Emmy Noether and the Theorem Linking Symmetry and Conservation
In 1918 the German mathematician Emmy Noether proved a theorem that connects two of the most basic ideas in physics. Wherever the laws governing a physical system remain unchanged under a continuous transformation, such as a shift in time, a displacement in space or a rotation, a corresponding quantity is conserved. The fact that the laws of physics are the same today as yesterday implies the conservation of energy; the fact that they are the same in one place as in another implies the conservation of momentum; the fact that they do not depend on direction implies the conservation of angular momentum. Noether’s theorem, as the result is now known, has become one of the organising principles of theoretical physics. Its author is also regarded as one of the founders of modern abstract algebra, and she achieved both while being refused, for most of her career, the formal academic standing that her work merited. This article describes Noether’s life and education, the problem in general relativity that led to the theorem, the content of the theorem and its significance, and her later work in algebra, her emigration in 1933 and her death in 1935.
Erlangen
Amalie Emmy Noether was born on 23 March 1882 in Erlangen, a university town in Bavaria. Her father, Max Noether, was a professor of mathematics at the University of Erlangen and a respected algebraic geometer. Her mother, Ida Amalia Kaufmann, came from a prosperous merchant family of Cologne. The family was Jewish. Emmy was the eldest of four children; her brother Fritz also became a mathematician.
As a girl Noether attended the municipal school for higher daughters in Erlangen, where she studied German, English, French, arithmetic and the piano. In 1900 she passed the examinations of the Bavarian state for teachers of English and French, which would have qualified her to teach languages in girls’ schools. She chose instead to study mathematics at the university. Women could not then matriculate at Bavarian universities, but they could attend lectures as auditors with the permission of each lecturer. Noether did so at Erlangen from 1900 to 1902, one of only two women among nearly a thousand students. In 1903 she passed the matriculation examination at Nuremberg, and she spent the winter semester of 1903 to 1904 at the University of Göttingen, where she attended lectures by the astronomer Karl Schwarzschild and the mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein and David Hilbert.
In 1904 Bavaria opened its universities to women, and Noether enrolled at Erlangen as a regular student. She wrote her doctoral dissertation under Paul Gordan, a colleague and friend of her father, known for his work in invariant theory, the study of algebraic expressions that remain unchanged under certain transformations of their variables. Gordan’s methods were computational, and Noether’s dissertation, completed in 1907, consisted largely of the explicit calculation of a complete system of invariants for ternary biquadratic forms, a list of 331 covariant forms. She later described this work as rubbish, a judgement reflecting the shift in her mathematical outlook rather than any defect in its execution.
From 1908 to 1915 Noether worked at the Mathematical Institute of Erlangen without pay or formal position, supervising doctoral students, sometimes lecturing in place of her father, and publishing papers. After Gordan’s retirement she came under the influence of his successors, Erhard Schmidt and especially Ernst Fischer, who introduced her to the more abstract methods of David Hilbert. In 1909 she was invited to join the German Mathematical Society and spoke at its meeting in Salzburg.
Göttingen and General Relativity
In 1915 David Hilbert and Felix Klein invited Noether to the University of Göttingen, then the leading centre of mathematics in the world. Hilbert and Klein were at that time deeply engaged with the general theory of relativity, which Albert Einstein was completing in Berlin, and they needed someone expert in invariant theory to help them with its mathematical problems.
Einstein’s general theory of relativity, published in its final form in November 1915, describes gravitation not as a force but as a curvature of space and time produced by matter and energy. Its equations have the property known as general covariance: they take the same form in every system of coordinates, however the coordinates are chosen. Hilbert had derived the field equations of the theory almost simultaneously with Einstein from a variational principle, an expression whose stationary value yields the equations of the theory, and he had found that the theory satisfied certain identities that seemed to replace the familiar law of conservation of energy with something of a different character. In classical physics and special relativity, the energy of a closed system is conserved in a straightforward sense: there is a quantity, defined at each instant, that does not change. In general relativity it was not clear how to define the energy of the gravitational field at all, and the conservation laws that could be written down appeared to be mathematical identities true for any configuration of the field rather than physical laws derived from the equations of motion.
Noether’s task was to clarify this situation. Her work went far beyond the specific problem. In a paper titled Invariante Variationsprobleme, Invariant Variation Problems, presented to the Royal Society of Sciences at Göttingen by Felix Klein on 26 July 1918 and published in the same year, she proved two general theorems connecting the symmetries of variational problems with the conservation laws or identities that follow from them.
The Opposition to Her Habilitation
While she was doing this work, Noether was attempting to obtain the right to lecture at Göttingen as a privatdozent. This required habilitation, the formal procedure by which a scholar with a doctorate qualified for university teaching in Germany, and Prussian regulations at the time did not permit women to habilitate. Hilbert and Klein supported her application, but it was opposed by members of the philosophical faculty, which included the humanities as well as mathematics. One objection attributed to them was that soldiers returning from the war would find themselves learning at the feet of a woman. Hilbert is reported to have replied at a faculty meeting that he did not see that the sex of the candidate was an argument against her admission, since the faculty was not a bathhouse. The remark is widely quoted but its exact wording rests on later recollection.
The application failed. Noether continued to lecture at Göttingen, but her courses were announced under Hilbert’s name, with the notation that she would assist. After the end of the First World War and the revolution of 1918, the new Prussian government removed the bar on the habilitation of women, and Noether habilitated in 1919, with the 1918 paper serving as her habilitation thesis. In 1922 she was given the title of extraordinary professor, which conferred no salary, and in 1923 she was granted a small paid teaching assignment in algebra. She never received a full professorship in Germany.
Symmetry and Variation
Understanding Noether’s theorem requires two ideas, the principle of least action and the notion of a continuous symmetry.
The principle of least action, more precisely the principle of stationary action, was developed in the eighteenth and nineteenth centuries by Pierre-Louis Maupertuis, Leonhard Euler, Joseph-Louis Lagrange and William Rowan Hamilton. In its modern form it states that the actual motion of a mechanical system between two moments of time is the one for which a certain quantity, the action, is stationary, meaning that it does not change to first order when the path is varied slightly. The action is computed by adding up, over the course of the motion, a quantity called the Lagrangian, which for many simple systems is the kinetic energy minus the potential energy. From the requirement that the action be stationary one can derive the equations of motion of the system, the Euler–Lagrange equations. The formulation applies not only to particles but also to fields, such as the electromagnetic field and the gravitational field, for which the Lagrangian is defined at each point of space and time and the action is obtained by summing it over a region of space-time.
A symmetry of a physical system is a transformation that leaves its laws unchanged. In the Lagrangian formulation, a transformation is a symmetry if it leaves the action unchanged, or changes it only in a trivial way that does not affect the equations of motion. A continuous symmetry is one that can be applied in arbitrarily small amounts, such as a rotation through any angle or a translation through any distance, as opposed to a discrete symmetry such as a reflection in a mirror.
The First Theorem
Noether’s first theorem concerns symmetries that depend on a finite number of constant parameters, which mathematicians describe as forming a finite-dimensional Lie group, after the Norwegian mathematician Sophus Lie, who had developed the theory of continuous groups of transformations in the late nineteenth century. The theorem states that to each such continuous symmetry of the action there corresponds a quantity that is conserved when the equations of motion are satisfied. Moreover, the theorem gives an explicit procedure for constructing the conserved quantity from the Lagrangian and the transformation. Conversely, under appropriate conditions, every conserved quantity of this kind corresponds to a symmetry.
The classical conservation laws follow from the most basic symmetries of space and time. If the Lagrangian of a system does not depend explicitly on time, so that an experiment performed today gives the same result as one performed tomorrow under identical conditions, the theorem yields a conserved quantity which is the energy. If the Lagrangian is unchanged when the whole system is moved through space, so that the laws of physics are the same everywhere, the conserved quantity is the linear momentum. If the Lagrangian is unchanged when the whole system is rotated, so that the laws do not single out any direction, the conserved quantity is the angular momentum. The theorem therefore explains why these quantities are conserved: their conservation is the expression of the homogeneity of time, the homogeneity of space and the isotropy of space.
The converse is also illuminating. If a system is placed in an environment that breaks a symmetry, the corresponding quantity is no longer conserved. A ball rolling on a sloping table, where the gravitational potential varies from place to place in a fixed direction, does not conserve momentum along that direction. In an expanding universe, which looks different at different epochs, the energy of light travelling across cosmic distances is not conserved in the ordinary sense, and photons lose energy as their wavelengths are stretched by the expansion.
Some of these relationships had been noticed before Noether in particular cases. Lagrange, William Rowan Hamilton and Carl Gustav Jacobi had connected conservation laws to the absence of particular coordinates from the Lagrangian or Hamiltonian, and Georg Hamel and Gustav Herglotz had studied the conserved quantities associated with the symmetries of special relativity. Noether’s contribution was a general theorem, valid for any finite group of continuous transformations of any system governed by a variational principle, together with a rigorous derivation and a precise statement of its converse.
The Second Theorem
Noether’s second theorem addresses symmetries that depend not on constant parameters but on arbitrary functions of space and time, which form what is called an infinite-dimensional group. The general coordinate transformations of general relativity are of this kind, since the coordinates can be changed by different amounts at different points. The second theorem shows that such symmetries do not yield conservation laws in the ordinary sense. Instead they imply identities among the equations of motion, relations that hold whether or not the equations themselves are satisfied. In general relativity these are the contracted Bianchi identities, which had puzzled Hilbert, Klein and Einstein.
The second theorem thus explained the peculiar character of energy conservation in general relativity. The conservation laws that Hilbert had found were consequences of the general covariance of the theory and were in this sense improper, holding identically rather than as physical constraints. The theorem showed precisely when a theory would have proper conservation laws and when it would have only identities, and it revealed that this difference distinguished general relativity from the mechanics of particles and from special relativity. Klein published a paper later in 1918 drawing on Noether’s results to settle the dispute.
The second theorem received less attention than the first for several decades, but its importance became clear with the development of gauge theories in the second half of the twentieth century, whose symmetries are also of the local, infinite-dimensional kind. The theorem explains, for example, the relation between the gauge symmetry of electromagnetism and certain identities satisfied by Maxwell’s equations, which are described in the article on Maxwell and the Equations of Electromagnetism.
The two theorems also clarified a long-standing confusion about energy in general relativity. In the theories of Hilbert and Einstein, the local conservation of energy and momentum of matter holds automatically as a consequence of the field equations, while a meaningful total energy for the gravitational field together with matter can be defined only under special conditions, for example in a space-time that becomes flat far from all sources. Noether’s analysis showed that this was not a defect of the particular theory but a general feature of theories whose symmetry group depends on arbitrary functions.
Noether’s second theorem has since been recognised as the general framework for the gauge theories of particle physics, in which the symmetry likewise depends on arbitrary functions of space and time.
A Worked Illustration
The content of the first theorem can be seen in a simple example. Consider a single particle moving along a straight line under the influence of a force that depends only on its position, such as a weight hanging on a spring. The Lagrangian of this system is the kinetic energy, one half of the mass times the square of the velocity, minus the potential energy, which depends on the position. The Lagrangian does not contain the time explicitly: the spring is the same spring at every moment, and the mass does not change. A shift of the origin of time, replacing every moment t by t plus a constant, is therefore a symmetry of the action.
Noether’s procedure takes the change in the Lagrangian produced by an infinitesimal version of this shift and, using the equations of motion, constructs from it a combination of the position and velocity whose rate of change is zero. The combination turns out to be the kinetic energy plus the potential energy, the total mechanical energy of the particle. The conservation of energy thus appears not as an independent law of nature but as a consequence of the fact that the system’s laws do not change with time. If the spring were slowly weakening, so that its stiffness depended on time, the Lagrangian would contain the time explicitly, the symmetry would be lost, and the energy of the oscillating weight would no longer be constant.
The same reasoning applied to a particle moving freely in three dimensions, with a Lagrangian that does not depend on its position, yields the conservation of each of the three components of its momentum, while applied to a particle moving under a central force, whose potential depends only on the distance from a fixed centre, it yields the conservation of angular momentum about that centre. In the second case the conservation of angular momentum implies that the line from the centre to the particle sweeps out equal areas in equal times, which is the second of the laws of planetary motion described in the article on Kepler and the Laws of Planetary Motion. A law that Kepler found empirically in 1609 is thus seen, through Noether’s theorem, as an expression of the rotational symmetry of the Sun’s gravitational field.
Beyond the Classical Conservation Laws
The theorem applies to symmetries of any kind, provided they are continuous and the system is governed by an action principle, and it has been used to discover conserved quantities that are far from obvious. The orbits of planets under an exact inverse-square force possess a hidden symmetry, beyond the obvious rotational symmetry, which corresponds to the conservation of a vector pointing along the major axis of the ellipse, known as the Laplace–Runge–Lenz vector. Its conservation explains why the elliptical orbits of the Newtonian two-body problem do not precess. When the force departs slightly from the inverse-square law, as it does in general relativity, the symmetry is broken and the orbit slowly rotates, which is the origin of the anomalous precession of the perihelion of Mercury. The same hidden symmetry, in its quantum form, accounts for the coincidence of energy levels of different angular momentum in the hydrogen atom, as Wolfgang Pauli showed in 1926.
In the physics of materials, the symmetry of a crystal under translations is not continuous but discrete, since the crystal looks the same only when shifted by a whole number of lattice spacings. The ordinary momentum of a particle moving through a crystal is therefore not conserved. A modified quantity, the crystal momentum, is conserved instead, up to multiples of a set of vectors determined by the lattice. This modified conservation law, which reflects a discrete rather than a continuous symmetry, governs the behaviour of electrons in semiconductors and metals and is basic to the theory of solids on which the electronics described in the article on The Transistor and the Birth of Solid State Electronics depends.
In the theory of elasticity and fracture, conserved quantities derived by Noether’s method from the translational symmetry of a uniform material were introduced by John Eshelby in the 1950s and by James Rice in 1968. The J-integral that Rice defined measures the energy released by an advancing crack and is used by engineers to assess the safety of pressure vessels, pipelines and aircraft structures. In numerical computation, methods known as variational or symplectic integrators are designed to preserve the symmetries of the equations they solve, so that the corresponding conserved quantities are maintained over very long calculations, which is important in the computation of planetary orbits over millions of years and in molecular simulations.
Women in German Universities
Noether’s career illustrates the conditions under which women entered academic mathematics in Germany. German universities admitted women as regular students only gradually, beginning with Baden in 1900 and Bavaria in 1903 to 1904, and with Prussia, which included Göttingen, in 1908. A few women had earned doctorates earlier, often abroad or by special dispensation; the Russian mathematician Sofia Kovalevskaya received her doctorate from Göttingen in 1874 without attending lectures there, and later became a professor in Stockholm. The right to habilitate, which opened the way to a university career, came only after the war and the political changes of 1918 and 1919. Even then the number of women who held university posts in mathematics remained very small, and Noether, despite her eminence, was never appointed to a salaried professorship or elected to the Göttingen Academy of Sciences.
Her position was further complicated by her Jewish origins and by her political sympathies, which were on the left; she had been a member of the Independent Social Democratic Party and later of the Social Democratic Party in the years after the war, and she expressed admiration for the Soviet Union after her visit to Moscow. These circumstances, together with her sex, contributed to the reluctance of the authorities to promote her, and they made her an immediate target of the measures of 1933. Weyl, who tried to obtain a better position for her at Göttingen, later said he had been ashamed to occupy a privileged position beside her whom he knew to be his superior as a mathematician in many respects.
Reception in Physics
Einstein, who read the paper, wrote to Hilbert in May 1918 that he was impressed by the general way in which such things could be treated and that the old guard at Göttingen should take some lessons from Miss Noether, who seemed to know her business. Despite this, the theorem was at first little cited by physicists. It was gradually incorporated into the textbooks of theoretical mechanics and field theory in the following decades, often without explicit attribution. Its general recognition as one of the central results of theoretical physics came largely in the second half of the twentieth century, and historians, among them Yvette Kosmann-Schwarzbach, who published a detailed study and translation of the 1918 paper, have documented the slow growth of its reputation.
In quantum mechanics, the theorem takes a related form. A symmetry of a quantum system is represented by an operator that commutes with the Hamiltonian, the operator representing the energy, and the observable quantity associated with the symmetry is then conserved. The relation between symmetries and conserved quantities became one of the main tools for classifying the states of atoms, nuclei and elementary particles. The rotational symmetry of an atom, for example, implies the conservation of angular momentum, and the quantisation of angular momentum underlies the structure of the periodic table, whose history is discussed in the article on Mendeleev and the Periodic Table.
The conservation of electric charge is associated, through Noether’s first theorem, with a more abstract symmetry: the invariance of the equations of quantum electrodynamics under a change in the phase of the wave functions of charged particles. When this invariance is required to hold separately at every point of space and time, a local symmetry, it requires the existence of the electromagnetic field. The generalisation of this principle to more complicated symmetries by Chen Ning Yang and Robert Mills in 1954 led to the gauge theories of the weak and strong nuclear forces that form the Standard Model of particle physics. In that framework the conservation of quantities such as electric charge, colour charge and, approximately, the numbers of baryons and leptons is traced to symmetries of the underlying equations. Where a symmetry is broken, the corresponding quantity is not conserved, and the study of broken symmetries led to the prediction of the Higgs boson, whose discovery is described in the article on The Discovery of the Higgs Boson at CERN.
Abstract Algebra
Within mathematics Noether is known above all for her work in algebra, which she pursued from about 1920 onward and which changed the way the subject was practised. Her approach, sometimes described as conceptual, sought to understand mathematical structures through their general properties and the relations among them, rather than through explicit calculations with particular formulae.
Her paper Idealtheorie in Ringbereichen, the Theory of Ideals in Rings, published in 1921, developed the theory of commutative rings, algebraic systems in which one can add, subtract and multiply, such as the integers or the polynomials. Central to it was the ascending chain condition: in certain rings, every increasing sequence of ideals, special subsets closed under addition and under multiplication by ring elements, must eventually stop growing. Rings with this property are now called Noetherian rings, and Noether showed that a great deal of the theory of polynomial equations could be derived from this single condition. In 1927 she characterised the rings in which ideals factor uniquely into prime ideals, now called Dedekind domains, by a set of abstract axioms.
From the late 1920s she worked on noncommutative algebras, in which the order of multiplication matters, and on their representations, establishing connections between the theory of algebras and the theory of group representations. In collaboration with Richard Brauer and Helmut Hasse she contributed to the theory of central simple algebras, and the Brauer–Hasse–Noether theorem of 1932, which states that every central division algebra over an algebraic number field is cyclic, was regarded as a major achievement of algebraic number theory.
Noether influenced topology as well. In the mid-1920s, in conversations and lectures at Göttingen, she pointed out to the topologists Pavel Alexandrov and Heinz Hopf that the Betti numbers and torsion coefficients by which they described the shapes of spaces could be understood as properties of groups, the homology groups. This algebraic reformulation became standard and laid part of the foundation of algebraic topology.
Teaching and the Noether School
Noether’s influence was transmitted not only through her papers but through her students and the circle of young mathematicians who gathered around her at Göttingen. Her lectures were informal and often difficult to follow, since she developed ideas as she spoke, and she discussed mathematics with her students on long walks and at her apartment. Many of her ideas reached publication through the work of others. Her students and associates, known as the Noether boys although the group included women, included Bartel van der Waerden, Max Deuring, Hans Fitting, Jacob Levitzki, Olga Taussky and Chiungtze Tsen. Van der Waerden’s textbook Moderne Algebra, published in two volumes in 1930 and 1931, was based in large part on lectures by Noether and Emil Artin, and it established the abstract axiomatic approach as the standard way of teaching algebra for the rest of the century.
She also spent time abroad. In the winter of 1928 to 1929 she was a visiting professor at Moscow State University, where she worked with Alexandrov and taught abstract algebra, and in 1930 she taught at Frankfurt. In 1932 she gave a plenary address at the International Congress of Mathematicians in Zurich, an indication of the standing she had then achieved. In the same year she and Emil Artin received the Ackermann–Teubner Memorial Award for their contributions to mathematics.
Dismissal and Emigration
In January 1933 the National Socialists came to power in Germany. In April the new government passed the Law for the Restoration of the Professional Civil Service, which removed Jews and political opponents from posts in the universities and the civil service. Noether, as a Jew, was dismissed from her teaching position at Göttingen. She continued for a time to hold informal meetings with students in her apartment. One of her students attended in the uniform of the paramilitary SA, and she is reported to have taken no notice of it.
With the help of the Emergency Committee in Aid of Displaced Foreign Scholars and the Rockefeller Foundation, Noether obtained a visiting professorship at Bryn Mawr College, a women’s college near Philadelphia in Pennsylvania, and she arrived in the United States in the autumn of 1933. At Bryn Mawr she taught a small group of graduate students and worked with the mathematician Anna Wheeler, who had studied at Göttingen. From early 1934 she also lectured weekly at the Institute for Advanced Study in Princeton, where Einstein and Hermann Weyl were then working. In the summer of 1934 she returned briefly to Germany to visit her brother Fritz and to arrange her affairs; Fritz, who had also been dismissed from his post, had emigrated to the Soviet Union, where he was later arrested and executed in 1941.
Death
In April 1935 doctors found a large tumour in Noether’s pelvis. She underwent surgery on 10 April, and for several days appeared to be recovering. On 14 April she lost consciousness, her temperature rose sharply, and she died at the age of fifty-three. The cause of the sudden deterioration was not determined with certainty. Her ashes were placed under the walkway around the cloisters of the M. Carey Thomas Library at Bryn Mawr.
Several of her colleagues wrote obituaries. Hermann Weyl delivered a memorial address at Bryn Mawr. Bartel van der Waerden wrote an obituary in Mathematische Annalen. Einstein published a letter in the New York Times on 4 May 1935 in which he described her as the most significant creative mathematical genius thus far produced since the higher education of women began. Pavel Alexandrov addressed the Moscow Mathematical Society on her life and work.
Legacy
Noether’s two bodies of work, the theorem of 1918 and the algebra of the 1920s and 1930s, have different audiences, and for a long time few people were familiar with both. Mathematicians regarded her chiefly as an algebraist and the author of a method that reshaped their subject. Physicists came to regard her theorem as one of the foundations of their discipline, sometimes without knowing her mathematical work. Since the late twentieth century her reputation has grown in both communities, and the centenary of the theorem in 2018 was marked by conferences and publications.
Many mathematical concepts bear her name, among them Noetherian rings, Noetherian modules, Noether normalisation and the Noether isomorphism theorems. A crater on the far side of the Moon and the asteroid 7001 Noether are named after her. The German Research Foundation named its programme for outstanding young researchers the Emmy Noether Programme. The relation she established between symmetry and conservation continues to guide physicists in constructing new theories: when they propose a new symmetry they know what quantity it will conserve, and when they observe a conservation law they look for the symmetry behind it.
Further Reading
The principal sources for this article include the Grokipedia entries linked in the text and the following: the biography of Emmy Noether in the MacTutor History of Mathematics archive at the University of St Andrews; the English translation of the 1918 paper by M. A. Tavel, available on the arXiv preprint server; and the profile of Noether maintained by Bryn Mawr College. Related articles on this site include Einstein’s Annus Mirabilis and the Theory of Special Relativity and The Man Who Solved Poincaré.
