Emmy Noether's Eye for Symmetries

On page 5 of his paper Archimedes – Das antike Jahrtausend-Genie, N. Froese writes:

"Vielleicht noch segensreicher war die Aufmerksamkeit, die Archimedes Symmetrien schenkt. Symmetrieargumente werden in den Beweisen teils explizit genutzt oder dienen als leicht aufzufassender Hintergrund bei der Anlage der Beweisgänge. Zudem tauchen Symmetrieforderungen immer wieder als Teil der Prämissen (Postulate) auf. Wie ungemein zentral Symmetrien für die theoretische Physik sind, hat im 20. Jahrhundert insbesondere Emmy Noether deutlich gemacht." (Seite 5)

[Translation: "Perhaps even more beneficial was the attention that Archimedes gave to symmetries. Symmetry arguments are sometimes used explicitly in the proofs or serve as an easily understood background in the design of the proofs. In addition, symmetry requirements appear repeatedly as part of the premises (postulates). How extremely central symmetries are for theoretical physics was made particularly clear in the 20th century by Emmy Noether."]

That is a wide arc: from Archimedes to Emmy Noether. But both used the search for symmetries in the hope of a better understanding of the world. And this is indeed an extremely productive approach. Its productivity can be seen not least in the Archimedes texts, which were increasingly read again from the Renaissance onwards. And in modern times, Noether's theorem on the connection between symmetries and conservation laws has had a lasting influence on theoretical physics.

Emmy Noether and Hilbert's Fight

Emmy Noether came to Göttingen in 1915, invited by David Hilbert and Felix Klein who recognized her expertise in invariant theory. They needed her help with problems in Einstein's general theory of relativity. Einstein later praised her "penetrating mathematical thinking" in his obituary for Emmy Noether.¹ Her solution to problems in general relativity led to the theorem bearing her name, establishing the fundamental connection between physical symmetries and conservation laws.

But her presence at Germany's leading mathematics center provoked fierce opposition. When Hilbert and Klein tried to get Noether appointed as Privatdozent in 1915, faculty members protested. The controversy exposed deep institutional resistance to women in academia. Historians and philologists particularly opposed her appointment, with one declaring: "What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?"

David Hilbert's response became legendary. During heated faculty meetings, he declared: "Meine Herren, die Fakultät ist doch keine Badeanstalt!" (Gentlemen, the faculty is not a bathhouse!)² This encapsulated his principle that intellectual merit, not gender, should determine scholarly standing. His unwavering support enabled Noether to lecture under his name from 1915 to 1919, circumventing official prohibitions. Social changes after World War I finally allowed her formal habilitation in 1919.

Noether's 1918 paper "Invariante Variationsprobleme" established the deep connection between symmetries and conservation laws that revolutionized theoretical physics. But this theorem was by no means her only outstanding achievement. She also gave algebra its modern face. Her student B. L. van der Waerden captured this in his 1935 obituary: "Und heute scheint der Siegeszug der von ihren Gedanken getragenen modernen Algebra in der ganzen Welt unaufhaltsam zu sein" (And today the triumphant advance of modern algebra, sustained by her ideas, seems irresistible throughout the world). Van der Waerden went on to disseminate Noether's ideas worldwide through his influential textbook Moderne Algebra, making his judgment particularly significant.

The Historical Context: Einstein and Energy Conservation

Noether's theorem emerged from a specific problem in Einstein's general theory of relativity. Between 1915 and 1918, physicists struggled with an apparent contradiction: general relativity seemed to violate energy conservation, one of the most fundamental principles of physics.

The difficulty arose because spacetime itself becomes dynamic in general relativity. Unlike Newton's absolute space and time, Einstein's spacetime curves and changes. This raised a troubling question: if spacetime itself can have energy, and if its structure changes, how can total energy remain constant?

Klein and Hilbert invited Noether to Göttingen specifically to address this puzzle. She approached it with characteristic conceptual clarity. Rather than attacking the specific problem of energy in general relativity, she asked a more general question: what is the relationship between symmetries of physical theories and their conservation laws?

Her answer, published in "Invariante Variationsprobleme" in 1918, resolved the energy conservation puzzle and revealed a fundamental principle that applies far beyond general relativity. Every continuous symmetry of a physical system, she proved, necessarily implies a conservation law. Time translation symmetry yields energy conservation. Spatial translation symmetry yields momentum conservation. Rotational symmetry yields angular momentum conservation.

This connection between symmetry and conservation was not merely a mathematical curiosity. It showed that conservation laws are not independent assumptions about nature but necessary consequences of the symmetries built into physical theories.

From Archimedes to Noether: Symmetries in Physics

When Archimedes used symmetry arguments in his proofs, he worked with geometric symmetries: a sphere looks the same from any direction, a line segment has two equivalent ends, a balance beam is symmetric about its fulcrum. These are visual, spatial symmetries that can be seen and drawn.

The symmetries in Noether's theorem are different. They are symmetries of the laws of physics themselves, not of particular objects. Consider a simple example: if you perform an experiment today and repeat it tomorrow under identical conditions, you get the same result. This is not a symmetry of any particular object, but a symmetry of the laws governing the experiment. The laws do not change over time. This is what physicists call time translation symmetry.

Similarly, the laws of physics are the same here as they are ten meters to the left. This spatial translation symmetry means that empty space has no preferred location. The laws are also the same regardless of which direction you face. This rotational symmetry means that space has no preferred orientation.

These symmetries of physical laws have profound consequences through Noether's theorem. But to understand these consequences, we must first understand what conservation laws mean and why they matter so deeply to physics.

Conservation Laws: The Pillars of Physics

A conservation law states that some quantity remains constant over time, regardless of what happens to the system. The total energy of an isolated system never changes. The total momentum never changes. The total angular momentum never changes. These are not approximations or convenient assumptions. They are exact principles that have never been observed to fail.

Conservation of energy means that energy can change form but cannot be created or destroyed. A falling ball converts gravitational potential energy into kinetic energy, but the total remains constant. A battery converts chemical energy into electrical energy. A lightbulb converts electrical energy into light and heat. In every case, careful accounting shows that total energy is preserved.

Conservation of momentum explains why a gun recoils when fired. The bullet gains forward momentum, so the gun must gain equal backward momentum, keeping the total at zero. Two ice skaters pushing apart move in opposite directions with momenta that sum to zero.

Conservation of angular momentum explains why a figure skater spins faster when pulling in her arms, why planets move faster when closer to the sun, and why a falling cat can rotate to land on its feet. These are not separate phenomena requiring separate explanations. They are all manifestations of a single principle.

These conservation laws provide the foundation for analyzing physical systems. Engineers designing machines, physicists studying particle collisions, and astronomers tracking planets all rely on conservation laws. They work because they are exact and universal.

Before Noether, conservation laws were empirical principles established by observation and experiment. Physicists knew they worked but did not fully understand why they held so universally. Noether revealed that conservation laws are not accidental features of nature but necessary consequences of the symmetries that structure physical theories.

The Noether Connection: Symmetry Implies Conservation

Noether's theorem establishes a precise mathematical connection: every continuous symmetry of a physical system implies a conserved quantity. To understand this connection, we examine three fundamental examples in detail.

Time Symmetry and Energy Conservation

If the laws of physics are the same today as tomorrow, then energy must be conserved. This is not an assumption or approximation. It is a mathematical necessity that follows from time translation symmetry.

To see why, consider a system that follows some physical law. Suppose we run the system today and measure how it behaves. Tomorrow we run it again with identical starting conditions. Time translation symmetry means we will observe identical behavior. The law governing the system has not changed.

Now suppose energy were not conserved. A system starting with 100 joules of energy today might end with 95 joules, losing 5 joules. But if we run the identical system tomorrow, time translation symmetry requires identical behavior. It must again lose exactly 5 joules. The amount lost does not depend on what day it is.

This consistency requirement, applied rigorously through the mathematical machinery that Noether developed, forces the existence of a quantity that never changes. That quantity is energy. Energy conservation is not an independent law of nature but a consequence of the fact that physical laws do not change over time.

If the laws of physics did change over time, energy would not be conserved. Cosmologists seriously consider this possibility when studying the very early universe or the large-scale structure of spacetime. Any detection of energy non-conservation would signal that time translation symmetry breaks down under those extreme conditions.

Space Symmetry and Momentum Conservation

If the laws of physics are the same here as they are one meter to the left, then momentum must be conserved. This follows from spatial translation symmetry in exactly the same way that energy conservation follows from time translation symmetry.

Consider an isolated system: two balls colliding, for instance. We perform the collision here and measure the momenta before and after. Then we move our entire apparatus one meter to the left and repeat the experiment. Spatial translation symmetry requires that we observe identical behavior. The laws do not care where in space the collision occurs.

Suppose momentum were not conserved. The two balls start with total momentum of 5 kilogram-meters per second and end with 4 kilogram-meters per second, losing 1 unit. If we move the apparatus one meter left and repeat the experiment, spatial translation symmetry requires the same loss of 1 unit. The amount lost cannot depend on where in space we perform the experiment.

This consistency requirement, applied through Noether's mathematical framework, forces the existence of a quantity that does not change. That quantity is momentum. Momentum conservation is not an independent law but a consequence of the fact that empty space has no preferred location.

This reveals something profound about the structure of space itself. If space had a preferred location, perhaps because some absolute reference frame existed, then momentum would not be conserved. The precise conservation of momentum we observe experimentally tells us that space is homogeneous, with no special places.

Rotational Symmetry and Angular Momentum Conservation

If the laws of physics are the same regardless of which direction we face, then angular momentum must be conserved. This follows from rotational symmetry.

Consider again two colliding balls, but now we measure their angular momentum about some chosen axis. We perform the collision with the apparatus facing north and measure angular momenta before and after. Then we rotate our entire apparatus to face east and repeat the experiment. Rotational symmetry requires identical behavior. The laws do not care which direction we face.

The same consistency argument that worked for energy and momentum now yields conservation of angular momentum. The amount of angular momentum cannot depend on our choice of orientation in space.

Angular momentum conservation has striking consequences. A figure skater spinning with arms extended has certain angular momentum. When she pulls her arms in, her rotational inertia decreases. To keep angular momentum constant, her rotation rate must increase. She spins faster. This is not a force acting on her but a necessary consequence of rotational symmetry.

Similarly, planets orbit faster when closer to the sun not because some force speeds them up but because angular momentum must be conserved as their distance from the sun changes. The precise mathematical relationship between orbital radius and velocity, described by Kepler's second law, follows from rotational symmetry through Noether's theorem.

The Mathematical Background

To understand how Noether proved her theorem, we need two pieces of mathematical machinery: the Lagrangian formulation of mechanics and the concept of symmetry groups. Following Richard Feynman's approach in The Character of Physical Law, we can grasp the essential ideas without heavy formalism.

The Lagrangian Formulation

Classical mechanics can be formulated in several equivalent ways. Newton's approach uses forces and accelerations. The Lagrangian approach, developed by Joseph-Louis Lagrange in the late 18th century, instead uses energy.

For any mechanical system, we define the Lagrangian as the difference between kinetic and potential energy: L = T - V. For a falling ball, the kinetic energy is ½mv² and the potential energy is mgh, where m is mass, v is velocity, g is gravitational acceleration, and h is height. So the Lagrangian is L = ½mv² - mgh.

The key principle is that the actual path a system follows is the one that makes the action stationary. The action is the integral of the Lagrangian over time. This principle of least action seems abstract, but it has a profound advantage: it makes symmetries manifest.

In Newton's formulation, forces and accelerations depend on the coordinate system we choose. If we shift our origin or rotate our axes, we must recalculate everything. In the Lagrangian formulation, symmetries become obvious. If the Lagrangian does not depend on position x, then the system has spatial translation symmetry. If it does not depend on time t, the system has time translation symmetry. If it does not depend on angle θ, the system has rotational symmetry.

Feynman emphasized this point in his discussion of physical law: symmetries are more easily seen in the Lagrangian framework. The Lagrangian makes the structure of the physical theory transparent.

What Noether Proved

Noether's achievement was to prove rigorously that every continuous symmetry of the Lagrangian implies a conservation law. If the Lagrangian does not depend on some variable, then the quantity conjugate to that variable is conserved.

If the Lagrangian does not depend explicitly on time, then energy is conserved. If it does not depend on position, then momentum is conserved. If it does not depend on angle, then angular momentum is conserved.

The proof uses the calculus of variations, the mathematical framework underlying the principle of least action. Noether showed that the condition for the action to be stationary under a symmetry transformation is precisely the condition that some quantity remains constant over time.

The mathematics involved is sophisticated, but the conceptual point is clear. Symmetries of physical laws necessarily produce conservation laws. This is not a coincidence or approximation but a mathematical theorem of the same status as the Pythagorean theorem in geometry.

Symmetry Groups

Modern physics describes symmetries using the language of group theory. A group is a mathematical structure capturing the idea of transformations that can be combined and reversed.

Consider spatial translations. We can translate by one meter to the left, then by two meters to the left, which is equivalent to translating by three meters to the left. We can reverse a translation: if we move one meter left and then one meter right, we are back where we started. These operations form a group.

Similarly, rotations form a group. We can rotate by 30 degrees and then by 60 degrees, which is the same as rotating by 90 degrees. We can reverse a rotation by rotating the opposite direction by the same amount.

Time translations also form a group. Advancing time by one second and then by another second is the same as advancing time by two seconds. We can conceptually reverse time, running events backward.

The mathematical structure of these groups determines the properties of the corresponding conservation laws. The fact that spatial translations form a three-dimensional group (we can translate in three independent directions) corresponds to momentum being a three-component vector. The structure of the rotation group determines the properties of angular momentum.

This group-theoretic perspective, which Noether helped pioneer, became essential for 20th-century physics. Modern particle physics is largely organized around symmetry groups. The Standard Model of particle physics is built on certain symmetry groups, and the particles and forces emerge as consequences of these symmetries, much as conservation laws emerge from spacetime symmetries through Noether's theorem.

Notes

¹ Einstein's obituary for Emmy Noether was published as a letter to the editor in the New York Times on May 5, 1935.

² From 1819 to 1931, there was a university bathhouse that was reserved for male members of the Göttingen University (students, lecturers, professors, and their sons).