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-...
I wrote an article targeting the average lay person that teaches this way; https://rubberduckmaths.com/eulers_theorem
Hopefully it's helpful and gives people good intuition for this. Group theory is extremely fundamental and can and should be taught after basic arithmetic and modulus operations. There's really no reason it can't be taught in childhood.
"The multiplicative group of integers modulo n that we saw above gets more interesting when you consider a composite number such as 15 which has factors of 3 and 5. Repeated multiplication by 2 will never produce a multiple of 3 or 5 and this time there are only 8 numbers, {1,2,4,7,8,11,13,14} less than 15 that are not multiples of 3 or 5."
I understood the earlier example of "mod 3" because you only have {1,2} but then it becomes a lot more complicated but there's no explanation of it. Multiplying by 2 repeatedly under mod 15 only yields {1,2,4,8}.
After writing this, I saw you explained it a bit later in the document, so perhaps a note to that effect would help other readers.
Wow you start going into the deep end and are already needlessly over-complicating everything.
I personally would have explained the concept of groups by writing the number symbols upside down and as words, count of things, etc. Then you force the students to prove the group properties. After that you should tell them to come up with a group isomorphism between the groups.
There is something off putting about being given definitions from a higher authority and having to wade through the mud and emerging with a poor intuition about the thing in question. Modular arithmetic is something that the students will have to learn on top of group theory, not something that acts as a learning aid.
It's kind of difficult to put into words, but the moment you manipulate any physical quantity, e.g. filling a kettle with water and emptying it, you are already deep into applications of group theory. The reason why it is possible to record physical quantities with numbers is that the physical thing you are measuring also obeys the properties of group theory.
What I'm trying to get at is that the definition of groups is that way, not just for a good reason, it must be that way, because otherwise it doesn't make sense.
[1] https://www.youtube.com/watch?v=4n1BhWzdVsU
[2] https://people.math.harvard.edu/~jjchen/docs/Group%20Theory%...
[1] Like so much else in maths.
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.
https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Math...
https://link.springer.com/book/10.1007/978-1-4614-0776-8
it doesn’t say what a lie group is but it gets you down the road if understanding representations and what tou can do with them. dramatically easier than fulton and Harris for self-study.
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?
I assume (for the lay physicist) it's the Hodge decomposition mentioned in here (pp6-8)
To be fair and looking back at history, the discovery of Maxwell equations, relativity and quantum theory are so intertwined with the discovery, invention and application of new Mathematical ideas, in particular emanating from the work of Hamilton, Grassmann and then Lie, Levi-Civita, Cartan, etc. that is difficult to separate at what extent those concepts influenced over each other in their attempt to explain and describe reality. The ability to express Maxwell equations in a compact form with quaternions before vector calculus was even a thing provides some evidence. One can argue that the classical formulation for electromagnetism could be expressed that way because Hamilton was trying to find the proper framework that could capture his ideas about physics. Fast forward some 60 years and you also have a similar thing happening with Pauli matrices in quantum theory, and the work of Noether in modern physics.
It is fantastically long, but still fascinating !
Charge is conserved => symmetry (though not capturing exactly the "(non-Noetherian) localization" that is special to it)
GP suggested the opposite thought process-- as you rightly imply:
disagreement between 2 observers whether charge is conserved or not => discovering that _something else_ is conserved
Alternativey, geometry is how we choose to formulate our understanding of the Universe's behavior.
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)
Update: better ref, but paywalled
https://pubs.aip.org/aapt/ajp/article-abstract/48/1/5/235124...
> 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.
Would love to hear some recommendations!
Maybe I’m misunderstanding the implication here but wouldn’t it be much more surprising if that weren’t the case?
The energy is a ratio between "action" and time, where "action" is a primitive quantity that does not depend on the system of coordinates.
While energy can be computed with various other formulae, like the product of force by length, all the other formulae obscure the meaning of energy, because they contain non-primitive quantities that depend themselves on time and length.
So energy depends directly on time, thus the properties of time transfer to properties of energy.
Similarly, the momentum is a ratio between "action" and length, so the symmetry properties of space transfer to properties of momentum, resulting in its conservation.
The same for the angular momentum, which is a ratio between "action" and phase (plane angle of rotation).
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.
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:
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.
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.
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.
https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory...
This article is the shallowest I have read from quanta magazine. I expected more, give there articles in mathematics.
> 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.