“The Mother of Modern Algebra” is the title that the mathematician Irving Kaplansky bestowed on Emmy Noether in 1973.

 While it is true that Noether remained a pure mathematician throughout her career, physicists today make use of her famous theorem that demonstrates the fundamental relationship between symmetries and conservation laws. In physics, symmetries play a crucial role because they help, in addition to simplifying problems (an indispensable skill in itself), to understand how the laws of nature that govern the universe work.

Steven Weinberg, Nobel Prize winner in physics and author of some of the most authoritative volumes on quantum field theory, relativity and cosmology, wrote in the article Symmetry: a “Key to Nature’s Secrets” for The New York Review:

“When I began doing research in the late 1950s, physics seemed to me to be in a bleak state. A decade earlier there had been a great success with quantum electrodynamics. But now we had to deal with newly discovered particles. And we had to deal with mysterious forces: the strong nuclear forces, which hold together particles inside atomic nuclei, and the weak nuclear forces that can change the nature of these particles. We had no theory to describe these particles and these forces, and when we tried a possible theory, we found that either we could not calculate the consequences, or when we could, we arrived at nonsensical results, such as infinite energies or infinite probabilities. Nature seemed intent on hiding its plan from us. At the same time, we had a precious key to understanding the secrets of nature. Its laws evidently obeyed certain principles of symmetry, the consequences of which we could work out and observe by observation. It was like having a spy in the enemy’s high command.”

A further clarifying example of how important the concept of symmetry is in physics is given by a work by Pierre Curie in 1894 where he exploited these properties to solve problems in crystalline media:

“Physicists often use conditions that derive from symmetry but, as a rule, they neglect to define symmetry rigorously because these conditions are simple and almost obvious a priori.”

It is within this conceptual framework that, in 1918, Noether published her article "Invariante variationsprobleme" destined to become a milestone, unfortunately, only in the decades following her death.

Here she demonstrated a result that is widely used today and is introduced in all university textbooks: almost all books, ranging from classical mechanics to quantum field theory, have a section entitled “Noether's Theorem”. Qualitatively, this explains that if the mathematical description of the system does not change after a transformation of the variables that describe it, there exist quantities that are conserved and that are, that is, constant over time. A capital example is that of the conservation of energy which is due to invariances under temporal translations. But, as Nina Byers discusses in, at that time the result was practically never cited because, perhaps, it was not fully understood and connected to the results achieved by other physicists and mathematicians. In fact, Eugene Wigner, winner of the Nobel Prize in Physics for his work on symmetry, wrote in 1972:

“We physicists praise with empty words the great results of Emmy Noether, but in reality we do not use her work. Her contribution to physics, which is often cited, was born from a suggestion by Felix Klein. It concerns the conservation laws, which she derived in an innovative way and which should have excited physicists more than it did. However, most physicists know little else about her."