Conservation laws are very useful for physics. Basically, when a particular measurable property of an isolated physical system does not change in time (and it is conserved), that is described by a conservation law.
Conservation laws are useful in a number of different fields of physics, but sometimes, they can become very complex. That’s where a 1910s theorem developed by a German mathematician comes in handy. Emmy Noether’s theorem connects symmetry and conservation laws, and its simplicity and elegance can be easily described as one of the most beautiful theorems in physics.
Amalie Emmy Noether was a German mathematician whose work focused especially on algebra. Her work was also extremely influential in a branch called mathematical physics — which deals with (you’ve guessed it) the development of mathematical methods in physics.
The development of many physics laws relies on something very abstract, the concept of symmetries. Symmetry can mean different things in different fields; in geometry, geometric shapes can be symmetrical, in chemistry, molecules can be symmetrical, but symmetry means something different in physics.
If you throw a ball upwards with an initial velocity, the kinetic energy turns into potential energy. Think of it this way: kinetic energy depends on motion, potential energy depends on position; when you throw the ball up, it starts by having a lot of motion energy — it’s going up. Then, when it reaches the maximum height, it has no motion energy, because it stops (and then goes down) — at that moment, it only has potential energy.
In other words, potential energy transforms into kinetic energy, and kinetic energy converts into potential energy, and then back again.
If you were to set up a kinetic energy-potential energy experiment, the experiment would work the same if you try it again under the same conditions. Tomorrow, the day after, and so on, you’d get the same thing — this means that the energy conservation is the same regardless of the time. In other words, the time transformation doesn’t change the system at all. This circumstance in which nothing changes in a law of physics/ feature of a system even with the action of a transformation is called symmetry.
The two theorems
In 1918, Emmy Noether published a seminal article connecting symmetries and conservation laws.
The German mathematician’s career was already showing signs of brilliance. Her interest in mathematics came from her father who was a mathematics professor at the University of Erlangen. Later on, Noether would go on to become a student and also a professor — but without a salary. Little had she known her most famous contribution, the Noether Theorem, would unify ideas that developed slowly century by century by René Descartes, Gottfried Leibniz, Isaac Newton, and Joseph-Louis Lagrange.
Noether published a key mathematical paper discussing transformation groups, but without discussing a direct relation to physics. Her work was divided into two theorems, the first of which gave a global relationship between continuous symmetries and conservation laws, and the second applied to systems with local symmetries.
The ball experiment regarding the conservation law is the simplest example. The energy is conserved because there is symmetry under time translation. Now consider a more sophisticated idea about the conservation of angular momentum (momentum of a rotating object). Ice skaters are the masters of this conservation: they know that if they spin with their arms tucked in, this makes them rotate faster — and if they open their arms, their rotation speed decreases.
Think of it this way: angular momentum is mass times velocity times radius (L = m*v*r). The mass of a skater remains constant, so if the radius increases, the speed must decrease to compensate for that, and vice versa.
So what is the symmetry in this case? If momentum is conserved, then space has to be isotropic. That means if there is no extra torque (angular force), the angular momentum does not change. In simple words, isotropy means things are the same even if you rotate them. For example, a perfect sphere, without any marks, is invariant under rotation, no matter which way you turn it, it looks the same.
This, believe it or not, relates to one of the most important theories in physics: general relativity.
Between 1915 and 1918, she got involved in debates with David Hilbert, Felix Klein, and Einstein himself about conservation equations in GR. Einstein had arrived at an expression that was quite similar to the energy conservation we are more familiar with. However, one component of the equation was invariant under a specific transformation called affine transformation. Affine transformations preserve parallel structures like a symmetric leaf. This is not expected from a general equation which is supposed to incorporate the big picture of energy conservation.
In 1918, Klein pointed out the difference between Einstein’s field equations and the usual conservation of energy. Einstein didn’t agree, he believed the conservation was analogous to the classical mechanics at least in part of his equations.
For Hilbert, a leading mathematician of his time, the fact that GR didn’t show a similar type of energy conservation was simply because …it is what is. Noether, and later Klein, agreed that there was no conservation law in General Relativity, and that conclusion came from her famous theorem.
From Noether’s second theorem, the matter in GR shows a conservation similar to energy and momentum conservation, but without equations of motion. As a matter of fact, any theory that is the same for any coordinate system shows a weird conservation law, one that does not give solutions with direct physical interpretations.
Noether’s symmetry theorem spills into several other fields of physics.
The Standard Model of particle physics divides particles into two categories: fermions and bosons. All known fermions (the particles that make up ordinary matter) have half-integer spin. The spin number describes how many symmetrical facets a particle has in one full rotation; a spin of 1/2 means that the particle must be rotated by two full turns (through 720°) before it has the same configuration as when it started. Meanwhile, bosons, which are not matter particles, have integer spin (0, 1, 2, etc). Spin is a property in particles that indicates where they would point if immersed in a magnetic field. A proposed theory called Supersymmetry (SUSY) suggests that for every discovered fermion there would be a superpartner boson with similar property.
Why would that be good? It makes things elegant for sure, but it also makes things much simpler. SUSY’s lightest particle would make an excellent dark matter particle. Moreover, other particles can be predicted from SUSY, not of them were observed experimentally though.
According to Noether’s theorem, it is possible to find conservation if it is connected to symmetry. SUSY’s conserved quantity is the supercharge, some sort of general idea of charge as we understand it.
Noether struggled in a sexist society like most other female science figures. For her to study, she couldn’t officially be a student. To give lectures, she couldn’t be an official professor. After finally being recognized as a mathematician, she became a professor, but without a salary.
As one of the most brilliant scientists, her talent was fundamental in convincing her peers to be respected. But that wasn’t enough. Unfortunately, she needed the support — or maybe we should call it approval –of important mathematicians to be taken seriously by universities. Hilbert took part in that, he convinced others to accept her at Göttingen, saying “I do not see that the sex of the candidate is an argument against her admission as privatdozent. After all, we are a university, not a bathhouse.”.
Noether’s theorem has been called “one of the most important mathematical theorems ever proved in guiding the development of modern physics”. Her contributions to mathematics were immense, especially relative to her short life. She discovered one of the most elegant, and quite possibly the most profound ideas in modern physics, often facing great adversity from male-dominated fields. It’s one of those that are absolutely worth knowing if you’re interested in science.