What I always miss from this introductory abridged explanations, and what makes the connection between Lie groups and algebras ('infinitesimal' groups) really useful, is that the exponential process is a universal mechanism, and provides a natural way to find representations and operators (eg Lie commutator, the BCH formula) where the group elements can be transformed through algebraic manipulations and vice-versa. That discovery offers a unified treatment of concepts in number theory, differential geometry, operator theory, quantum theory and beyond.
Such a bad (AI written?) article. These kind of introduction to advanced topics feels like how to draw an owl tutorial where they spent so much time diving into what group is.
> 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.
If, like me, you're not a real mathematician but suffered through linear algebra and differential equations, you can still totally understand this stuff! I started off teaching myself differential geometry but ultimately had far more success with lie theory from a matrix groups perspective. I highly recommend:
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!
There is also Naive Lie Theory by Stillwell, which is targeted at an undergraduate level. I haven't read it yet, but it's been on my radar for a while.
> For instance, the fact that the laws of physics are the same today as they were yesterday and will be tomorrow — a symmetry known as time translation symmetry, represented by the Lie group consisting of the real numbers — implies that the universe’s energy must be conserved, and vice versa. “I think, even now, it’s a very surprising result,” Alekseev said.
Maybe I’m misunderstanding the implication here but wouldn’t it be much more surprising if that weren’t the case?
> For instance, the fact that the laws of physics are the same today as they were yesterday and will be tomorrow
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.
The surprising thing isn’t that physics remain the same from one day to another, it’s that that fact is the reason for conservation of energy. There are lots of different symmetries for the laws of physics: the laws don’t change from one day to another, they don’t change from one part of the universe to the next, and they don’t change based on angles (e.g. if you snapped your fingers and rotated the entire universe by 10 degrees around some arbitrary point, the universe would continue exactly the same as before, just 10 degrees rotated). From Noether’s theorem, you can take any symmetry on the laws of physics, and use that to derive a conservation law. In those examples, that gives you conservation of energy, conservation of momentum, and conservation of angular momentum, respectively.
I hate statements like this due to their imprecision and their contribution to making mathematics difficult to learn.
> 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.
Including this near miss for Lie group E8 which at least had a pretty diagram and made some predictions about new particles. It looks like it was disproven.
There's an amazing way to derive Maxwell's equations, and the equations for 3 of the other fundamental forces of nature, directly from Lie group symmetry. You try to write down a theory that is symmetric under -local- symmetry transformations, meaning that the theory should give the same predictions even if you rotate (in some abstract space that you tack into the theory) by an angle that depends on position arbitrarily. At first this seems impossible, because any derivatives with respect to position will depend on the spatial variation of the rotation angle. But if you add an additional field that subtracts off the variation in the rotation angle, you find that this field is a dynamical object that coincides with the electromagnetic field! (Or to be more correct, it's the vector potential, which is directly related to electric and magnetic fields).
So there's some strange sense in which these laws of nature seem to arise from, or are at least deeply connected to geometry.
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[ 124 ms ] story [ 1213 ms ] thread> 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.
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://link.springer.com/book/10.1007/978-1-4612-0979-9
https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Math...
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.
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:
https://plato.stanford.edu/entries/induction-problem/
> 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.
https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory...
So there's some strange sense in which these laws of nature seem to arise from, or are at least deeply connected to geometry.
This article is the shallowest I have read from quanta magazine. I expected more, give there articles in mathematics.