Guest post by Dwight E. Neuenschwander
Noether’s Theorem of 1918, which is celebrated in Natalie Angier’s splendid biographical article about Dr. Emmy Noether, is developed in detail in my book, Emmy Noether’s Wonderful Theorem. The theorem makes explicit the connection between a system’s symmetries and conservation laws. Emmy Noether was a mathematician. For her, the theorem’s usefulness to physics was merely one of its applications. But she taught the physicists a thing or two. Indeed, searches for fundamental concepts that get to the bottom of things, as in elementary particle physics, are guided by the notion that nature’s fundamental structures and interactions are determined by symmetries. The spirit of Emmy Noether—unsung but influential—permeates fundamental physics today.
To appreciate her theorem’s content, let us consider “conservation” and “symmetry” separately, then put them together.
Among the most robust and productive principles of physics are the “conservation laws.” To say that a quantity—such as energy, electric charge, or momentum—is “conserved” means that, in a process or reaction, the quantity “before” equals that quantity “after.” Energy, for instance, can be transformed from one form to another, as when the gravitational potential energy of water stored in a reservoir is converted into electrical energy by falling through the turbine and spinning a generator. But throughout the transformation, the amount of energy carried into the process equals the amount coming out.
The notion of “symmetry” finds precise expression in the definition of “invariance.” For example, if you rotate a wine glass about its stem axis, I cannot tell that the glass has been rotated; the appearance of the glass is “invariant” under the rotation. This invariance occurs because the wine glass is symmetrical about its stem axis. Other transformations include a shift of location in space, an elapse of time, changing relative velocities between the observer and the observed, or changing the phase of a signal.
Emmy Noether’s theorem uncovers a deep connection between invariance (and thus symmetry) and conservation laws. If a system is invariant under a designated transformation, Noether’s Theorem spells out the conditions under which conservation laws result, and articulates the conserved quantities precisely.
Before Noether’s Theorem, conservation laws in theoretical physics could only be derived from dynamical principles case-by-case, using specialized tools such as Newton’s laws of motion for mechanical motion, or Maxwell’s equations for the electromagnetic field. One might have wondered, “Why does nature ‘cherish’ energy (for instance) so much as to conserve it, both in mechanical motion and in electromagnetism?”
It’s a bottomless question, but Noether’s Theorem takes a deepening step by unifying diverse sets of conservation laws into a common vocabulary.
Energy conservation results from a physical system exhibiting invariance under a time translation: the oscillations of a pendulum conserve energy because the system behaves the same today as it did yesterday. Electric charge conservation arises from the invariance of quantum fields under phase changes. One can even turn the theorem around, using it to discover new transformations that leave a system invariant, and thereby deduce previously overlooked conservation laws.
Natalie Angier mentions that Albert Einstein called Emmy Noether “the most ‘significant’ and ‘creative’ female mathematician of all time.” Einstein had written letters of support for Noether during her struggles to obtain a permanent teaching position. When Noether came to Bryn Mawr College, she also lectured at the new Institute for Advanced Study in Princeton, NJ, where fellow refugee Einstein had recently accepted a position. One wistfully wonders what might have been accomplished in mathematical physics if Noether and Einstein had been able to collaborate over an extended time. When Emmy Noether passed away, the Einstein comments that Angier recalled appeared in an obituary that Einstein wrote for Noether. That letter was published in the May 4, 1935, issue of the New York Times.
The title of Angier’s Times article, “The Mighty Mathematician You’ve Never Heard Of,” is aptly chosen. In this writer’s experience of asking, the few physicists who recognize Noether’s Theorem by name perceive it to be an important but esoteric topic tucked away in one or two advanced subjects, notably general relativity or the “gauge theories” of particle physics. Even fewer seem to realize how the sweep of Noether’s Theorem applies throughout almost all of undergraduate physics as well, from mechanics to electromagnetism to optics.
Emmy Noether’s Wonderful Theorem, one of the few books focused on the subject, was written to be accessible to anyone with a solid background in introductory physics and calculus. It strives to offer a thorough introduction to the logic and applications of Noether’s Theorem. May it contribute to helping others recognize Emmy Noether as a mighty mathematician, and one whom we have come to appreciate.
Dwight E. Neuenschwander is a professor of physics at Southern Nazarene University and editor of the Society of Physics Students publications of the American Institute of Physics. 
The views expressed in this guest post belong solely to the author and in no way reflect the official opinion of the Johns Hopkins University Press.
