Emmy Noether. A not well known female scientist

ORIGINAL: Yonatan Zunger

Portrait of Emmy Noether.  (1910). nhn.ou.edu/

There are some people who are simply insufficiently famous, and Emmy Noether is one of them. Now, mathematicians may claim her as one of their own, merely because she was a professor of mathematics for several decades and made more foundational and revolutionary contributions to the study of groups, rings, and algebras than I could easily list in a post, pioneering numerous basic techniques and having important mathematical structures named after her. But all of this manages to pale next to her contribution to physics; and being an ex-physicist, and thus feeling no reason not to be horribly biased about this, I want to tell you about Noether's Theorem: her 1915 proof that is, perhaps, the single most important result in mathematical physics. And when I say the single most important result, I mean not only that most of modern physics is built on top of it, but working theoretical physicists actually use this theorem several times a day.

Her theorem relates two seemingly unrelated things: symmetries and conserved charges. A physical system has a symmetry if there's some way to transform it which leaves all of its behavior the same. For example, if you took the entire Solar System and moved it three feet to the left, nobody could tell the difference. Similarly, if you rotated a sphere around any of its axes, nothing about them would change. Some symmetries are discrete, which means that they can only take on a fixed set of jumps: for example, you can rotate a square by 0°, 90°, 180°, or 270° and it will look the same, but if you rotate it by 260° it will look different. Other symmetries are continuous, which means that there's a continuous range of transformations you can make: e.g., you can rotate a circle by any angle, rotate a sphere by any angle along three different axes, or translate (shift) a physical system by any distance along any of three axes, or forward and backwards in time. Noether's Theorem actually applies to both kinds of symmetry, but the continuous ones are the most interesting ones.

There's an obvious reason that symmetry is important in physics: it tells you about things you can ignore, and strip out of your calculation, e.g. the overall angle of your experiment. 

A conserved charge is a quantity which doesn't change as a system evolves. Probably the most famous one is energy: conservation of energy means that you can't create energy out of nothing, nor can energy simply vanish. All you can do is change its form. There's also conservation of momentum: momentum is mass times velocity, and measures how much inertia something has. Angular momentum (also conserved) measures with how much oomph something is spinning. Electric charge (conserved) measures how much something can produce (and be affected by) electric and magnetic fields. Strong and weak charge (conserved) measure how things are affected by the fields which hold atomic nuclei together. 

Conservation laws are extremely important in physics for two major reasons. The first one is that they let you calculate things, because the amount of that charge before a reaction has to be the same as the amount after the reaction. To take a really simple example, say that a car (mass M, velocity v) runs in to a smaller car. (Mass m, velocity zero because it's at rest) After the collision, they're moving at two new velocities, v1 and v2. Before the collision, the total energy is the kinetic energy of the car, 1/2 M v^2; after the collision, it's 1/2 M v1^2 + 1/2 m v2^2. Before the collision, the total momentum is M v; after the collision, it's M v1 + m v2. Two equations, two unknowns, and you can immediately solve for v1 and v2. (I've cheated a little bit here – I assumed that all of the energy went into making things move, rather than into other things like bending metal. But even when you do the more elaborate calculations, you still start from conservation of energy)

The second reason is that if you have a conserved charge, you know that it's going to stay the same over all time, and therefore that value is a way to characterize the state of the system. That's especially important in quantum mechanics, where it turns out that any system can be described as a combination of certain special states – "eigenstates" – which are described entirely by their conserved charges. Once you've solved the equations to describe these eigenstates, you can use some simple algebra to describe any other possible state of the system.

So what did Noether do?
She proved that these two seemingly unrelated things are actually exactly the same thing. Specifically, whenever you have a continuous symmetry, there's a corresponding conserved quantity, and whenever there's a conserved quantity, there's a corresponding continuous symmetry. Not only that, but she gave a very explicit formula for how to relate the two. 

Some examples:

• string theoryPretty much every system has the "total translation invariance" symmetry, which means that if you shift the entire system over by any arbitrary distance along any of three axes, things stay the same. The corresponding charge is the total momentum of the entire system.
• Similarly, systems tend to have total rotation invariance: if you rotate everything by any angle around any axis, things stay the same. The corresponding charge is total angular momentum.
• Time translation invariance – moving the entire experiment into the past or the future – corresponds to the conservation of energy.

Electric charge (and those other charges) correspond to a much subtler symmetry, called "gauge symmetry," which has to do with the ways that electromagnetic fields can be transformed into each other. I don't have a particularly simple explanation of that one, unfortunately, but instead I have an interesting piece of history. The gauge symmetry of electromagnetism, which is related to electric charge, was only discovered shortly before Noether's theorem. Nearly a century later, when trying to understand the nuclear forces, people started to experiment with other types of gauge symmetry which were mathematically possible: combining gauge symmetry with noncommutative algebras, another subject where Noether made major contributions. It turned out that some of these new gauge symmetries, and the new charges they implied, perfectly described the interactions of all particles. 

The Standard Model of particle physics followed from this.

In Relativity, space and time intermix in complicated ways, but Noether's Theorem still applies. Instead of separate laws of conservation of energy and momentum, these combine into a single "conservation of four-momentum." Einstein's famous relationship E=mc^2 is actually a special case of that conservation equation.

So what does this law mean for modern physics? 

Nearly everything. I already mentioned the Standard Model, which is the backbone of particle physics. All of high-energy physics actually uses this: string theory depends on it in even more complex and subtle ways, and much of the subject has to do with understanding subtle, mathematical symmetries of the system, and then transforming those into conservation laws which can characterize real systems. In solid-state physics, the subject behind the existence of all the semiconductors, LEDs, laser diodes, and so on, the symmetries of crystals (and the related conserved quantities) are what determine the quantum mechanics of the electrons inside them – which is why we can predict how new materials will behave. In fact, the entire quantum theory of atoms, the basis for pretty much all of the study of matter since the 1920's, is really only tractable because the very first thing you do in any calculation is apply Noether's Theorem: each symmetry lets you eliminate one variable (an angle, a position, etc) and replace it with a conserved quantity, which is just a number. 

There are very few mathematical results which are this important to physics. This one, in particular, is studied by every physics undergrad, is used every day by every theorist, is so basic to the way we understand the universe that there's simply no way to imagine modern physics without it. 

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