How to rotate in higher dimensions? Complex dimensions? | Lie groups, algebras, brackets #2
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- Опубліковано 1 лип 2024
- Part 3: • What is Lie theory? He...
Around 11:50, can't imagine that this error got in - it should have been SU(n) = {U in U(n), det U = 1}.
Orthogonal and unitary groups. Rotational symmetries, real and complex, are particularly useful in the field of Lie theory, because their (complexified) Lie algebras, together with that of the symplectic group Sp(n), are the only infinite families of simple Lie algebras. This video is to familiarise with the SO(n), SU(n) notations, and provides further motivation to study Lie theory.
Files for download:
Go to www.mathemaniac.co.uk/download and enter the following password: orthogonalunitary
Video chapters:
00:00 Introduction
01:04 Real rotation in n dimensions
07:03 Complex rotation
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Originally I wanted to include at least the definition of a group in this prerequisite video, but really couldn’t fit it in properly, so I will postpone it to the next video (together with a rough definition of a manifold). This is an extremely standard video, but I will promise a lot more visuals in the upcoming ones - as said, this is just a prerequisite, and mainly for me to use the notation SO(n), SU(n) without explaining again.
Aww
Just fine for me.
This is not the first video of the series.
In any case, groups in general are like sets but they use steroids.
Direct product of SU(3)xSU(2) for standard model of particles, where you have a SU(2) With 3 degrees of freedom, 3 axis, but it transforms into a 4 axis object, and it gets worse in SU(3), 8 linearly independent transforms to 9 D🤓
we can have relatively easy to use tools, thanks god, but we need linear algebra as a good prerequisite.
I love my (almost) Riemannian mannigfaltigkeit manifold, for relativity, special I mean, I'm a quantum grad bloke... The abstract maths, I didn't fancy them until I really needed them, and I got the power of the dark side.. 🤔 😬 I mean, the power of an abstract framework like quantum 🤓🤓🤓🤓🤓🤓
have you considered adding quaternion into the mix as well? instead of just real and complex like in this video
Tip: careful while using "standard notations"; it's a novel language for anyone's first contant, and words/symbols mean nothing at all in the first glance.
When piling label on label, the resulting tower becomes "pattern recognition" for oldcomers, yet for people who actually need explanation, nothing meaningful forms.
It's kind like the "Chinese Room" thought experiment. Concrete context is imperative. Both before and after the video, you can already use the notation without explaining. Yet only to speak with those who already used it.
Can you do a video on base 6 and it properties, and how every prime number after 3 is end in either 1 or 5, since every prime is adjacent to a multiple of six?
Base 10 (2,3,5,7,11,13,17,19...)
Base 6 (2,3,5,11,15,21,25,31...)
@@abdjahdoiahdoaiI hope so; that's how you get SP(n).
Octonions would be nice, too; they come into play with the exceptional Lie groups. (At least, that's what John Baez says; I never got quite that far.)
This was the clearest explanation for O(n)/U(n)/SO(n)SU(n) I've seen so far. Well done! Looking forward to the rest of this series!
Yeah really, I literally said "wow" when i finished the video
For real, had to study them for my exam in nuclear and subnuclear physics and this 10 minutes video explained it better then the 20 pages in my professor's notes.
I completely agree! Amazing explanation! 🤩
That section should be shared to in schools
THIS is what teacher should do.
Never drop an definition without explaining WHY. Amazing job, i can't wait !!
This is the most natural introduction to the rotational and unitary groups I've ever seen. It makes me feel like I've missed out on the geometry of these groups for years! Thanks for your hard work.
I'm pretty sure the most natural introduction to rotational groups that I've seen was that they're compositions of an even number of reflections, where a reflection composed with itself is the identity.
Something I noticed is that the definitions given here only seem to work for spherical rotations around the origin. The origin is important for linear transformations, but is there really anything special about it geometrically?
@@angeldude101well if you describe a general rotation that also moves the origin, then you can decompose it into a translation and a rotation about the origin. So yes, in that sense the rotation around the origin is special because more complicated rotations can be made out of it, with a simple translation.
(And by the way two rotations about two arbitrary points can also be reduced to a translation and rotation around the origin)
@@karolakkolo123 Not a rotation about the origin and a translation. A rotation about _any_ point and a translation. The axis need not be include the origin in any way.
I'd just like to confirm that I'm thinking about this geometrically rather than algebraically. You can take any two planes in space, reflect across each and get a rotation about a predictable axis, without realizing the origin of your geometric space was halfway across the planet a few miles into the mantle. Rotations do not depend on an origin; only an axis.
@@karolakkolo123 Edit: Sorry, I was thinking of arbitrary handedness-preserving rigid transformations. A single rotation can indeed be translated from any axis to any parallel axis regardless of dimension.
Old comment:
Oh, and two rotations being reduced to a translation and a rotation is only true in 3D and below. It might be true in 4D, but I'm pretty sure it isn't true in 5D.
i think there are many paths to navigate Lie theory and it largely depends on preference. differential geometry and manifolds, rotations and algebraic operations, representation theory and generators/root systems, probably some other things i'm missing
Interesting, so now I know what an SU(n) group is! I've heard that it's used to describe the behaviour of forces in Quantum Mechanics.
It does! Idk where your math knowledge is, but essentially, it comes down to the fact that some value that’s conserved in a theory is invariant under SU(n) transformations. For basic understanding of what this means, notice that the derivative of a function is invariant under adding a constant to the function. The invariant transformation can get more complicated as you change your differential equation. For example, notice first order linear differential equations are invariant under multiplication by a constant. Every conserved value in physics has a related differential equation (which you will learn about when you learn about Noether’s theorem and Lagrangian mechanics), and that differential equation will be invariant under some transformation. For example, it turns out that the laws of physics wouldn’t change if electric charge were multiplied by U(1) elements, the complex numbers on the unit circle in the complex plane. For the other fundamental forces, they all have their own corresponding charges, with growing complexity, and so that’s where SU(n) shows up
@@joshuagrumski7459great answer! I'm a little confused, though, by the statement that electric charge can be phase rotated. I was under the impression that it was the (local) phase of the wave function that could be rotated freely without changing the momentum, and that electric charge and the EM field were a consequence of that invariance. In other words, the Schrodinger equation alone is insufficient to preserve momentum in the case of a phase shift, and subtracting the offending term that appears after integration requires the original equation to feature a field and charge with all the familiar properties of the EM field and charge. With the exception of relativistic effects, in the case of the Schrodinger equation, of course.
@@davidhand9721 Yeah, you're totally right! My bad!
@@joshuagrumski7459SO(n)U(n)😂
These videos make higher level maths so much more approachable. Looking forward to the rest of the series!
Glad you like them!
Cutting all the fluff and educating in a direct, simplistic and elegant manner, exactly what maths education videos should be. Look forward to the rest of the series and recapping some good old QFT.
Amazing. As a student of engineering, the walls to learn pure math are very high. Thank you for providing a passage through!
I was the person who asked if you could cover the topic in the community post asking fod suggestions.
Thank you so much. Your style is fantastic, and I can't wait to continue watching.
Thanks for your amazingly good introduction of orthogonal group.
You were a lifesaver! I was reading Visual Complex Analysis by T. Needham and I was stuck at this part; and you just popped up in my recommendations and answered all my questions
What a fascinating video about Lie group this is! I have waited for this kind of videos for decades.
GREAT video! Looking forward to seeing the rest :). Superb clarity! First time I see this all exposed so clearly..
Ooh this is a really good motivation! I'm on the edge of my seat for the next video!
best video on the topic , I've seen so far . Really loved it
Superb clarity! First time I see this all exposed so clearly.
Oh I’m SO hype for this series!
Just discovered this golden channel! Keep up with the great work Trevor!
Magnificent, clear explanation. Great job!
This was so clearly explained I can't wait for your future videos. Thank you so much.
Great video as always!
Looking forward to seeing the connection between SU(2) and SO(3).
Great video! I recently fell into the quantum mechanics rabbit hole on Wikipedia and "lie group" was one of the first terms where I had no idea what it meant. I was really happy when you announced your lie group series not long after - great timing ^^. Now I am really looking forward to the rest of the series, to get one step closer to understanding the math behind QM
You're in for a hell of a ride. If I may suggest, Wikipedia is not the first place I would go when trying to learn a science or math topic, math especially. For whatever reason, those wiki pages tend to stay at expert level, giving as many details as possible while not explaining a lot of things conceptually very well. I know I personally have a very hard time learning a new topic this way.
Fortunately, UA-cam is a much better resource. To get warmed up, the channel PBS SpaceTime has a large number of episodes that go over a lot of things, including the math, at the conceptual level. Or, you can skip straight to the plethora of university intro level lecture series that are available here, for free. Stanford University is a great example; there are full courses with good video quality so you can see the writing on the blackboard on a cell phone, if that's your thing. If you really want a full grasp of some more advanced topics, I highly recommend the channel xylyxylyx. The detail and depth he provides, as well as conceptual clarity, is second to none if you have the time to invest in his lecture series on GR, Lie algebra, tensor algebra/calculus, QFT prerequisites, and other topics. I honestly would never have fully understood tensors and differential forms without him, every other source failed to mention the one critical thing that made it click.
@@davidhand9721 thanks for the suggestions!
If you're still interested in learning quantum mechanics, I would recommend you a book "A modern approach to quantum mechanics" by John S Townsend. Probably the only prerequisites are some basic understanding of calculus, complex and linear algebra, but the book still gives a very solid understanding of the basics of quantum mechanics
@@dominikbaron9267 any time. Have fun.
Yo my friend I think you dropped this : 👑
No but seriously this series is off to a great start !
Thanks! I finally learnt what SO and SU stand for 🙂
Nice video, waiting for the next one!
GREAT video! Looking forward to seeing the rest :)
Clear Explanation, great video. And background sound takes me to the complex dimension..🙂
Man I was missing something like this a few years ago, this is awesome
Your voice is the best
Clearly calm and nice
Thanks for understand who love mathematics phisic and more science but not English first language or native English
Thank you !! waiting for part 3 !
Thank you so much for the high quality content.
I'm literally doing some research on optimization on Lie Groups, this is fantastic; quite literally the best introduction I've seen on the topic. If I could ask for something in this series, would be to have an example of how to evaluate functions (non linear functions, and ideally with non exponential growth) on Lie Groups: for example if you have a function over complex vectors (could be holomorphic) how could you extend it to the Lie Group.
Also talking about some calculus on the group could be super useful!
Thank you so much for this explanation. I was reading some papers related to generating novel protein backbones through diffusion models and kept on seeing the term SO(3). This video clears up the terminology and has saved me from a lot of frustration. Looking forward to the rest of the videos!
Uhm, very interesting, I guess the mathematical apparatus behind this kind of phenomenon appears as a differential equation system isn't.
Excellent as always
I knew that R^T R = I for orthogonal matrices, and knew the algebraic proof but didn't realize that just by noticing that angles and dot product are preserved, you can get there quicker. So intuitive. Thanks 😊
Originally I wanted to go through this in a more "traditional" way just by noticing the column vectors are orthonormal, but this preserving dot product is far more useful in QM - it is how we considered (rotational / translational) symmetries in the first place in QM!
Thank you for this valuable lecture.
This is already becoming a great series of videos (not that I expected any different)!
I'm concurrently learning geometric algebra, which that caused me to almost scream at the screen at 0:20 - "no no no! we need a plane for rotation, not its orthogonal vector!" ;-)
What an excellent video!
As someone who’s been out of undergrad for 6 years, I truly appreciate your content.
I've been out of undergrad for 17 years, and I can't even begin to explain how incredible UA-camrs are as a resource for higher learning. Spoiler: it's going to get a lot harder as you age to soak up new info, especially math, and especially if you don't keep at it most days of the week. Even if you work in a technical field, you can easily get into a routine of using the same processes and the same knowledge every day, and your ability to learn new things can stagnate.
Tl;Dr It's good to see that you are appreciating this resource for the miracle that it is, and I highly recommend that you use it while it's still easy.
Really nice explanation
Thanks
Another top tier video; a secret run of the power mill; thus, we know that we know what is known by those that know what is not known by those that know not what is what; and what is known now!
Very nice explanation.
yaaaaaas cant wait for the next one!
VERY GOOD AND VERY INTRESTING - THANK YOU🙏🙏🙏👍👍👍
My exam is tomorrow and was hoping for the Golden Content of Lie Algerba , but thanks and all the best !!
Great work.
Great video!
Superb!
just Superb
@mathemaniac, hoping you can discuss about Representation Theory
Excellent! ❤
Outstanding explanation of O, SO, U, SU zoo for the layman. Thank you. Just one remark, what about revisiting this with GA (geometric - clifford algebra) and the generalization of vectors (into multivectors)?
I want to be assured at 11:50 for the complex portion U belongs to O(n) or it would be U belongs to U(n)...
Thanks for your effort❤️ Making such complex things easier to be understood🙏
It belongs to U(n) as previously affirmed; in 11:50 there is a typo.
Thank you very much.
Super interesting ! Thank you for making theses videos 🙏🏼 i have a question .. In the end of the video you mentioned that with Lie theory we can have a formula for rotation matrices. Can you direct me to where to find this formula ? Thank you in advance
Wow, thanks!
I'm really hype for this series on Group theory
Thank you
Thanks
Nice video
awesome video! does anyone know of any intuitive difference between U(n) and SU(n)? I'm guessing physics deals mostly with SU(n) because it only has one component
11:24 I may be misunderstanding the concept of orientation, but why do we not have the notion of orientation in the complex vector space?
I've watched _very_ detailed lecture series on Lie groups and algebras and I feel like I get the guts of it, but I'm just a little too lazy and not confident to take that and figure out what I really need to know. I guess I should say that I only need to know for my own satisfaction, but I just want to understand why there are 8 SU(3) matrices and why they mostly look like reflections instead of rotations. It's similar to the SU(2), but in that case I understand how the 3D basis vectors are mixed, so to speak. With SU(3), I have never seen it spelled out. I hope you'll cover this in the future.
SU(n) in general have n^2 -1 generators that are the fundamental blocks that you can use to build any element of the group. In the case of n =2 you have 3 matrices; with this 3 matrices You can write any 2X2 complex valued matrix. In physics this is useful, because this 3 matrices coincide with the three Pauli spin matrices. For n = 3 you have the number of gluons that are needed to study interacting quarks; mathematically speaking you'll need 8 matrices to write any element of SU(3).
Can't wait to take this stuff, next year in abstract algebra?
I remember the Rubik cube project in group theory course, it was a S54 at first, but after a little bit of examination it gets WAY Smaller!
Let's just say any configuration can be solved in maximum 26 movements. Of course there are things like pi/2 rotation, and pi /4, that counts as one move 2/4 or 1/4 turn.
Pretty cool, I wrote it in python, with graphics (2D) super useful and easy, computer-friendly stuff 🤓🖖
Rubik’s cube group is a subgroup of S_48 since the centers never move
@@-minushyphen1two379 that's right, and those are the only independent, that cross is the vector base.
That S48 can be reduced lots cause the other squares are not independent, they are linear combos
@@misterlau5246 the rubik’s cube group is non-abelian
What is the interpretation of det on complex matrices?
I will definitely steal parts of this if I wver give TA classes with this material again
I will send the students the video, but after I ~~stole~~ took inspiration from it
Also this is the first time I've seen dagger used for this notation, we usually used an asterisk/star for it where I studied
2:23 'cz of linearity prop; rotation is a linear transformation (i.e. can be rep as a matrix)
waowww...
11:51 Typo? Should it not be SU(n) = {U € U(n), det(U) = 1}?
12:10 I guess by "quadratic", you mean a polynomial of multiple variables where the maximum order of terms is 2?
In this case I think its a system of n^2 equations of the form
Σ(k=1 to n)(u_ki* u_kj) = {0 if i≠j, 1 if i=j}
if that makes any sense.
ah nice after using latin letters, greek letters, hebrew letters, we now uses daggers isnt that wonderful
💡11:50 On the right, you want U in U(n), not U in O(n).
complex reflection groups
YAY!
Lorentz Rotation L, g_(μν)L^μ_ρ)L^ν_σ = g_(ρσ) , instead of (R^T)R=I, here the g_(μν) is the Minkovsky matrix.
I still dnt know why determinate preserves orientation, need some hlp😮🤔
Can you do a video on base 6 and it properties, and how every prime number after 3 is end in either 1 or 5, since every prime is adjacent to a multiple of six?
Base 10 (2,3,5,7,11,13,17,19...)
Base 6 (2,3,5,11,15,21,25,31...)
3B1B has a video where he explains why this must be the case among other stuff. ua-cam.com/video/EK32jo7i5LQ/v-deo.html
Primes in base six can't end in 0,2,3 or 4 because then they would be a multiple of 2 or 3 and thereby not prime. It's basically the same reason why there are no primes ending in 0,2,4,5,6 or 8 in base ten (those are automatically divisible by 2 or 5).
@@lunkel8108 Thank you! I will watch it!
@3:35 I guess you assume that your vectors are represented in an orthonormal basis. Because only in am orthonormal basis you can write v.w=v^T w.
Best best best
Nice title
I love daggering to w
11:27 For me this is hard to come to terms with. This is where I struggle with higher math -- the motivations become entirely abstract.
We extend rotation to the complex case by "fixing" the definition for the real case to satisfy some similar properties.
But does this complex rotation really deserve the name rotation? It's very difficult for me to just accept that we also require det(U) = 1 if orientation isn't a thing, or if I no longer understand these "vectors" with their complex components. What the hell am I looking at is what goes through my mind. Where is the geometric interpretation of these complex vectors? Given their applications in physics, it seems like a reasonable question to me.
Maybe mathematicians have a way to go in terms of interpretability... or maybe my tendency to cling to geometric meaning is just slowing me down.
The det(U) = 1 is to preserve orientation
Rotation matricies always have positive determinants. And since the lengts are preserved the value = 1
how to like more than once, a complex like?
When the next video will be uploaded?
I would like to know as well. I can't promise any timeline but I am working on it. Please be patient.
Someday someone may experimentally prove extra dimensions to exist at quantum realms. There is a long record of maths proceeding physics, after all.
The subtitles are scuffed, they're all at once in the first few seconds.
It works fine on my end, though...?
@@mathemaniac Idk man, might be a new firefox bug or something. Just checked it on chromium and that worked, but Cachy browser doesn't.
Can confirm, the entire transcript of the video flashes on screen at the beginning and then the subtitles are gone
Edit: Turns out UA-cam auto-translated the subtitles from UK English to just "English" and introduced the problem, the original subtitles are fine.
i dislike rotation matrices; have to be one of my least favourite ways of rotating vectors
(exponentiated bivectors are far easier to work with, and they reduce confusion about spinors too, but the human world seems caged by convention)
hi
In 1973 I wish I say that video.
All the people liking the long explanations.. okay, great. But I came here because the title is "How to rotate in higher dimensions". I'm a programmer; to be honest, I dont really care about why, I just want the HOW. Like in the title of the video.
There was so much matrix math in here, I couldnt understand it. Is there any point in this video that actually says,"If you have an n dimensional vector, and you want to rotate it, apply this formula" ?
I do want to watch your animations, I have done some stuff in Python, lots of libraries!
Anyways, so how does one rotate the famously infamous quantum number s for spin, since it's a 4pi rotation 😆 one cycle gets it from up to down or viceversa 🤔 though up and down are terrible names because those wibbly wobbly stuffs are ORTHOGONAL, up and djsn🤔 😬 up and...lateral.. 🤔 😬 🤣 🤓
1st comment (excluding the OP)