NOETHER, EMMY

Emmy Noether was a very important mathematician whose work profoundly influenced twentieth-century physics. Her 1918 paper "Invariante Variationsprobleme" contains theorems and their converses that reveal deep, fundamental connections between symmetries and conservation laws. Although she knew that these theorems were of great importance for physics, she regarded the work as something of a departure from the main line of her research. The work was done just after the completion of the general theory of relativity, when Albert Einstein, her Göttingen colleague David Hilbert, and others were seeking a principle of conservation of energy in the general theory of relativity. Her work clarified the issue of conservation of energy and solved the problem of the apparent absence of the law in Einstein's theory. After completing this work, she returned to her main line of research, which was development of abstract algebra. In the publication of her Collected Works, editor Nathan Jacobson writes, "Abstract algebra is one of the most distinctive innovations of twentieth century mathematics, and it is largely due to her" (Jacobson, 1983). Concepts and methods of modern algebra are now widely used in all areas of physics. Noether is distinguished as a contributor to physics not only for her work on symmetries and conservation laws but also for her contributions to modern abstract algebra whose importance for twentieth-century physics cannot be overstated.

Emmy Noether was registered as a student in the University of Erlangen in 1904, the first year women were admitted. She is an outstanding example of the fact that, after women had been excluded for centuries, female scientific genius emerged as soon as women were allowed to study in institutions of higher learning. Noether was born March 23, 1882, in Erlangen, Germany. Her father, Max Noether, was Professor of Mathematics at the University of Erlangen. His father was the first in the family to take up academic studies, but he became a merchant, and the family assumed that Max's mathematical talent came from his mother's side. Max Noether was a distinguished and highly respected mathematician who became even more distinguished in later years as the father of Emmy. Emmy attended the Municipal School for Higher Education of Daughters in Erlangen from age seven to age fifteen, when her formal schooling ended, as was normal for girls at that time. She then studied French and English and was certified to teach in girls' schools. To further her education, she sought permission to audit lectures at the University of Erlangen. This was granted, but at the discretion of the lecturer; some professors refused to lecture when a woman was present. In the winter of 1903–1904, she went to Göttingen to attend the lectures of the great mathematicians there. She returned to Erlangen in 1904 when, after years of debate, the university finally admitted women. She wrote a doctoral thesis under the direction of Paul Gordan and was awarded Ph.D. summa cum laude in 1908.

For more than a decade after receiving her Ph.D., Emmy Noether worked unpaid at the University of Erlangen, and later at Göttingen, teaching and doing mathematical research. During this period, she published fifteen papers in important mathematical journals, became a member of the prestigious Cicolo Mathematico di Palermo and the Deutsche Mathematiker Vereinigung (German Association of Mathematics [DMV]), and gave two lectures to the DMV. David Hilbert and Felix Klein invited

her to join their group at the University of Göttingen, then a world center for mathematics and physics. In 1915, she went to Göttingen after her father died. Prior to his death, his health had not been good, and she had been filling in for him as lecturer at the University of Erlangen. She was refused appointment as lecturer (Privatdocent) by the University of Göttingen in spite of the very strong recommendations of mathematicians such as Hilbert and Klein who were two of the university's most distinguished scholars. The reason the university refused to appoint her was because she was a woman. This so enraged Hilbert that he stormed out of a faculty meeting saying, "I do not see that the sex of a candidate is an argument against her admission as Privatdocent. After all, we are a university, not a bathing establishment."

On July 16, 1918, Noether's paper "Invariante Variationsprobleme" was read to the Königliche Gesellschaft der Wissenschaften zu Göttingen (Royal Society of Sciences of Göttingen) by Felix Klein. Presumably Klein presented it because Noether was not a member of the Society; it seems likely she wasn't even there when the paper was read. Records of the Society were lost in World War II, and it is not known when women were first admitted; counterpart societies in London and Paris did not admit women until after World War II. For example, the Royal Society (London) elected its first female member in 1945, and the Académie des Sciences of Paris did so in 1962.

This paper was immediately seen to be of enormous importance because it shows with great generality logical connections between symmetries and dynamical properties of the fundamental forces of nature. The results led to a deeper understanding of the principle of conservation of energy and of a vast variety of other conservation laws as well. Since the paper gives proofs of theorems and their converses, the insight it provides has led to discoveries of new symmetries of nature following empirical discoveries of new conservation laws. Examples include gauge field symmetries in the Standard Model of particle physics. In 1919 Noether was given Habilitation and was finally able to lecture as Privatdocent and be paid. In 1922 she became Lehrauftrag für Algebra und nicht-beamteter ausserordentlicher Professor and was paid a salary, although this was not an ordinary faculty appointment.

Historians of mathematics date the creation of modern abstract algebra to the work of Emmy Noether and Emil Artin and their school during 1921 to1933. Prominent mathematicians came from all over Germany and abroad to consult with Noether and attend her lectures. It was in 1921 that she published "Ideal Theorie in Ringbereichen," which is regarded as a truly monumental work, the first paper published on this vast subject. A pillar of this subject is her 1929 paper "Hyperkomplexe Grössen und Darstellungstheorie," which gave a general representation theory of groups and algebras, valid for arbitrary ground fields.

In 1933, when Adolf Hitler came to power, all Jews were dismissed from the University of Göttingen, and she could no longer teach there. A world-renowned mathematician, she had only two job offers. They were from two leading women's colleges, Bryn Mawr in Pennsylvania and Somerville College in Oxford, UK. Lacking financial resources, they could not offer her a secure position. Somerville offered her room and board and a small stipend. The Bryn Mawr job was subsidized in part by the Rockefeller Foundation. She went to Pennsylvania and was also invited to lecture at the Institute for Advanced Study in Princeton, New Jersey. Weekly, she took the train from Bryn Mawr to Princeton to lecture there. As in Göttingen, her lectures were very well attended. Presumably, if she had not been female, she would have been invited to be a member of the institute.

Following what was expected to be a minor surgery, Emmy Noether died at Bryn Mawr in April1935. In memoriam Albert Einstein wrote, "In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance . . . Pure mathematics is, in its way, the poetry of logical ideas. . . . In this effort toward logical beauty, spiritual formulas are discovered necessary for deeper understanding of laws of nature" (Einstein, 1935). Einstein may have been referring, in part, to formulas Noether gave in her 1918 paper. They do indeed yield deeper understanding of laws of nature. What must have been most impressive to Einstein shortly after he completed the general theory of relativity was the deeper understanding of the law of conservation of energy her work offered. Conservation of energy is one of the most important conservation laws in physics. In the early days of the general theory, Einstein, Hilbert, and others were perplexed by the apparent absence of a law of conservation of energy in Einstein's theory. Noether's theories solved this problem. Her Theorem I shows that space-time symmetry implies local energy conservation in classical field theories that are nonrelativistic or governed by the special theory of relativity—a familiar result. On the other hand, in the presence of gravitational fields, the general theory applies, and there is no such local energy conservation law. Instead, her Theorem II shows that space-time symmetry implies a different formula. This is necessary to accommodate gravitational radiation and the principle of equivalence. These results give deeper meaning to the principle of conservation of energy. What is truly extraordinary about her theorems is that they are very general and apply to a vast variety of symmetries and conservation laws.

See also:Conservation Laws; Symmetry Principles

Bibliography

Bourbaki, N. Elements of Historical Mathematics, translated by J. Meldrum (Springer-Verlag, Berlin, 1994).

Byers, N. "The Life and Times of Emmy Noether: Contributions of Emmy Noether to Particle Physics" in History of Original Idea and Basic Discoveries in Particle Physics, II, edited by B. Newman and T. Ypsilantis (Plenum, New York, 1996).

Byers, N. "E. Noether's Discovery of the Deep Connection between Symmetries and Conservation Laws." Israel Mathematical Conference Proceedings12 , 215–230 (1999).

Dick, A. Emmy Noether (1882–1935), translated by H. I. Blocher (Birkhauser, Boston, 1981).

Einstein, A. "Letter to the Editor." New York Times (May 5, 1935).

Jacobson, N. Introduction to Emmy Noether, Collected Papers, edited by N. Jacobson (Springer-Verlag, Berlin, 1983).

UCLA. "Contributions of 20th Century Women to Physics." <http://www.physics.ucla.edu/~ewp/>.

Nina Byers