Noether Shows the Equivalence of Symmetry and Conservation Summary

  • Last updated on November 10, 2022

Emmy Noether wrote one of the most beautiful chapters in mathematical physics by demonstrating the necessary and sufficient connection between a symmetry and a conserved quantity.

Summary of Event

As an achievement of theoretical physics, Emmy Noether’s 1918 theorems were the upshot of developments in mathematics and physics along several lines. With the proof of these theorems, Noether succeeded in bringing together various threads that had close mutual connections scientists had not suspected. The physics threads included conservation laws, the “symmetry principle,” and Albert Einstein’s theory of gravitation (general relativity). General relativity Relativity;general Gravitation theories On the side of mathematics, the threads were the theories of invariants, the notion of transformation groups, and an abstract concept of geometries. The idea of symmetry, which theoreticians were processing into a precise, abstract, and useful formulation, was another theorem. Mathematics;symmetry and conservation [kw]Noether Shows the Equivalence of Symmetry and Conservation (1918) [kw]Symmetry and Conservation, Noether Shows the Equivalence of (1918) [kw]Conservation, Noether Shows the Equivalence of Symmetry and (1918) Mathematics;symmetry and conservation [g]Germany;1918: Noether Shows the Equivalence of Symmetry and Conservation[04420] [c]Mathematics;1918: Noether Shows the Equivalence of Symmetry and Conservation[04420] [c]Physics;1918: Noether Shows the Equivalence of Symmetry and Conservation[04420] Noether, Emmy Hilbert, David Klein, Felix Einstein, Albert

Noether was the daughter of Max Noether, professor of mathematics at the University of Erlangen and a specialist in algebra. For mathematicians, Erlangen had been famous for Felix Klein’s Erlanger Programm, which was proclaimed in 1872. Klein unified the proliferating family of geometries by declaring that, defined by a group of transformations, a geometry investigates everything that is invariant under the transformations of this given group. Nevertheless, Noether did not start with an abstract approach. Her Ph.D. adviser was Paul Gordan, an old-fashioned specialist on the algebraic theory of invariants. After 1907, Noether became known as a theoretician on invariants. In 1915, Klein and David Hilbert invited Noether to join them at Göttingen University because they were interested in Einstein’s theory of general relativity and because they wanted Noether, with her expertise in invariants, to make contributions in this regard.

“Invariant” means constant, unchangeable. From the nineteenth century, in both physics and mathematics, scientists became increasingly interested in unchangeable characteristics in both physical systems and mathematical entities. Beginning from the law of the conservation of matter, a list of conservation laws began to take shape, concerning energy, momentum, angular momentum, electric charge, and so on. Physicists realized that it was important to ask under what conditions what quantities maintain constant values. The conservation of energy was officially titled the first law of thermodynamics. At the same time, physicists became aware that heat itself is hardly a conserved quantity (even under most usual physical conditions). The conservation of electric charge was confirmed as a basic principle; in the meantime, scientists realized that “magnetic charges” did not exist, much less as conserved quantities.

In mathematics, by the middle of the nineteenth century, Arthur Cayley Cayley, Arthur and James J. Sylvester Sylvester, James J. began to develop the idea of invariants. For example, Cayley showed that the order of points formed by intersecting lines is always unchangeable, regardless of spatial transformation. It is ironic that in the late 1820’s, the seminal idea of group theory occurred independently to two tragically short-lived mathematical prodigies: Niels Henrik Abel, Abel, Niels Henrik a Norwegian, and Évariste Galois, Galois, Évariste a Frenchman. When they attacked the same problem of algebra, their approaches took similar paths. In 1824, Abel proved the impossibility of solving the quintic equation—general equation of the fifth degree—by “radicals” (algebraically). Galois devised a more general solution: the clarification of the conditions that an equation must satisfy in order for it to be solvable algebraically. By the middle of the nineteenth century, Cayley was most responsible for furthering the idea of transformation groups. Later, Marius Sophus Lie, in collaboration with Klein, discovered the fundamental importance of the concept of continuous transformation groups, which can be used, for example, to classify mathematical theories. Moreover, in physics, one has to make transformations from one reference system to another; these are continuous transformations.

At the beginning of the twentieth century, Einstein became concerned with a fundamental question of physics: A physicist measures physical phenomena and formulates physical laws in relation to a reference system, yet an infinite number of possible reference systems exist, all equally good. This line of inquiry led to the relatively late idea of the “symmetry of physical laws,” Symmetry of physical laws or the “symmetry principles.” Einstein noticed that there existed a certain incongruity between dynamics (Newtonian mechanics) and electrodynamics (Maxwellian electromagnetism): Dynamic laws are valid for all the “inertial systems,” whereas electrodynamic laws appear to be valid in only one of these infinitely many “inertial systems.” He solved this problem by establishing the theory of special relativity. Special relativity Relativity;special Now, dynamics and electrodynamics are unified on a common basis: They are both valid for all inertial systems. In other words, the symmetry principle of the theory of special relativity requires that physical laws are symmetrical—invariant, unchanged—under any transformation from one inertial system to another.

After establishing the theory of special relativity in 1905, Einstein saw the need to generalize his result to include all possible reference systems. Previously, he had found electromagnetic laws in his way, as if they posed a hurdle; now, he found a barrier in the traditional gravitational law. Prior to 1916, Einstein made a great effort to reform the theory of gravitation. Göttingen mathematicians were aware of and very interested in his work. In 1915, Hilbert independently derived the basic equations of the theory of general relativity.

Both Hilbert and Klein welcomed Noether at Göttingen because she was able to help them with her invariant-theoretic knowledge. In fact, her work on the theory of general relativity ended in its most general mathematical formalism. On the basis of such eminent work, in 1918 Noether proved two theorems. One of these theorems is about the symmetry of a physical system. If it is symmetrical, there must exist a conserved quantity corresponding to this specific symmetry, and vice versa. The second theorem is about the symmetry of a physical law. Starting from a certain physical condition, which is decided by physical measurements, the requirement that the physical law is invariant under a transformation group leads to the quantitative law itself. Noether’s theorems thus represent a high degree of generalization. Physicists had already begun to realize the equivalence between a symmetry and a conservation law; Noether collected all these specific cases under the designation of her first theorem. Physicists had begun to realize the symmetry principle; Noether stated the “principle” in a general, rigorous, and succinct mathematical formulation.


For three decades, most physicists were not aware of Noether’s significant contribution. For several years following 1918, only the mathematicians at Göttingen introduced or extended her theorems. Ernst Bessel-Hagen applied Noether’s results to dynamics and electrodynamics (1921). Hermann Weyl mentioned them in his book Raum, Zeit, Materie (1918; space, time, matter), and so did Wolfgang Pauli in his Relativitätstheorie (1921; Theory of Relativity, 1958). Hilbert included them in his Methoden der Mathematischen Physik (1924; the methods of mathematical physics), which was coauthored by Richard Courant.

Around 1950, physicists began to cite Noether’s theorem—a trend in which Soviet mathematicians and physicists played an influential role. Since the middle of the 1950’s, because of the revolutionary impact of the confirmation of parity-nonconservation, “Noether equations” have been applied frequently in the context of general theory of relativity and in the exposition of quantum field theories. Textbooks sometimes do not explicitly mention either the title of the theorem or its inventor.

When Richard P. Feynman, Feynman, Richard P. Nobel laureate and distinguished teacher, gave a series of lectures on the new college physics between 1961 and 1963 at the California Institute of Technology, he discussed the “symmetry and conservation laws.” Symmetry is important, according to Feynman, because one can examine “some of the even more remarkable symmetries of the universe—the symmetries that exist in the basic laws themselves.” In quantum mechanics, for each of the rules of symmetry, there is a corresponding conservation law. Feynman noted that there is “a definite connection” between the laws of conservation and the symmetries of physical laws. This “definite connection” is the Noether theorem.

In 1983, the First International Meeting on the History of Scientific Ideas took place; the title chosen for this scholarly gathering was “Symmetries in Physics (1600-1980).” The meeting’s major speaker on the Noether theorem was Hans A. Kastrup from Aachen, Germany. His paper, “The Contributions of Emmy Noether, Felix Klein, and Sophus Lie to the Modern Concept of Symmetries in Physical Systems,” was published in the proceedings of the conference in 1987. In 1981, an anthology, Emmy Noether: A Tribute to Her Life and Work, was published to commemorate Noether’s one hundredth birthday. This great scientist is now very well known, and her personal stories have been told through a number of magazine articles, one of which was titled “Emmy Noether: She Did Einstein’s Math.” That part of her math is her 1918 achievement, but many specialists point out that it represents only a minor part of her total mathematical contribution. Mathematics;symmetry and conservation

Further Reading
  • citation-type="booksimple"

    xlink:type="simple">Brewer, James, and Martha Smith, eds. Emmy Noether: A Tribute to Her Life and Work. New York: Marcel Dekker, 1981. Includes articles by several specialists, Noether’s former colleagues. Valuable for its history of twentieth century mathematics.
  • citation-type="booksimple"

    xlink:type="simple">Dick, Auguste. Emmy Noether, 1882-1935. Translated by Heide I. Blocher. 1970. Reprint. Cambridge, Mass.: Birkhäuser Boston, 1981. The earliest biography of Noether in a single volume. Authoritative. Includes Weyl’s often-quoted memorial address.
  • citation-type="booksimple"

    xlink:type="simple">Feynman, Richard P. The Feynman Lectures on Physics. Rev. ed. 3 vols. Reading, Mass.: Addison-Wesley, 2005. Text of lectures Feynman delivered in the early 1960’s; geared for college physics students. Reads like a verbatim record of Feynman’s lively language.
  • citation-type="booksimple"

    xlink:type="simple">Kastrup, Hans A. “The Contributions of Emmy Noether, Felix Klein, and Sophus Lie to the Modern Concept of Symmetries in Physical Systems.” In Symmetries in Physics (1600-1980). Proceedings of the First International Meeting on the History of Scientific Ideas, edited by Manuel G. Doncel, Armin Hermann, Louis Michel, and Abraham Pais. Barcelona: Universitat Autonoma de Barcelona, 1987. Although somewhat technical, provides an excellent view of the “internal history” of science.
  • citation-type="booksimple"

    xlink:type="simple">Kramer, Edna E. The Nature and Growth of Modern Mathematics. 1970. Reprint. Princeton, N.J.: Princeton University Press, 1983. Discusses outstanding mathematicians around Noether as well as the topic of invariants. Unfortunately, does not include the two theorems or the topic of conservation.
  • citation-type="booksimple"

    xlink:type="simple">Weyl, Hermann. Symmetry. 1952. Reprint. Princeton, N.J.: Princeton University Press, 1983. As the late 1950’s witnessed feverish discussion on symmetries among physicists, this brief popular book can be described as timely, even prophetic. Introduces the general, abstract notions of group, geometry, symmetry, and so on by discussing the prevalence of symmetry in nature and the arts.

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