not at all easy to answer.

General Relativity during World War I
When I first wrote about how Noether's work was situated within the immediate reception of general relativity in Göttingen, I stressed that her seminal paper was not about one theorem but rather about two "Noether theorems" and that the less familiar   [4] in which she proved the Noether theorems.
second theorem was her main result at that time. [9]She wrote the paper in order to clarify the distinction between conservation laws in special relativity and other types of derived relations in general relativity.In both cases, one utilizes an invariant variational principle to derive these results.In theories governed by the principle of general covariance this yields formal relations which have no analogue in special relativity.Thus, Noether's first theorem establishes a framework for deriving physically conserved quantities from an invariant variational system.Her second theorem, on the other hand, describes how additional relations arise when that invariance holds under general coordinate transformations.
These types of relations were used by both Einstein and Hilbert in constructing two different formulations of energy conservation in general relativity.In a letter to Einstein, written in May 1916, Hilbert conjectured that their two approaches were either identical or closely related, and he informed him that he had given this problem to Emmy Noether. [10]This marks the beginning of her involvement with problems relating to the physical status of energy laws in general relativity.Very little is known about her work from that time, though she was able to show formal similarities between Hilbert's approach to energy conservation and Einstein's.More than a year later, the discussions in Göttingen on general relativity became much more intense after Klein and Einstein began corresponding about these matters.Indeed, Noether's two papers were written in close consultation with Felix Klein. [11,12]So how did this all begin?
In the spring of 1915, Emmy Noether came to Göttingen from Erlangen, where her father was the well-known professor of mathematics, Max Noether.Emmy's brother, Fritz, had studied under Arnold Sommerfeld in Munich before gaining a position as a private lecturer in Karlsruhe; he later became an accomplished applied mathematician in Breslau.Both she and her brother had spent time in Göttingen, where their father and Klein first met in the late 1860s, the beginning of a lifelong friendship.Emmy's interests in algebraic invariant theory overlapped with Hilbert's, whereas both he and Klein had ample opportunity to appreciate her abilities as a researcher.Moreover, it was wartime.On 1 December 1914, Noether wrote a letter to Hilbert that ended with the news that her brother was stationed north of Reims and doing well. [13,19]When Hilbert and Klein decided to invite Emmy Noether to habilitate in Göttingen the following year, quite a few younger men were engaged in war service.This was probably the one and only time in Emmy Noether's life when she benefited from being a woman.
Hilbert and Klein also both had longstanding interests in physics, and they had funds available from the Wolfskehl Foundation to invite prominent foreign guest speakers.Henri Poincaré delivered the first series of Wolfskehl lectures in 1909, followed a year later by H.A. Lorentz's lectures, which the young Max Born wrote up for publication.During the early summer of 1915, the invited speaker was Albert Einstein, who spoke about his new theory of gravitation.There, Einstein gave an account of the state of the theory before his final breakthrough to full general covariance in the fall of 1915. [14]And although very little is known about what he said, Hilbert was genuinely excited by what he heard in those six lectures and probably even more by his discussions with Einstein during that week.But what about Emmy Noether?Did she interact with Einstein at all back then?Unfortunately, like many other aspects of Noether's life, this relationship is not well documented, even though at the time of her death Einstein wrote a well-known letter for the New York Times in which he underscored her unique talents. [15,16]Sometimes people have even asked whether he knew her personally at all, and I recently discussed the possibility that they only first met in Princeton, after she was forced out of her position in Göttingen. [17]Of course, she would have met him already in the summer of 1915 had she attended his lectures, but until recently that seemed rather doubtful in view of the fact that she went back to Erlangen in May, at which time her mother passed away.Her father was an elderly invalid, who had been stricken with polio as a child, and Emmy was his only daughter.So we have no direct evidence that Emmy Noether was present when Einstein delivered his Wolfskehl lectures.
Thanks to recent research undertaken by Cordula Tollmien, however, there is no longer any doubt that Noether must have met Einstein during the week of his visit, because she had already returned to Göttingen the month before.This is important to know, because it helps to anchor some of the exchanges that took place afterward between Einstein and the mathematicians in Göttingen.During his visit there, Einstein stayed with the Hilberts, and so he was clearly made aware of an impending plan to petition the Prussian Ministry for an exemption from explicit regulations that prohibited women from being allowed to teach.He was surely also told about Noether's background and research interests, especially those aspects connected with Hilbert's own work.How she thereafter gradually took up various questions relating to the mathematics of general relativity is a complicated story.The literature listed below deals with it in considerable detail.

Emmy Noether steps back Onstage
In recent years, many new studies have appeared that have led to a better understanding of Emmy Noether's life, work, and times.Philosophers of modern physics have undertaken probing new investigations of the Noether theorems and their implications in ref. [18].Moreover, the historian Cordula Tollmien has embarked on a multi-volume project, which will illuminate many new facets of Noether's life.Her first two books in this series, [19] and, [13] document in vivid detail the final obstacles Noether had to overcome in order to attain the lowly position of a private lecturer at a German university.The mathematicians who supported her caseabove all David Hilbert and Felix Klein-were famously influential figures.Yet even they encountered such deeply entrenched resistance to the very notion of allowing a woman to teach that her appointment proved impossible.Only after the collapse of the German monarchy and the appointment of a Social Democrat as Prussian Minister of Education did this last barrier fall as well.Beginning in 1919, Noether taught her own courses, eventually attracting students from all over the world.Not once, however, was she ever offered a professorship in Germany.
Beginning in 2019, many people have gained a new appreciation for Emmy Noether's unique career through a play produced by Portraittheater Vienna, an ensemble that specializes in bringing famous or lesser-known women back to life onstage.The original German version premiered on the 100th anniversary of Noether's final qualifying lecture, after which she gained the right to teach courses as a member of the Göttingen faculty.Since then, "Diving into Math with Emmy Noether" (the English version) has been performed at fourteen colleges and universities throughout the United States (see ref. [20]) and at numerous localities in Europe.For information about past and future performances, see https://www.portraittheater.net/.

Figure 1 .
Figure 1.Emmy Noether in 1930 aboard a ship crossing the Baltic to attend a conference of the German Mathematical Society held in Knigsberg.Photo by Helmut Hasse.