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:<p><a href="https://www.amazon.com/Lie-Groups-Introduction-Graduate-Mathematics/dp/0199202516" rel="nofollow">https://www.amazon.com/Lie-Groups-Introduction-Graduate-Math...</a><p>and<p><a href="https://bookstore.ams.org/text-13" rel="nofollow">https://bookstore.ams.org/text-13</a><p>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!
My experience with groups and linear algebra is similar. I made real progress only after I got past the initial fear and intimidation, making a point of understanding those beautiful equations. Now I find myself agreeing with those who argue that mathematics education could profitably begin with sets and groups instead of numbers.<p><a href="https://d1gesto.blogspot.com/2025/11/math-education-what-if-we-started-with.html" rel="nofollow">https://d1gesto.blogspot.com/2025/11/math-education-what-if-...</a>
Super easy to explain sets and groups once you've learnt how modulus works too. Start with the additive group and how it behaves under mod m, then go into the multiplicative group and the differences it has and the show why x^y = 1 mod m for certain values due to behavior of the multiplicative group. It's reasonably easy to grok how those two groups work and this gives people an intuitive understanding for the additive and multiplicative groups and they can go further from there.<p>I wrote an article targeting the average lay person that teaches this way; <a href="https://rubberduckmaths.com/eulers_theorem" rel="nofollow">https://rubberduckmaths.com/eulers_theorem</a><p>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.
Second this! And if you want a part memoir part history of this subject as it relates to physics (through Langlands Program) part ode to the beauty of maths, I recommend reading Edward Frenkel's Love & Math:<p><a href="https://en.wikipedia.org/wiki/Love_and_Math" rel="nofollow">https://en.wikipedia.org/wiki/Love_and_Math</a><p>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.
I recommend this intro graduate text on Lie representation theory:<p><a href="https://link.springer.com/book/10.1007/978-1-4612-0979-9" rel="nofollow">https://link.springer.com/book/10.1007/978-1-4612-0979-9</a>
for those who need an easier introduction to the subject (no general integration theory required, just finite sums) i can highly recommend<p><a href="https://link.springer.com/book/10.1007/978-1-4614-0776-8" rel="nofollow">https://link.springer.com/book/10.1007/978-1-4614-0776-8</a><p>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.
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).<p>So there's some strange sense in which these laws of nature seem to arise from, or are at least deeply connected to geometry.
If anyone is interested in this - checkout Richard Behiel's video on this. <a href="https://www.youtube.com/watch?v=Sj_GSBaUE1o" rel="nofollow">https://www.youtube.com/watch?v=Sj_GSBaUE1o</a><p>It is fantastically long, but still fascinating !
Indeed you can use symmetry, but it feels more like a mathematical hack, and the fact that it agrees with reality could be a coincidence. You can state that, and there is a lot of evidence for, that nature follows some basic geometrical rules. Applying that through a Lie theory framework on a symplectic manifold to see how charges behave differentially will eventually get you to Maxwell equations because of how those Lie algebras operate. However for me the real revelation was just using the Lienard–Wiechert approach to calculate how charged particles should behave in a relativistic field, which is as simple as it gets, and then see that you can build the full electromagnetic theory on top of that, with the bonus that the formulation is already relativistic. The same resulting symmetry in a corresponding Lie group is consequence of that (nicely captured by Hodge's equation), and invariance or operator rules don't need to be forced.
In the "opposite" direction, you might discover quantum mechanical "spin" from the Maxwell equation. Suggesting that coincidence is a kind of historical artifact :)<p>Thanks for the postclassical angle on this, I missed that in the comment below, which was only "charge"<p>Not sure what you mean by Hodge equation, care to elaborate?<p>I assume (for the lay physicist) it's the Hodge decomposition mentioned in here (pp6-8)<p><a href="https://arxiv.org/pdf/1305.6874" rel="nofollow">https://arxiv.org/pdf/1305.6874</a>
Somewhat related is noether's theorem (from Emmy Noether) that draws direct correspondence between symmetries and conserved quantities. E.g. conservation of linear momentum corresponds to a system that is invariant to translations. So you can find some of the fundamentals of a system by looking at symmetries and Lie groups/algebras give you tools to look at symmetries.
Yup, gauge theories can be understood geometrically as connections on vector bundles (and in a deeper sense as connections on principal bundles).
To add on to your mention of the rotation in abstract space , this is a local transformation of the electromagnetic potential. Not saying that "rotation" is a terrible thing to call it. it's just not usually thought of as a literal rotation. How about "twisting the potential"? Eg "twist" electric field into magnetic field? Rotation would connote that this is not 1D.<p>Some also think of this additional Lie as a ("central") extension of the Galilei group?<p><a href="https://physics.stackexchange.com/questions/281485/how-did-maxwells-theory-of-electrodynamics-contradict-the-galilean-principle-of" rel="nofollow">https://physics.stackexchange.com/questions/281485/how-did-m...</a><p>(Sorry, couldnot get Gemini to give a ref for that)<p>Update: better ref, but paywalled<p><a href="https://pubs.aip.org/aapt/ajp/article-abstract/48/1/5/235124/If-Maxwell-had-worked-between-Ampere-and-Faraday" rel="nofollow">https://pubs.aip.org/aapt/ajp/article-abstract/48/1/5/235124...</a>
> So there's some strange sense in which these laws of nature seem to arise from, or are at least deeply connected to geometry.<p>Alternativey, geometry is how we choose to formulate our understanding of the Universe's behavior.
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.
We are running a live online bootcamp, Group Theory 360: <a href="https://quantumformalism.academy/group-theory-360" rel="nofollow">https://quantumformalism.academy/group-theory-360</a>.<p>Lie groups are central part of the bootcamp where we will cover their applications beyond physics including geometric deep learning!
> 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.<p>Maybe I’m misunderstanding the implication here but wouldn’t it be much more surprising if that <i>weren’t</i> the case?
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.
> For instance, the fact that the laws of physics are the same today as they were yesterday and will be tomorrow<p>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.
It's funny you say that, because energy actually isn't conserved in general.<p>One somewhat trivial example is that light loses energy due to redshift since photon energy is proportional to frequency.
What "loses energy" actually means here depends on what kind of redshift you're talking about.<p>If you're talking about gravitational redshift, because the light is climbing out of the gravity well of a planet or star, there actually <i>is</i> 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.<p>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).
Where does the energy go then?<p>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.
The consequence of Noether's theorem is that if a system is time symmetric then energy is conserved. On a global perspective, the universe isn't time symmetric. It has a beginning and an expansion through time. This isn't reversible so energy isn't conserved.
The typical example people use to illustrate that energy isn't conserved is that photons get red-shifted and lose energy in an expanding universe. See this excellent Veritasium video [0].<p>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!<p>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.<p>Alan Guth's book, The Inflationary Universe, is a great book on the topic that is very readable.<p>[0] <a href="https://youtu.be/lcjdwSY2AzM?si=2rzLCFk5me8V6D_t" rel="nofollow">https://youtu.be/lcjdwSY2AzM?si=2rzLCFk5me8V6D_t</a>
There are certain answers to the above question<p>1. Lie groups describe local symmetries. Nothing about the global system<p>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.<p>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.<p>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.
That symmetries imply conservation laws is pretty fascinating (see the Noether theorem). I guess it seems only strange it you assume already that the conservation law holds.
It is surprising that you can derive conversation laws entirely from the symmetry of lie groups, and that every conservation law can be tied to a symmetry.
>the laws of physics are the same today as they were yesterday and will be tomorrow<p>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:<p><a href="https://plato.stanford.edu/entries/induction-problem/" rel="nofollow">https://plato.stanford.edu/entries/induction-problem/</a>
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.<p><a href="https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory_of_Everything" rel="nofollow">https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory...</a>
Original title is “What Are Lie Groups?”.<p>This article is the shallowest I have read from quanta magazine. I expected more, give there articles in mathematics.
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.<p>> 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.<p>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.
That is very strange. It's certainly not an academic level explanation, but that's not what the magazine is for. But the blatant incorrect statement is beyond the pale. Dim(SO(N)) = N(N-1)/2. Thus SO(4) has dimension 6.
The quality of this article is par for the course for Quanta Magazine, sadly. I do not need to accuse the author of using AI to explain the data I'm seeing here. It feels like every submission on HN from Quanta garners the exact same discussion: The article is almost worthless because it presents complex ideas in such a cheap, dumbed-down, and imprecise way that it ceases to communicate anything interesting. (Interested readers can fare much better by reading other sources.) It's been this way for years. The phenomenon is almost Wolfram-Derangement-Syndrome-like.
The “tangle of spheres and circles” is probably a reference to the Hopf fibration.
Correct. I have all of this worked out if anyone wants to check my work. I validated it through John Baez.
See also: <a href="https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory" rel="nofollow">https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory</a>
I hate statements like this due to their imprecision and their contribution to making mathematics difficult to learn.<p>> Though they’re defined by just a few rules, groups help illuminate an astonishing range of mysteries.<p>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).<p>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 <i>certain</i> groups that illuminate these areas.
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