The Mystery of Spinors
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- čas přidán 9. 05. 2024
- In this video, we explore the mystery of spinors! What are these strange, surreal mathematical things? And what role do they play in physical reality? We'll talk about the algebra of SO(3) and SU(2), and the profound physical implications of spinors, particularly as it relates to spin-statistics and the stability of matter!
Video notes PDFs available for download on Patreon:
/ richardbehiel
All support is highly motivating and greatly appreciated! :)
Recommended reading: "An introduction to spinors" by Andrew M. Steane: arxiv.org/abs/1312.3824
For a more advanced and comprehensive treatment of spinors, see "Spinors and Space-Time" by Penrose. The homotopy class animations in SO(3) were based on Section 1.5 of that book.
To learn more about the Spin-Statistics Theorem, see "Pauli and the Spin-Statistics Theorem", by Ian Duck and E. C. G. Sudarshan.
Also, check out the wonderful CZcams series "Spinors for Beginners" by EigenChris! / @eigenchris
Chapters:
0:00 Intro
3:08 Topology Warmup
9:22 Axis-Angle Representation of 3D Rotations
13:15 Homotopy Classes of Loops in the Axis-Angle Space
22:50 The Algebra of Rotations, SO(N)
33:48 SU(2)
39:35 SU(2) Double Covers SO(3)
49:15 Exploring the Mystery
1:01:20 Superconductivity
1:05:00 Let's get Existential
1:07:50 Conclusion
#math #physiccs #quantum #quantumphysics #spinors
Been patiently waiting for this one. Welcome back Richard. You made up your absence by a literal 70 minute giant, I'm happy.
Glad to hear that! :) Yeah sorry, I would have posted sooner but it took forever to make 😅
I know how hard it is to work with manim, assuming that’s at least partly what you used. And to explain one of the hardest topics ever with it? To the public? With a 70-minute video? ~3 months is honestly an impressive duration. I am sure it’s a great video too, I haven’t finished it yet but that’s my first thing to do tomorrow. Thank you and keep up the good work 😁
@@RichBehielbakers gotta bake
Richard is great, what a beautiful representation of spinors.
@@RichBehiel Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality.
Bosons are dual to Fermions -- atomic duality.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Inclusion is dual to exclusion -- the Pauli exclusion principle is dual.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Antipodal points identify for the rotation group S0(3) -- Dual perspectives!
Points are dual to lines -- the principle of duality in geometry.
"Always two there are" -- Yoda.
Duality creates reality!
Spinors are mobius loops.
The Klein bottle is composed of two mobius loops -- self intersection.
The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
anyone who's ever tried to plug in a USB cable/stick intuitively knows about having to rotate an object more than 360º for a full rotation
Ahh yes, the superposition of USB 😂
lol, brilliiant, yes! almost always only on the third rotation
I was told USB cables were 4 Dimensional objects... but perhaps they're spinors too?😉
@@markzambelli why_not_both.jpg
I wondered from the first one I ever saw, “how could you start with a DB-9, and get to THIS, and call it better?” Guess they didn’t employ any mechanical engineers…
This to me is another stellar example of why youtube may be one of the greatest libraries of human knowledge ever collected in a single place. Congrats on offering up your tome, and for supporting the spread of quality information.
Like in the Time Machine H.G wells🎉🎉🎉🎉🎉🎉🎉🎉🎉
The internet CAN Be used for good.
You tube videos will disappear one day. Information on the Internet is ephemeral.
That too available at all levels of understanding
Hehe if ya love CZcams so much maybe you should marrrry it then 😊
I recently got a PhD in atomic physics and found this extremely enlightening on concepts that I took for granted all this time. I wish videos like this existed when I was in grad school.
Indians need not apply.
Lets go jackson we love to hear it keep up the good work
"A wiggle is homotopic to an octopus" beats "a donut is topologically a coffee cup" six ways to Sunday. Excellent presentation!
My favorite is a human being topologically a donut
@@ExpandDong420 "7 holes" - VSauce
A human being is actually topologically equally to a Honey Comb cereal@@ExpandDong420
I'm sorry you had a poor education.
@@ExpandDong420 No. It is things you can squeeze together to a ball, and things you cannot squeeze together to a ball. Anything that has a hole in it, cannot be squeezed into a perfect ball, the hole will still be there, even if it is smaller, a rift or whatnot.
And so comes the conclusion, there are two shapes, and every item is one or the other. And so the question: Is the universe one or the other, which shape is the universe.
In formality, a priori, knowing nothing about the universe, it's a coinflip. It could just as well be one as the other. The chances the universe is like a teacup (with a handle, that makes a hole in the shape, so you cannot squeeze it into a perfect ball), is 50%, and the chances that it is like the coin that you flip, that can be squeezed into a ball, is also 50%.
The shape with the handle, is what gives birth to the idea of a wormhole. Which we can then say, has a 50% chance to be a possibility, in our universe, a priori, knowing nothing about the universe.
It really goes to show, how farfecthed and thinly veiled in philosophy, the idea of a wormhole is, given that it has become a thing in the imagination, put beside established fact in the minds of man. If the universe has a handle like a teacup, can you then travel along that handle, to show up somewhere completely different than where you started?
This is SO. MUCH. WORK. How did you get this video out the door? My god the animations!! Here's hoping that if there is a day job in your life, it is paying really well. This is waaaaaaaaaay more valuable education than I can get from paid sites.
Thanks! Yeah it was a lot of work 😅 But the ratio of time spent working on it, to the time that people will spend watching it, is a really good deal! It’s a great way to put positive vibes out into the world.
At least one of the animations is stolen from pbs spacetime, the complex 3d one
@Shplinkinshploinkin good point! That animation was by Jason Hise. He published it copyright free, so anyone can use it. I put his name in the thumbnail and later on in the video when that animation comes up. Credit where credit is due! :) I rarely borrow from others, but his animation was too beautiful not to include. I did all the other animations though.
@@RichBehiel Is everything in your video made in Manim? If so that's seriously impressive
@123string4 Matplotlib actually :)
Wow. This is what youtube should have been. Not youtube shorts that rot my brain chaining me to scroll endlessly for miniscule amounts of dopamine and serotonin. Thank you. Honestly. Thank you.
Thanks Drake, that means a lot! :)
Your are absolutely welcome sir. You have damn well earned it. 👏 I will be checking out your other content feverishly. Haha
You can get it to stop showing you shorts all the time if you tap the three dots on all the ones that show up on your homepage and tap "not interested". It'll eventually start recommending them again, but less and less often the more you tell it "not interested" in every short it puts on your homepage.
"If you get this concept about the two homotopy classes, if you really feel it, then instinctively you'll suspect that maybe there might be some mathematical object that is sensitive to the homotopy class of rotations... you'll yearn for it"
I can tell you've done an incredible job of setting up the intuition for this subject because that was exactly what I was thinking by this time in the video.
Thank you my friend. I just took the message you sent me. We are going to Evolution
I don’t normally comment on posts but this deserves a bump in the algorithm. Well done.
Thanks! :)
I was just looking to learn more about quaternion and now I have some existential crisis over the fact that their "square root" hold the universe together by preventing some atomic collapse. Great job
I'm sorry you had a poor education.
I...I think I'm in the wrong lecture.
All are welcome here! Embrace the mystery of spinors! :)
Nah dude. Just listen and apply the first principles to everything you can.
The knowledge will be self revealing.
...backs out slowly, apologetically
Oppenheimer Reference
HAHAHAHA I should be studying Economics ye
Am i day-dreaming. This is so ridiculously good. You are a grandmaster educator. Thank you.
Thanks, I’m glad you enjoyed it! :)
I know this was for a primarily physics audience, but I have never had SO(3), SU(2), quaternions, and spinors explained to me so clearly in any video ever. As someone from more of a programming background with interest in rotations and vectors from an algorithmic perspective, I've vaguely known about quaternions and matrices and their relation to rotation. But never have I ever had these objects explained in a way that I well and truly understood in a way that I could explain to others. I still am not 100% on the link between quaternions and spinors since you kind of glossed over it here, but I feel like I've definitely taken a major step in being able to get it.
The mathematicians out there should learn that rigor is not explanation! I've seen videos that rigorously explain what spinors are, precisely, and I kind of got it. But I never made the connections on how all the parts really fit together until this video. So thank you! For me, it's all about understanding the motivations and framing the concepts in a way that you "discover" them on your own. That's how you build true understanding. You did an amazing job of that here.
3blue1brown has been real quiet since this dropped. The animations are astonishing and I'm impressed by how efficiently and continuously you were able to explain this difficult idea. Good job
Hes going to become the 3blue1brown of physics
Lmao is it a competition
What is 3blue1brown?
@@nicolasolton hes a youtuber who explains math consepts with great visualizations
Again, the most pellucid explanation on the topic you cover. Last time complex numbers, and the Dirac equation. This time, spinors. Bravo, Richard. Bravo.
Thanks Curt, that means a lot! :)
Hey what u doing here, curt? 😅
I've learnt a new word, "pellucid" : ) nice
This was perhaps the best CZcams video I've ever watched. Thank you so much for creating this. CZcams is brimming with vapid, GPT4-summary-level, pop-sci, padded word count content. Finding a creator with a genuinely deeply subject mastery, who also passionately shares their insight with the rest of us, makes my day. People like you, Karpathy, Innuendo Studios, Technology Connections, Sasha Rush, 3Blue1Brown, This Old Tony, etc. make CZcams a platform worth visiting.
I can't presently afford to support you on Patreon, but hopefully the like/subscribe/watch time from a CZcams Premium user will help. You earned at least one new fan today. : )
Amazing work! This is such a great service to the physics community to see this discussed so lucidly and with a friendly tone.
Thanks Thomas, that means a lot! :)
Not just the physics community. This is awesome 🤩
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality.
Bosons are dual to Fermions -- atomic duality.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Inclusion is dual to exclusion -- the Pauli exclusion principle is dual.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Antipodal points identify for the rotation group S0(3) -- Dual perspectives!
Points are dual to lines -- the principle of duality in geometry.
"Always two there are" -- Yoda.
Duality creates reality!
Spinors are mobius loops.
The Klein bottle is composed of two mobius loops -- self intersection.
The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
@@hyperduality2838 Thank you 🙏🏻 for the comment. It was stimulating. I liked the parts about spinors, Mobius loops, and Klein bottles from topology. I have thought about these things as definitely related, I didn’t think anyone else had. 😂🤷♀️
Another tip of the hat to your brilliant pedagogy. Without you even trying the first few minutes of your video will surely be the ah ha moment for those struggling to grasp simplicial complexes, vietoris rips and persistent homology. It’s better than anything I’ve seen on CZcams actually trying to “explain” the subject! Let alone the rest of the video. Just WOW
Thanks for the kind comment, and I’m glad you enjoyed the video! :)
"I want to show you a subtle thing that will open up a crack in reality, which we can use to smuggle this into our imagination"
The rawest line that has ever been dropped since at LEAST 1687
BUT BEFORE THAT LETS DO 30 MINUTES ON VOCAB WITH NO REFERENCE TO REALITY AND U NEED TO REMEMBER IT
You have just given me the extremely rare and thus appreciated sensation of: "Yeah, I know where this is going, I have dealt with this befo... OOOoooooh, that's a neat way of looking at it!"
I’m very glad to hear that! :)
I've been wanting to understand the spin-statistics theorem, but all the explanations I've come across look very complicated. One of those cases where a simple theorem doesn't have a simple proof, I guess. I might check out that book you recommended.
Love your videos!
Hey I know you, you made that spinors for beginners video 🎉
Hey! :) Yeah, proving spin-stat turns out to be super complicated, it’s a rabbit hole but definitely worth diving into. That spin-stat book was what convinced me that spinors are actually genuinely mysterious.
Btw I should have given you a shout-out in this video. Your spinors series is awesome, and very helpful. I’ll be sure to mention your series in the next video!
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality.
Bosons are dual to Fermions -- atomic duality.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Inclusion is dual to exclusion -- the Pauli exclusion principle is dual.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Antipodal points identify for the rotation group S0(3) -- Dual perspectives!
Points are dual to lines -- the principle of duality in geometry.
"Always two there are" -- Yoda.
Duality creates reality!
Spinors are mobius loops.
The Klein bottle is composed of two mobius loops -- self intersection.
The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Not just the belt/plate trick.
Respect.
The "belt/plate trick" isn't a trick. It's a fair intuitive/visual account of spinors. A spinor isn't a rotation of a free object. It's a rotation of a tethered object, as illustrated in the CZcams. An electron seemscto be tethered by its attached electromagnetic field lines.
@@christophergame7977 it's a possible account... but are electrons really "tethered" to electric field lines?? What does that even mean? At rock bottom, that "explanation" is simply a heuristic.
The Dirac equation for a free field contains no reference to the electric or magnetic potentials. So, for me, that is an insufficient explanation for the physical nature of a spinor and why they necessarily appear in the mathematics of fermions.
@@jmcsquared18 Thank you for your valuable response. You may be right about electrons. But they are just a conceivable physical example of my point about spinors, considered as geometrical objects. There is a geometrical difference between a free rotor and a tethered rotor. Is it fair to say that this is a fair intuitive geometrical picture for spinors?
A spinor is a geometrical object, not a physical object. It may or may not give a nice account of some physical object.
@@christophergame7977 true, spinors exist outside of physics. But physical intuition could provide some help because, again, we're looking for intuitive guidance which is difficult to come by with spinors.
For instance, we don't always say that electrons are just quantum particles with certain charges or spins; we also say that electrons (or any fixed-mass elementary particles) are irreducible unitary representations of the Poincaré group. The mathematical structure of Yang-Mills QFT's gives intuition for quantum particles, and vice versa.
One reason we have so much trouble with spinors is that, historically, we found the quantum theory first, rather than e.g., with electrodynamics in which we had a rich history of the vector potential theory (Maxwell) long before it was quantized. Some are researching the foundations classical spin-1/2 field theory in hopes to gain insight into the nature of spinors.
Understood almost nothing of the physics parts but I'm inspired, my brain got spun for a good reason. Appreciate your Hard Work!!! 😻
I don't know enough to comment on the quality of the content, but this is amazingly presented and VERY interesting. Thank you for the obviously large amount of effort you put into conveying this abstract necessity.
Thanks, I’m glad you enjoyed the video, and I appreciate your kind comment! :)
50:21 Damnn, bro pulled this out of nowhere, that's epic lol
I get it! I finally get it! All the talk of a belt, or the Balinese candle dance, or a little cube inside a big cube and connected by rubber bands, I could never visualize. But I can see that a wiggle is the same as an octopus!
You got it! Once you see that wiggle = octopus, everything else clicks into place.
I cried from the happiness that your lecture gave me. Thank you sir.
Here is some intuition on the two rotation types hopefully:
Grab some object ( not needed, you can use an empty hand, but an object makes visualisation way easier )
Without re-gripping the object, If you make a class II rotation, the object will end up in the same orientation ( by definition ), and your hand can also end up in it's starting orientation.
If you make a class I rotation, the object will end up in the same orientation, but your hand will end up twisted, and the only way to fix that is to make a second class I rotation ( or re-grip the object ).
In all cases, you can just make the exact same class I rotation again.
( may not be obvious at first how to do that though )
The two class I rotations together form a class II rotation, which means that your can end up how it started
If you do that a few times with different rotations, there is a 90% chance that you can now intuitively differentiate which rotations are class I and which are class II just by looking at them.
You have also just demonstrated having to turn an object ( your hand ) around twice for it to end up in its original state. This happens because it is connected ( with specific constraints ) to an object which itself cannot rotate.
Ok I’m playin with this but it doesn’t make total sense. It seems like class II rotations I have to move my entire body around the object, like walk in a circle, whereas class I rotations my body stays still and my arm rotates (rather uncomfortably) and then I have to regrip to get my arm and hand back to its original configuration. Is this correct?
@@floydjaggy translations (moving the object) don't matter for this
a well written and animated 1 hour video essay on abstract mathamatical objects and their relaiton to computer graphics and quantum mechanics? i subscribed within about 3 minutes.
I’m glad you enjoyed the video, and thanks for subscribing! :)
I finished my master's in physics eng. about a year ago and sadly I don't think I will come back to academia after that. But this video is so inspiring it makes me want to be a student again
21:38 Class I on top is the best loading icon I've ever seen and I need it!
This is so dope. Like, easily one of the most impressive videos I've seen on youtube. It's elegantly concise, with relatively minimal assumptions about prior knowledge. Plus the whole thing's informed by an simply communicated, profound intuition for the originary principles of topology. Amazing way to frame an introduction to spinors. Not to mention so many amazing labor intensive animations. Awesome aesthetic intuition throughout.
Rare to see so much work go into a video that truly privileges' a generous pedagogy. Been a huge help for me today in creating tangible mental images as I pursue self-education in an intimidating subject. Sure thousands of other folks will feel the same gratitude. To me, nothing's cooler than someone who works so hard to share their hard earned skills like this. Especially when it's given to everyone for free.
Thanks dude!
Regarding 44:52. Gimbal Lock is not an issue inherent to SO(3), and using SU(2) does not avoid it. That's a common misunderstanding that gets repeated a lot. Gimbal Lock can happen when you naively attempt to compose a rotation using Euler Angles. It's an issue with the *mapping* from Euler Angles to the space of rotations. It doesn't matter if you use SU(2) or SO(3); if you try to compose a rotation from Euler Angles, you can run into Gimbal Lock.
For real?? Bah, for people like me who have very limited visual imagination these things can be so unintuitive!
The only upside of Euler angles is that they are invertable in the 18th C century.
Interesting! Thanks for the correction!
I’m confused though. Won’t any representation of SO(3) have poles, which would present a problem for a 3-gyroscope system? Whether using Euler angles or axis-angle vector.
Or are you saying that, suppose we had a four-gyroscope system, we could still use SO(3)? I see how we could do that in a roundabout way through SU(2), but I’m struggling to see how to do it directly in SO(3) without the poles being a problem.
@@RichBehiel Tait Bryan angles, AKA "roll pitch yaw" used in aircraft attitude. Since I've written tons of code for rotations in quantum mechanics, remote sensing and mars landings...I've used many reps (and spacecraft GNC ppl use "quaternions" and if you said SU(2) repression they would have no clue what you're talking about).
Anyway, any rep that has 3 different axes shouldn't be degenerate, and requires a computer to invert. I am convinced the only reason Euler (an absolute genius, ofc) used repeated axises, thus allowing degeneracy, is that his iPhone had a lousy chip and he wanted to be able to invert rotations, which for Euler angles is (alpha, beta, gamma) --> (-gamma, -beta, -alpha)..no chip required.
In a related note, doing geodesy, I also learned we have the first, second, and even 3rd eccentricity, flattening, etc..is because early cartographers couldn't do definite elliptic integrals with parchment and an inked up feather.
@@RichBehielI would see the Stack Overflow post titled "Quaternion reaching gimbal lock". The first and second answers are very elucidating.
The way a rotation matrix stores a rotation is not via axis angle or Euler angles but by storing the orthonormal basis that results from the rotation.
only seven minutes in, but I have to say, personifying the loop is a stroke of genius. No we cannot hurt the loop!
Thank you for mentioning the excellent "Spinors for Beginners" series by the "eigenchris" CZcams channel in the video description.
This is an unbelievably well-made video. The way you present this information makes it a lot easier to somewhat grasp insanely bizarre concepts like this and I really wish more people put this much effort into these kinds of videos.
Have you explored the interpretations and formalisms for spinors within Geometric Algebra? It's pretty cool, especially the stuff in Projective Geometric Algebra!
Clifford/geometric algebras are some of the most pleasing things I've ever set my eyes on in mathematics. It's like learning complex numbers all over again, except you can do perform operations in high dimensional spaces or space(times) with nonpositive signatures. It's absolutely stunning.
This was my first intro to GA: citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=97f522fa52db9f8ac08bce28f38179e88a67524b
In the context of regular 3D rotations, his treatment utterly demystifies the whole topic (Quaternions represent a pair of reflections). And the reinterpretation of the pauli spin matrices is fascinating, but I'm not 100% sure I fully understood it or that it was ever fully ironed out. It raises serious doubts that the usual approach is the best way though.
I love the cohesiveness of GA, it's too bad that interest was pretty low until computer graphics and robotics came around.
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality.
Bosons are dual to Fermions -- atomic duality.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Inclusion is dual to exclusion -- the Pauli exclusion principle is dual.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Antipodal points identify for the rotation group S0(3) -- Dual perspectives!
Points are dual to lines -- the principle of duality in geometry.
"Always two there are" -- Yoda.
Duality creates reality!
Spinors are mobius loops.
The Klein bottle is composed of two mobius loops -- self intersection.
The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Check out the channel xylyxylyx . He has been going through a foundational spacetime algebra paper for the last dozen videos or so. He just got done with how STA represents spinors. It... makes more sense than this... Unless you actually want to do math with it by hand lol. SU(2) is mathematically _much_ simpler, but seeing spinors spelled out in a totally geometric form gives you a much better intuition than the flagpoles and scatter plots.
Maybe the most interesting youtube video I've ever watched.
As a layman I've always had a casual interest in these topics, this tied together so many things I was curious about but never quite grasped. Thank you for your work!
This is, hands-down, the best explanation of spinors I have seen anywhere. The flag trick alone made 3-D rotation much more intuitive and showed me aspects I hadn't appreciated compared with just moving dots around. The visual depiction of the double cover and what the spinors are doing in that space was much easier to follow and see how the two related. Turning a non-simply-connected ball into a simply-connected (hyper-)sphere was the really astonishing part for me and made it obvious why spinors are so valuable.
A+, keep up the good work!
I wasn’t expecting such a deep/philosophical dive at the end with the spin-statistics theorem. I left inspired after watching the whole video. Appreciate such masterpiece.
This is the first video I've seen of yours, and it's impressive. I have some background in this, but I was still re-thinking concepts with the help of you're exceptional visualizations and explanations. This is a much better format than just on the math side, and you've done a service to people here.
I feel honored to have this recommended before 10k views
Just a quick comment on how much I love your channel and really appreciate the amount of effort you must put in to make these high quality videos. I have been trying to understand spinors for awhile now and was so happy when I found your video on it. I can tell you I have watched this video 3 times now and feel like I understand spinors about 50% which is the most I have EVER understood them. Appreciate your time and efforts on all these excellent videos you put together.
Thanks Brian, that means a lot! :) Comments like yours make it all worthwhile.
This is a very friendly walkthrough establishing mental relations for a sound understanding of basic concepts. Kudos mate! 👍
You have that comment on never being able to illustrate quaternions as well as 3Blue1Brown, but wow this video looks great! It shows a lot, but draws your attention to whay you should look at, and is just beautiful and elegant!
Thanks, I appreciate that! :) Nobody does it better than Grant though 😅 But I’ll keep trying to improve!
I'm only 3 minutes in but i just wanted to comment, my god is this presentation style good, great voice, enthusiastic while still formal and informative, clear graphics that are visually appealing but (semi presumably) mathematically valuable. I can just tell this is going straight into my favourites and my video essays playlist when I'm done!
It makes me feel like having been this 5 years old kid on xmas eve. Opening the next gift (video) gives an ever bigger grin in my face. Thank you!
I was thinking of how does something move from one Planck length to the next because of the discreet nature of it. Your presentation on Spinors here gives me some food for thought! I'm super amateur, but I love thinking about this stuff! I can't wait to finish this video after work and get to the part about the complex numbers. There may be a connection to a thought I was having about higher dimensions and branes. Thank you very much!
Loved the visuals, they were incredibly helpful. While they seem to be inherently mysterious, after watching this, spinors are no longer abstract. Thank you so much for taking the serious amounts of time and effort needed to make this video and then be so kind as to share it with us all. Bravo good sir! Bravo!
Excellent presentation! I have a phd in theoretical physics too. It was a long time ago and I am rusty, so I have reverted to be only a little past the denial stage :)
I love how you can consistently make some of the best physics videos out there.
You should do a video some time on why the w and z bosons have mass and/or the Higgs mechanism. I've seen it described before, but never in a way that's as intuitive yet rigorous as yours.
Engineer here, run for cover. It seems that the spaghetti animation is showing something interesting that I'm not sure has been made explicit here or elsewhere. The 720 rotation has two phases where the first phase has the spaghetti strand is going over the origin and the second its going under the origin, and that might visually explain the double cover. I've seen that type of animation a good number of times, but haven't noticed that connection until watching this video, and haven't yet heard it explained in that way. No doubt this may be obvious to those familiar and comfortable with the ideas. Thanks for the video, good stuff.
Guess I found a new channel to binge while I sew. This video was so fascinating.
If anyone tells me that the internet bill is too high, or CZcams premium is a waste of money.. I think they don't realize that beeing able to find, watch, rewatch, review, discuss and share this information is so so, SO valuable... Atleast to me. 🤷♂
CZcams premium gives more vids?
Seriously good video. I can comfortably say that you are the 3b1b of physics. As in I feel like your way of explaining things in a visual manner really tingles my brain in a way no professor ever could using a simple blackboard. Giving me for the first time a sensation that I not only know how to apply the math I learned but truly understand it intuitively. You have no Idea how thankful I am, really... Keep up the good work and keep taking your time with these masterpieces. You have no Idea of how helpful your videos have been.
this is an incredible video. i'm a computational chemist that regularly does quantum mechanical calculations but i've never had a chance to really peek under the hood of the code i run. this is a great starting point!
Great video, and not just because it’s a fun surprise that my animations made a guest appearance!
Definitely going to go back and watch your earlier videos next now that I’ve discovered this channel.
Thanks Jason, that means a lot coming from you! :) Your animations are beautiful. By the way, what software do you use?
I use C++ to write Maya plugins which extend the ways I can generate and deform the geometry. From within Maya I can then keyframe the parameters which drive those custom deformations and render the resulting animation out as a bunch of still frames. I compile the frames into final gif animations or mpeg videos using the ffmpeg command line utility.
The animation you showed here is actually pretty easy to recreate. All you need is a way to write a function which can map each given vertex Q in your original mesh to a new position P based on a time parameter T. Here’s the algorithm!
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First, compute the distance D of the vertex from the center of rotation C. This will determine how much of the rotation should be applied to the vertex. You can define an inner radius R0 within which the full rotation will be applied and an outer radius R1 where no rotation will be applied.
V = Q-C
D = clamp(mag(V), R0, R1)
K = (D-R1) / (R0-R1)
Next, compute which direction to rotate the core. Note that the amount the core is rotated is always 180 degrees, and what is actually being animated is not the core itself, but the axis of rotation B which is used to turn the core upside down.
θ = T * 2π
B = (cos(θ), 0, sin(θ))
Finally, just rotate by some percentage of 180 degrees about your spinning animated axis based on the radial distance to the vertex!
Φ = smoothstep(K) * π
P = C + V * exp(Φ * iB)
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In case you haven’t seen it, there’s an animation on my channel which hopefully shows how this works a bit more intuitively. There are a bunch of nested hemispheres with a rubber band stretched across them and a spindle sitting above them. The spindle rotates, the shells turn upside down around it, and then a snapshot is taken of the state of the rubber band. The shells are then rotated back before the spindle moves to its next position. The sequence of snapshots can then be used as a twisting animation of the rubber band.
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Truly underrated creator. You deserve to be up there with Numberphile, 3Blue1Brown, etc. Loved the video!
Thanks for the kind comment, and I’m glad you loved the video! Hopefully I can be up there with those guys someday. I’ve got some catching up to do, though! Working on the next video now :)
Excellent. I've been trying to wrap my brain around spinors for a while now, and this video is the best explanation I've seen yet.
Being recommended your channel via this video was great. I never knew how relatively straight-forward the things I learned in order to understand machine learning could be applied to learning physics.
Spinors are intriguing mathematical entities, defined within structures like vector spaces and Lie groups. These frameworks give them behaviors and characteristics that might not always directly correspond to what we experience in the physical world. Nevertheless, spinors are incredibly useful for making sense of real-life phenomena, despite their abstract nature. It's important to remember, though, that while they provide valuable insights, they're still just mathematical representations and shouldn't be confused with the concrete realities they describe.
Reads like ai wrote this
@@GetSwifty As English isn't my native language, I've put in some effort to express myself as fluently as possible. Apologies if my writing comes across as a bit robotic! ;-)
Is it a coincidence that the alternative that makes computation easiest, SU(2) rather than SO(3), also is kind of chosen by nature, chosen by reality?
As if an analogy to The path of least resistance shines through?
We have been waiting for your comeback anxiously. Thank you for sharing your knowledge
This is an insanely good introduction to a particularly challenging topic. I know very little about abstract algebra and nothing about topology and yet I was able to follow all the way through. Thanks for taking the time and effort to make this!
Thanks for watching, and I’m glad you got something out of the video! :)
Wow!! Just blow my mind! In the best of all aspects... Understanding ❤
I love the way this is simplified by the host!! The comedic relief is a blessing as well. And the mystery of the spinors is my favorite part!! I think spinors could easily explain the mental and astral planes in relation to the physical plane. But first I need people to understand these formulas. lol.
this video is great! Well done. It goes in deep and hard but feels approachable at the same time!
I'm proud of myself that I thought that this feels similar to quaternions in a way before you mentioned it ^^
This is a really wonderful video. Thank you for making subjects like this so approachable to visual learners.
Thanks, I’m glad you enjoyed the video! :)
This is what CZcams is for. ❤
I'm an undergrad student in physics and I just saw the Dirac equation for the first time in my lectures and was so awe struck. Seeing it for the first time, I just thought: "This is so incredible" even tho we just started with the very basics and I don't know the full scope of its implications. I immediately wanted to dive deeper and the alluding usage of the word "spinor" that wasn't explained in my lecture at all brought me here. I am so happy, youtube showed me this as a first result searching for spinors, your video is absolutely incredible. You have sparked a tremendous fascination for this subject in me, which I am very grateful for! Thank you for making this video :)
I saw a glimpse of this in my dream the other day. No idea what I was a looking at until I found this video! Thank you for all the information and visual representation!!
Extremely epic and legendary
Very relaxing after an exhausting day at work.
These visual representations that you have created for this video are incredibly good. I'm a computer scientist and not a physicist and yet I feel that you've made it easy for me to follow along with the concepts.
I feel like I have watched every video on the spinors out there and this one was the best for me. it was really approachable and explained very clearly. Great job.
Thanks, I’m glad you enjoyed the video! :)
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality.
Bosons are dual to Fermions -- atomic duality.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Inclusion is dual to exclusion -- the Pauli exclusion principle is dual.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Antipodal points identify for the rotation group S0(3) -- Dual perspectives!
Points are dual to lines -- the principle of duality in geometry.
"Always two there are" -- Yoda.
Duality creates reality!
Spinors are mobius loops.
The Klein bottle is composed of two mobius loops -- self intersection.
The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Superb pedagogy. Thanks for making this.
Thanks for watching! :)
thank you for your hard work, this is awesome. The average world IQ is going up because of these videos
Topological holes cannot be shrunk down to zero -- non null homotopic (duality).
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality.
Bosons are dual to Fermions -- atomic duality.
Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation.
Inclusion is dual to exclusion -- the Pauli exclusion principle is dual.
Syntax is dual to semantics -- languages or communication.
If mathematics is a language then it is dual.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Antipodal points identify for the rotation group S0(3) -- Dual perspectives!
Points are dual to lines -- the principle of duality in geometry.
"Always two there are" -- Yoda.
Duality creates reality!
Spinors are mobius loops.
The Klein bottle is composed of two mobius loops -- self intersection.
The left handed spinor is dual to the right handed spinor synthesizes the Klein bottle.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
so about the axis-angle rotation: the unit vector points to the direction of the _axis_ around which our physical object rotates, and the magnitude is the angle of rotation of our physical object?
I need to process the properties of our physical object, the 3D space it lives on, and the abstract space of rotations so sometimes the "thing that does the stuff" get mixed up
I was reluctant to watch the video due to its length.. thank god I changed my mind! it wasn't just a math video, it was an experience. [I barely understood anything after SO(N) chapter]
Axions!!! Do Axions!!!! This is so exhilarating- this channel is like 3b1b for advance physics. LOVE IT
When he said "a wiggle is homotopic to an octopus" he unlocked my memory of Henry the Octopus ........ from the Wiggles
This inspires me to step back in awe. Thank you, for pointing me in this direction, you are so right on!
What a beautifully done video. Your attention to visuals, details (right down to the time stamps at the beginning), your easy to digest script topped with your soothing voice, couldn’t be better 👌🏼 *chefs kiss*
Thanks, I’m glad you enjoyed the video! :)
this is a lot, your videos are changing the way I see reality itself
“Can’t hurt the loop”😂
❤❤❤
Outstanding video - subscribed! I was previously looking into whether there is any relationship between spinors and the holographic principle (e.g. SU(2) on the holographic boundary -> SO(3) within the bulk). This video really helped clarify what may (and may not) work with that line of thinking. Thank you
Went back and rewatched your hopf fibration video after this one. I can actually see the phase space of the spinor in the hopf now. Thankyou!
Thank you for the exquisite video. And thank you for your generosity.
very satisfying to watch
I'm a bit surprised to hear your conclusion about the mysteriousness of fermions. I would think that the mathematical consistency of fermions with the QFT framework is enough to predict them. For example, when building a QFT Lagrangian, after positing fields and symmetries, we expect every allowed term to appear. Similarly, since fermions are a mathematically consistent kind of field, we expect them to appear in the spectrum of particles in the universe. Indeed I think we would have to ask ourselves why they should be absent, if they are allowed.
But I'm surely missing something if Atiyah and others don't agree with this. I should read the book.
Incredible video btw! I think it's worth discussing how the double cover in this case is actually the universal cover, and how the universal cover of SO(3) is only isomorphic to SU(2) by coincidence and that this doesn't generalize to other dimensions.
Wow a new video about spinors! ok this time ima try understand them
Me too 😂😂😂
Bro, have I been waiting for this, I started almost rediscovering the stuff from scratch, thank you for your effort
You put so many fields together in one video. Awesome.
Very nicely put together!
Thanks!! :)
Wow, that was incredibly helpful. Very good video
Thanks, I’m glad you enjoyed it! :)
Great stuff. Tying together rotations in GNSS firmware and Cooper Pairs is incredible.
Such great videos. One after another. Complex topics simplified and visualized, along with the wonderful explanations.
A question though, would you mind describing the software stack you are using for those videos visualization and animation?
Great job explaining!
Thanks! :)
So all those nights watching that old how to turn a basketball inside out video finally paying off I see
That’s a great video!
The graphics alone are amazing . Such a lot of work but it does make for a fascinating video .