Posted by ibobev 6 hours ago
https://www.amazon.com/Lie-Groups-Introduction-Graduate-Math...
and
https://bookstore.ams.org/text-13
My friends were all putnam nerds in college and I was not, and I assumed this math was all beyond me, but once you get the linear algebra down it's great!
https://d1gesto.blogspot.com/2025/11/math-education-what-if-...
https://en.wikipedia.org/wiki/Love_and_Math
and if you went to school in maths but now have left that world, this book engenders an additional spark of nostalgia and fun due to reading about some of your professors and their (sometimes very difficult) journey in this world.
So there's some strange sense in which these laws of nature seem to arise from, or are at least deeply connected to geometry.
Thanks for the postclassical angle on this, I missed that in the comment below, which was only "charge"
Not sure what you mean by Hodge equation, care to elaborate?
Some also think of this additional Lie as a ("central") extension of the Galilei group?
https://physics.stackexchange.com/questions/281485/how-did-m...
(Sorry, couldnot get Gemini to give a ref for that)
Maybe I’m misunderstanding the implication here but wouldn’t it be much more surprising if that weren’t the case?
Don’t we just commonly assume this axiomatically but there’s no evidence one way or the other? In fact, I thought we have observations that indicate that the physics of the early universe is different than it is today. At the very least there’s hints that “constants” are not and wouldn’t that count as changing physics.
One somewhat trivial example is that light loses energy due to redshift since photon energy is proportional to frequency.
If you're talking about gravitational redshift, because the light is climbing out of the gravity well of a planet or star, there actually is a conserved energy involved--but it's not the one you're thinking of. In this case, there is a time translation symmetry involved (at least if we consider the planet or star to be an isolated system), and the associated conserved energy, from Noether's Theorem, is called "energy at infinity". But, as the name implies, only an observer at rest at infinity will actually measure the light's energy to be that value. An observer at rest at a finite altitude will measure a different value, which decreases with altitude (and approaches the energy at infinity as a limit). So when we say the light "redshifts" in climbing out of the gravity well, what we actually mean is that observers at higher altitudes measure its energy (or frequency) to be lower. In other words, the "energy" that changes with altitude isn't a property of the light alone; it's a property of the interaction of the light with the observer and their measuring device.
If you're talking about cosmological redshifts, due to the expansion of the universe, here there's no time translation symmetry involved and therefore Noether's Theorem doesn't apply and there is indeed no conserved energy at all. But even in this case, the redshift is not a property of the light alone; it's a property of the interaction of the light with a particular reference class of observers (the "comoving" observers who always see the universe as homogeneous and isotropic).
Edit: I just looked into this & there are a few explanations for what is going on. Both general relativity & quantum mechanics are incomplete theories but there are several explanations that account for the seeming losses that seem reasonable to me.
But there's a much more striking example that highlights just how badly energy conservation can be violated. It's called cosmic inflation. General relativity predicts that if empty space in a 'false vacuum' state will expand exponentially. A false vacuum occurs if empty space has excess energy, which can happen in quantum field theory. But if empty space has excess energy, and more space is being created by expansion, then new energy is being created out of nothing at an exponential rate!
Inflation is currently the best model for what happened before the Big Bang. Space expanded until the false vacuum state decayed, releasing all this free energy to create the big bang.
Alan Guth's book, The Inflationary Universe, is a great book on the topic that is very readable.
1. Lie groups describe local symmetries. Nothing about the global system
2. From a SR point of view, energy in one reference frame does not have to match energy in another reference frame. Just that in each of those reference frames, the energy is conserved.
3. The conservation/constraint in GR is not energy but the divergence of the stress-energy tensor. The "lost" energy of the photo goes into other elements of the tensor.
4. You can get some global conservations when space time exhibits global symmetries. This doesn't apply to an expanding universe. This does apply to non rotating, non charged black holes. Local symmetries still hold.
We do not actually know that the current laws of physics will still hold tomorrow, we just assume they will. That's the entire problem of induction:
Lie groups are central part of the bootcamp where we will cover their applications beyond physics including geometric deep learning!
https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory...
> The group of all rotations of a ball in space, known to mathematicians as SO(3), is a six-dimensional tangle of spheres and circles.
This is wrong. It's 3D, not 6D. In fact SO(3) is simple to visualize as movement of north pole to any point on the ball + rotation along that.
> Though they’re defined by just a few rules, groups help illuminate an astonishing range of mysteries.
An astute reader at this point will go look up the definition of groups and come away completely mystified how they illuminate anything (hint: they do not).
A better statement is that many things that illuminate a wide range of mysteries form groups. By themselves, the group laws regarding these things tell you very little. It's the various individual or collective behaviors of certain groups that illuminate these areas.