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Physicists like Isaac Newton, Albert Einstein, and Robert Oppenheimer are widely recognized and celebrated in science and history books. However, the name Emmy Noether often draws blank stares and confusion: “Never heard of her.”
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This is unfortunate because Emmy Noether was a phenomenal thinker who transformed both mathematics and physics. Her work underpins many of today’s fundamental physical theories, from the Standard Model of particle physics, which explains the basic particles of the universe, to the theory of relativity, which addresses the universe on cosmic and subatomic levels.
The extraordinary aspect of Noether’s contributions lies in their purely mathematical nature. Unlike physical laws, Noether’s theorem is formally proven and remains valid as long as the principles of mathematics are accepted, which lends it significant power. In the 1950s and 1960s, for instance, scientists were able to predict some elementary particles—the fundamental components of matter—by analyzing symmetry alone.
A world of hidden symmetries
Physicists perceive “symmetry” as a type of consistency or uniformity. Even when an object is transformed, such as through rotation or reflection, symmetry indicates that its fundamental characteristics remain unchanged.
Noether established that every continuous symmetry of a system corresponds to a conserved quantity—an invariant over time. Consider a car moving on a straight road with no friction from the wheels and no engine power. Once set in motion, this car will continue indefinitely. Moving the car 10 meters forward or backward does not alter the scenario: it is symmetrical concerning displacement.
According to Noether’s theorem, this symmetry leads to a conserved quantity, which in this case is momentum, the product of mass and velocity. Thus, the car cannot gain or lose velocity without external influence because momentum remains constant. However, if the road is uneven with hills and valleys, the symmetry is disrupted. Moving the car could mean traveling uphill or downhill, altering its speed. The system loses symmetry with respect to displacement, and momentum is no longer conserved: the car accelerates downhill and decelerates uphill.
Another common example in physics is the elastic collision of two spheres: as they approach each other, collide, and then separate, physicists rely on the conservation of total energy and momentum before and after the collision. Noether’s theorem supports this assumption of conservation.
Additional continuous symmetries exist. For example, satellites orbiting Earth are rotationally symmetric, maintaining constant angular momentum as long as their distance from Earth remains unchanged. More abstract symmetries can also be identified, such as the phase in the wave function of a quantum object.
Few realize that Noether formulated not one but two crucial theorems for physics. The second theorem addresses more abstract symmetries, particularly relevant in particle physics.
Noether’s exploration of these theorems drew physicists’ attention to symmetries and the related field of group theory, which greatly aided the development of the Standard Model of particle physics. Her work also contributed to the understanding of relativity.
In 1915, mathematicians David Hilbert and Felix Klein approached Noether because they observed that energy seemed not to be conserved in Einstein’s newly published general theory of relativity. Acknowledging her expertise, they sought her help with this conundrum. Noether’s subsequent work provided clarity: energy is not conserved in Einstein’s theory because time is not static; it can dilate and compress. Consequently, energy conservation only holds in specific cases.
Despite the critical nature of her work and her strong reputation among mathematicians, Noether never secured a permanent academic role. As a woman, she had to consistently battle for recognition, even with influential supporters like Einstein and Hilbert. Regrettably, her contributions have not received the widespread acknowledgment they deserve.
Noether also encountered obstacles due to her Jewish heritage. Born Amalie Emmy Noether in 1882 in Erlangen, then part of the Kingdom of Bavaria in the German Empire, she was likely inspired by her father, a notable mathematician. She began auditing math courses at the University of Erlangen and later enrolled as a full student when Bavarian laws permitted women to attend universities.
After earning her doctorate, she remained at the university for eight years without official status, occasionally lecturing in place of her father. In 1915, Hilbert and Klein invited her to Göttingen and advocated for her teaching position, though it took four years before she was approved as a lecturer and she received no pay for years.
By all accounts, Noether was passionate about mathematics and its advancement. She generously shared her ideas, contributing significantly to fields like algebraic topology. Her students, known as the “Noether boys,” went on to have successful careers of their own.
In 1932, she became the first woman to deliver a plenary lecture at the International Congress of Mathematicians in Zurich. The next year, she was expelled from Germany during a purge of Jewish professors after Adolf Hitler’s rise to power. She moved to the U.S. and taught at Bryn Mawr College, mentoring the “Noether girls” for two years until her death at 53 following surgery for an ovarian cyst.
With this introduction to Noether, the upcoming newsletter will delve into the calculus and physics concepts that enabled her to formulate her groundbreaking theorems.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
