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Emmy Noether’s Breakthrough: Mathematical Symmetries Are Equivalent to Physical Conservation Laws

Written by Kat Jivkova. Emmy Noether's contributions to mathematics and Einstein's theory of relativity have been undervalued. What did her work involve? And why has it taken so long to be properly recognised?

Emmy Noether was essential to twentieth-century mathematics. She lived during a time when women were not mathematicians, where she, a German, was discriminated against for being Jewish. It is her identity which perhaps prevented her from being recognised as a renowned academic, in a period where the mathematical world was largely dominated by Felix Klein and David Hilbert. She was initially isolated from the mathematical world, but that was how she liked to work anyway. Noether once wrote “I always went my own way in teaching and research”. She preferred to see mathematics as a series of structures, or abstractions, rather than networks of relations between entire sets of objects. This enabled her to create proofs for more general structures and, in turn, revealed previously unnoticed connections.

This approach, which we will call the “Noether approach”, changed twentieth-century algebra. Noether realised that algebra could influence other mathematical fields, such as topology, hence creating a new discipline called algebraic topology. It was after obtaining her PhD at the University of Erlangen that she began to catch the attention of fellow mathematicians. Hilbert and Klein invited her to the University of Gottingen in 1915. The two prominent men had been working on Einstein’s general theory of relativity, which still had a few gaps in its mathematics, particularly in how energy fitted into his equations. Klein believed that Noether, an expert on invariant theory, would offer a new unique solution with her penetrating mathematical thinking. However, Noether’s path to her triumphal paper was not an easy one. Upon her invite to Gottingen, she applied for certification to teach, however this almost immediately led to a controversy in the faculty over whether qualified women were capable of teaching. A compromise was eventually reached whereby Noether could teach courses under the name of Hilbert. The same could be said about her paper on invariants, which had to be submitted by Klein to the Gottingen Scientific Society in 1918.

Klein had been conferring with Carl Runge on the general relativity problem, the latter who had allegedly found a way to solve the problems associated with gravitational energy in Einstein’s theory. By particularising the coordinate system so that Einstein’s pseudo-tensor for gravitational energy would disappear, energy-momentum conservation would be possible for general, just as in special relativity. Klein described Runge’s solution as “a pure egg of Columbus” and wrote to Noether about the breakthrough. She was quick to criticise the idea by suggesting “one cannot, in any matter, obtain an energy law unless one postulates it in place of Runge’s condition.” Einstein soon also dismissed the idea since “the theory predicts energy losses due to gravitational waves” which means the conservation of energy would not necessarily happen in a closed system. While this failed proposal caused Runge and Klein to reconsider the matter , Noether was on the verge of obtaining key results and reassured Klein in a letter that “I have now seen that the energy law fails to hold for invariance under every extended group of transformations induced by z.” She came to the realisation that conservation laws had actually derived from various symmetries.

By showing that nature requires symmetries that are accompanied by conserved physical quantities, such as energy, she “clarified this difficult matter [general relativity] fully” and upon studying her paper, Einstein stated that “everything is wonderfully transparent.” Not only did this give a solution to general relativity, but Noether’s theorem also built a new framework for the discovery of physical laws. Even the standard model of particle physics was created in terms of Noether’s symmetries. Dutch mathematician Van der Waerden writes:

“She came and at once solved two important problems. First: How can one obtain all differential covariants of any vector or tensor field in a Riemannian space? …. The second problem Emmy investigated was a problem from special relativity. She proved: To every infinitesimal transformation of the Lorentz group there corresponds a Conservation Theorem.”

The reception of Noether’s theoretical results were slow. Her paper submitted by Klein was in the Nachrichten of the Gottingen Scientific Society, mixed together with research from a variety of different scientific disciplines, which made it difficult to access. Despite Klein regarding her findings as definitive, one finds only a few dispersed references to “Invariante Variationsprobleme” in literature on relativity theory. Her theory’s importance remained largely hidden until the 1950s, and this was arguably due to a bias and discrimination against women in mathematics – one which persists today. Even Einstein, who admired Noether’s work, gave little generosity when it came to citing her work. Similarly, Hilbert keenly minimised the significance of her work in his paper of 1924 by only mentioning her in passing with only one footnote citing “Invariante”for a “general proof”. From this, it may seem that Noether was merely a genius at the mercy of her male colleagues. On the contrary, she was a self-assured mathematician who was able to overcome the hostility of a male-dominated world of mathematics, and who went on to successfully teach mathematics using her own name at the Institute for Advanced Study at Princeton. Her famous theorems showing the connection between symmetries and conserved objects in variational systems will no longer be dismissed. Nor will her general determination to change the scientific world against all odds.

Written By Kat Jivkova

Bibliography

“Celebrate the Mathematics of Emmy Noether.” Nature (London) 561, no. 7722 (2018): 149-50.

“Emmy Noether.” Physics Today, 2015, Physics Today, 2015-03-23.

Rowe, David E. Emmy Noether – Mathematician Extraordinaire. 1st Ed. 2021.. ed. Cham: Springer International Publishing : Imprint: Springer, 2021.

Rowe, David E, and Koreuber, Mechthild. Proving It Her Way. Cham: Springer International Publishing.

O’Connor, J J and Robertson, E F. ‘Emmy Amalie Noether.’ [Online]. [Accessed on 26 May 2021]. https://mathshistory.st-andrews.ac.uk/Biographies/Noether_Emmy/.

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