She Who Changed Physics Forever
An introduction to Emmy Noether and her genius
For a really long time, I’ve wanted to write about Noether’s theorem. But I never did. I figured now’s as good a time as any.
Noether’s theorem, postulated by Amalie Emmy Noether, is by far, one of the most beautiful mathematical ideas out there. It doesn’t receive nearly as much credit as it deserves. So here’s me doing my part.
Guttingen, Germany, 1915:
Einstein and Hilbert were trying to complete general relativity. They failed for a really long time. Running into walls of theoretical nonsense. However, one particular paradox stood between them and the completion of this remarkable theory. The paradox was this:
“If energy warps spacetime and spacetime contains energy, then spacetime should warp spacetime”
It took Emmy Noether to fix this. But how?
Enter: Noether’s Theorem.
What if everything in the universe was translated to the right of where it is now? Or what if the Earth skipped to half a rotation ahead? Or what if Tottenham won the league? None of these things would ever happen. But you’d still ask “what changes?”. But more importantly, what stays the same?
These thought experiments may seem like an indefinite abuse of time. But trust me, they really aren’t. And shortly, you’ll know why. The usual understanding of Noether’s theorem is:
Symmetries imply conservation laws.
But what is a symmetry? What is a conservation law? How are they connected?
The property of being symmetrical usually corresponds to the idea that when an object is translated, rotated, or flipped along some arbitrary axis, it would be yield the same orientation as before the flipping.
And undeniably, that is what symmetry means. But we could make it mean so much more than that. In fact, mathematicians took the idea of symmetry and — like they always do — generalized it to death.
As far as mathematics is concerned, a symmetry is established when applying an arbitrary transformation to an object that yields no orientational change. Rotate a sphere 90 degrees about its center and you’d have the same thing you started with. The sphere would be “rotationally symmetric”. Take an infinite line, move it 4 units to the right, and you’d still have the same line. This would be considered “translationally symmetric”. Admittedly, this may seem like an irrelevant idea with no use beyond fancy mathematical definitions and abstract algebra. But it really isn’t.
The symmetry that Emmy Noether considered is this:
Suppose the object is some system continuously transformed to your whims. The “system” is a part of the universe or if you’d like, the universe itself. Maybe stretch all distances by some value λ, rotate all angles by λ, or even shrink every distance by λ. The question Noether asked was whether, in any sense, the system stays the same.
A caveat: Noether’s theorem limits itself to continuous symmetries. Continuous symmetries are described by transformations that change continuously as a function. They aren’t discrete. A sphere is continuously symmetric but a triangle isn’t.
To kick this off, Noether was particularly interested in the energy. So the definition here is tweaked a bit. We say that a system has a symmetry if the total energy of the bodies within that system does not change under some arbitrary transformation. For example, if I isolated a mass and compared it with a shifted version, the energies would remain the same.
So this system would be considered “translationally symmetric” since the arbitrary transformation we made on it was a translation of λ units to the right. On the other hand, a small change to our system can result in this symmetry being rendered void. Suppose we now have a planet in this system. The mass closer to this planet will have lesser gravitational potential energy while the mass further away will have more. This system is then not considered translationally symmetric.
Well, that’s the symmetry part. What does the conservation law mean? If you’re familiar — which I hope you are — with physics at a fundamental level, you’d know that conservation laws are extremely important. Simply put, conserved quantities are those quantities that cannot be destroyed nor created; only transformed from one form to another. Some conserved quantities are energy, momentum, and charge.
A conserved quantity is something that remains constant in amount over time and cannot be created nor destroyed. Laws that describe these quantities are called conservation laws.
But why? Why are such quantities conserved? Why can’t we just create energy? Noether’s theorem answers all of those questions. It explains — to a great deal of accuracy — where conservation comes from.
Specifically, it postulates that translation symmetry implies conservation of momentum. In fact, if every conceivable atom in the universe shifted 1 meter to the right, we wouldn’t be able to tell the difference. But then the question arises. When isn’t momentum conserved?
Well, let’s suppose an apple is falling. If you shifted the apple 2 units closer to the ground, it would have less gravitational potential and the velocity would be greater. As such p=mv would not be the same for both instances. The system would be considered translationally asymmetric. But even consider the universe if you’d like. Shift everything a few units to the left or right and you wouldn’t notice a thing out of place. That must mean that the energy is conserved and as a result, momentum would be too. So with complete confidence, we have the understanding that the total momentum of the universe is constant. The point is that translational symmetry implies conservation of momentum.
Translational symmetry → conservation of momentum
But what about rotational symmetry? The same would hold true. Consider the Earth and the apple as our system. But instead of falling, the apple orbits the Earth. We say that the conserved quantity here is angular momentum. If the object follows a spherical orbit, like the one shown below, then the total energy of the object at any given position remains the same. And so it has rotational symmetry about that axis.
Rotational symmetry → conservation of angular momentum
We’ve talked about transforming systems in space. But what would it mean to translate a system in time? In other words, suppose you have a system doing something at a particular moment, t, and you compare the system some time later, t + λ. Well, if the system maintains the same energy then it is considered time translationally symmetric. What does Noether say is conserved? Energy.
Time symmetry → conservation of energy
You don’t need to bother about understanding the complete extent of what’s shown below but I’ve only included it to juxtapose the complexity. It’s a series of brilliantly elegant mathematical operations that yield one of the most profound proofs.
The lines of math above include the derivation of a single independent variable. There are also — like explored earlier — time, translational, and rotational invariances. For quantum mechanical systems, there is the field theory version that is the fundamentals of modern-day particle physics. But it doesn’t just stop there. From Maxwell’s equations to General Relativity, you’ll find it everywhere.
Einstein and Hilber’s relativistic conundrum was solved by Noether. As she postulated, not all energy warps spacetime. It’s only the energy in the stress-energy tensor that matters. A stress-energy tensor is a mathematical object that contains all the information about the energy that warps spacetime.
The tensor carries 4 types of information. Two of them are “charges”, in the sense that is something conserved:
1. The energy of the field
2. The momentum of the field
It’s after these two, by the way, that the tensor is named. Those are exactly analog of the energy and momentum of a particle, but for a continuous system: the field → spacetime itself.
So far we have 4 components: 1 for the energy, 3 for the momentum (since momentum is a vector, it can be broken down in the arbitrary x, y, and z directions). The other two types of information are “fluxes”:
3. The energy flux
4. The momentum flux
But what’s a flux? Think, for instance, of an electromagnetic wave. It’s well known that it carries energy, right? You can burn things with a laser… pretty dramatic. Less well-known, but equally true, it carries momentum. This may not make much sense since photons are massless and if m = 0, then in p = mv, you’d get p = (0)v. But the answer’s a little more complicated than just that. What’s important is that waves have momentum. And well, the fluxes are an exact expression for how much energy is flowing in each direction (therefore, 3 components x, y, and z again) and how much momentum vector is flowing in each direction.
That’s all the information inside the energy-momentum tensor.
It doesn’t “explain all matter”. It only summarizes the gravitational influence of matter. Gravitation couples universally to energy and momentum and the flux of energy and momentum. It doesn’t care whether it comes from matter or electromagnetic fields, or what kind of matter it comes from. It’s symmetric.
This tensor, in particular, is the result of applying Noether’s theorem to a general spacetime transformation, that is induced by the space-time symmetry. This means that you have conservation laws for spacetime too.
In Quantum Field Theories, however, Noether’s theorem can be violated due to quantum effects. A symmetry at the classical level that is broken by quantum corrections is referred to as “anomalous”. Anomalous symmetries are one of the many predictions of the Standard Model. The following article will explore the violations of symmetry and how they relate to the quantum mechanical world.
And with that, thank you for reading! I hope you learned something about Noether and her work. If you enjoyed it, I’d appreciate your support.
