NoahExplainsPhysics - 2021-08-23
ANSWERS TO FREQUENTLY ASKED QUESTIONS: https://scholar.harvard.edu/files/noahmiller/files/dirac_belt_trick_faq.pdf This is my submission to 3Blue1Brown's "Summer of Math Exposition 1" #SoME1. In this video, I explain what Dirac's famous belt trick has to do with the topology of rotating spin 1/2 particles, such as electrons. I created the 3D animations using Three.js/CCapture.js, and the math animations with Manim Community v0.8.0. 00:00 Introduction 4:14 The space of rotations 9:40 Paths through the space of rotations 18:48 Group theory & the fundamental group 31:30 Quantum spin and SU(2) 39:31 SU(2) as the double cover of SO(3) 48:26 Bringing it all together 52:22 Tying up loose ends Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u Largo From Concerto No 5 Exzel Music Publishing (freemusicpublicdomain.com) Licensed under Creative Commons: By Attribution 3.0 http://creativecommons.org/licenses/by/3.0/
A word of advice: if you discover a new phenomenon and immediately reach for a matrix representation to describe it, you should call your doctor and ask if the automorphism group of a vector space is right for you. Clifford Algebras provide a safe, sanitary, and intuitive alternative, and is recommended by nine out of ten dentists.
I was going to mention this! Especially in the beginning when he rotated about an axis rather than within a plane. Can we all please rotate in planes from now on? :)
This is pure genius.
@Nicolai Haas Giedraitis jesus it was disgusting to see
@Nicolai Haas Giedraitis hey, I know you wrote this comment 7 months ago, but I'm curious if you can tell me the timestamp of the part you're talking about. Because I'm curious about what you mean by "rotating in a plane".
Edit: I just looked it up and I found that specifying the plane of rotation is useful in higher dimensions (>4) and of course, you can't rotate about an axis in 4D or higher. But in 3D, specifying the plane of rotation is equivalent to specifying an axis of rotation, which is just a vector in the 1D subspace orthogonal to the plane. So I'm not sure what the problem is with talking about an axis of rotation for 3D. Unless I misunderstood your comment (still curious about the timestamp)
Too much work, can I hire someone to understand this?
This is a masterpiece. Thank you for making it. Please do more of it. Animations about the bloch sphere ans the Pauli matrices would be highly appreciated.
This is absolutely astonishing. Please keep making more mathematics/physics content like this. I have never seen these concepts explained so darn well! <3
Totally agree
This video deserves to be seen over and over by anyone interested in the mathematical insight of spin. You are the first person to ever convey to me the right intuition for the Dirac belt trick. Keep up the great work, mate!
When I first saw the Dirac belt trick I thought it was flippant and didn’t explain anything. I still think so, but your video explanation was beautiful 😍 thanks for making this!
There's a video around showing a spin half particle has having an infinitude of belts coming from it to connect it to all points of the universe, making a continuous fibrous field instead of a scalar field. It's a nice visualisation about how the state of each point of the field includes the half-spin properties. I don't know if the electron is supposed to be spinning all the time and thus emitting ripples in that field, being electromagnetic waves including circular polarisation inline with the electron's spin vector.
@Tristan Wibberley do you remember the title of the video? Sounds interesting
@DasItMane "electrons DO NOT spin" by PBS Spacetime. It didn't go into a lot of detail but it was interesting to see properties of their visualisation
@DasItMane it might have been "how electrons make matter possible" on the same channel
This is the best covering of this subject I’ve ever seen. Most of this material I’ve seen scattered across various courses like introductory topology (I had flashbacks when you put Hatcher on the screen), differential topology, non-relativistic quantum mechanics, or field theory, but no one’s ever put it all together like this with incredible visuals. Rotating an electron in the Black Lodge was just the cherry on top! I’m grateful for SoME 1 for putting this on my radar and I truly hope you do something like this again.
I felt very heavy vibes of the "turning a circle inside out" timeless video with the narration and imagery, especially at the first couple chapters.
Amazing work! As a layman i half half understood it, which is a gigantic feat!
not knot
that's a sharp corner
The color scheme of goldish-yellow and purple for the 2 sheets of the double cover seems like a direct reference to that video.
I can't express how absolutely taken away I am by this video! Fantastic animation, amazing narrative! I had that feeling of awesome math discovery throughout the whole video, thank you so much for putting in an immense amount of effort and love into this video!
I think this may be the single greatest video on physics I have ever watched.
I have no words to explain how good is this.
I mean, I am reading The Road to Reality, specifically chapter 15 which is dealing with these matters. My background is telecommunications, so group theory is a bit alien to me. How helpful is this, I think Sir Roger Penrose would be utterly pleased by this video. I am sooooo curious what he would say.
Thanks a quintillion!
Noah , this was truly excellent. Thank you for making it and doing the faq and video indexing too
This brought together in a single presentation many of the concepts I’ve encountered over the years creating a type of map or perhaps a trail of breadcrumbs to be followed. I still don’t understand the nature of electron spin, but you’ve provided a wonderful foundation for appreciating the mathematics not usually discussed when looking at Dirac’s solution of relativistic wave equation. ( Dirac , Principles of QM, 3rd Ed Chap XI ). As with any good map, to appreciate one must make the journey. I’m sure I’ll be looking at this video many times as I do that. Best EC D. ps : I see you have another video on this topic. Thanks in advance.
Wonderful video. Your animations and script are very methodical without being boring. Your video reminds me that, as David Hilbert once said, 'A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street'. Well done.
Awesome video! For section 2, it might help make clearer what you're saying/doing if you point out that you can translate and scale the belt any way you like. Demonstrate that, and it becomes clearer why a single twist can't shrink to the origin: because it cuts through the edge of the sphere one time, and can't "undo" that. A two-twist belt cuts through the edge twice, forming a loop. Distort that loop as you do in the video, to show it as a loop that cuts the edge in two places, then translate to remove the cuts, then shrink to the origin.
I've been thinking about this for years after reading Penrose book "Road to Reality" - is such a great entrance point to the higher math!
Thanks for summing up a lot of points! Would be great to see part2 with Quaternions and Clifford Algebras, hah
I cannot begin to give you enough compliments. No only did you explain the belt trick you also explained the basics of Algebraic Topology without using any arcane terminology. I went to the Fundamental Group page on Wikipedia after this and understood most of it and understand what homotopy is. Just like that. Keep it up. You write book = I buy book. You go to Patreon = I will subscribe and sponsor.
Me tooooooo!
Wow! That was so astonishingly beautiful... the kind of quality I have come to expect from 3Blue1Brown... While I have an MSc in Computing Science, I was actually pretty good at math and physics as an undergrad, and continue to try to better understand quantum physics. Spin is so hard to wrap my head around (pun intended), but this really gave me such a good feel for what might be going on, a glimpse in the nature of quantum mechanics. By the end of your video, I could really appreciate how particles have angular momentum, and why fermions are so special. Thank you so much for opening my eyes...
If you try to understand the spin of an electron by looking at it from all angles, you won't get it ;).
When I first watched this video, I remember being very confused. After reading an introduction on Lie Groups, being reminded of this video and rewatching it, I get it now. Amazing quality.
I love how even though this is complex mathematics it ensures you know the basics like radians and degrees lol
This might be the best physics lecture I've ever seen. Could you do one on entanglement? Specifically, I've yet to hear or see a convincing explanation of why the spin of entangled particle was NOT set at their creation long before measurement is taken.
Wow just wow. I am bit older and my math degree is from the 70's. Damn I wish we had these beautiful visualization back then. I did a little bit with knot theory and would love to see this covered in a video.
Thank you for making this! I was trying to wrap my head around the whole so3 and su2 thing and was just searching youtube for any visualization. Didnt expect to find something so high quality!
Very good video. Clear and motivacional. It is not an easy topic to explain for those with no basics on algebraic topology, but quite illustrating. Congratulations.
Thank you. I'm a senior maths student and just learned about group theory and have always been confused when I heard SU(2) and SO(3), thank you for this intuitive explanation!!
Great work. A worthy description of the content of the video can also be seen in John Baez's "Gauge, Knots and Gravity", or in the more brief lecture notes on spin.
This was really well paced, I had several moment of "oh that must mean ..." followed by the next section confirming it. Not had that experience in a while so it was an enjoyable journey.
This is the most fantastic video I've ever seen on youtube. I mean fantastic in the positive sense. It is absolutely mind-blowing. But the greatest miracle of all is that: It is understandable. Even someone like me, who struggles alone for years with these concepts, could follow everything in it.
You are an outstanding teacher! Part 2 is by far the best visual illustration of the 4 pi concept. I know that required a large amount of work on your part and your viewers thank you. If I understand correctly, there in part 6 there is a need also to introduce the sphere S^3 as a disjoint sum of upper and lower disks D^3 mod equivalence of common boundary; S^2 = shared boundary of upper and lower D^3 disks.
This is an absolutely brilliant video! I am so glad to find this channel thanks to the SoME1.
I never, ever encountered explanations of this quality!! This is absolutely exceptional! You should try to make a business from this supporting theoretical physics courses in universities!
I stunned... I know some of this but I have never seen it that way... I cannot believe what I see... shockingly good explanation...
Steady on.
Totally agree - my sentiments exactly
This has such a 3B1B feeling to it... NICE WORK!
Big complement there. Agreed
Grant did the audio for the vid clearly.
You belong to the Group of "Great Explainers".Thank you very much for a very clear explanation of a rather abstract concept.The best I have seen sofar.
Great presentation style. A bit mathematically heavy but very well done. I studied Physics but sadly never got into group theory - this is a nice summary. Thank you!
Your video is of outstanding quality. Maybe a bit advanced for a general audience, making it hard to appreciate if you are not a physicist like myself. Keep making videos like this !
This video was unbelievable, not only in the mathematics but also the pacing and narrative. Despite how abstract these concepts are, I was just able to keep up with everything said!
OMG! Amazing! I've not studies maths or physics beyond high school and i understand a lot of this. Thank you for making the fundamental principles of the universe accessible to me. <3 :D (~22mins) ... oh wait. 34 mins in and im getting very lost but still, i got much further than usual.
Had to take several days to watch this due to time but, the realization how everything he explains relates to the belt and quantum mechanics around 45:00 felt like a hit of heroine. The satisfaction of this just completely washed over me
Without doubt, this is one of the best (if not THE best) video on this topic that I've ever seen. A big thank you!
Absolutely fantastic video! Genius presentation - thanks so much for putting this together!
Thanks Bro. For the past 3 months I was struggling to understand what quantum spin is in terms of Topology and Group Theory. Thanks for connecting the dots with a clear explanation. Great teaching.
I was concerned when he said “for professionals” and he still explained everything he did beautifully
Fan-ta-stic! Thanks a lot for this amazing video. As a quantum mechanics teacher, I will strongly recommend it to my students and... my collegues too! This a really great job. Many thanks again :-)
What a beautiful video. I've know this fact for a few years now (proving pi_1(SO(n)) = C_2 for n>2), but never really felt I understood it because I never took the time to learn to visualize it. But you show the moduli space right there in front of you as a ball with the boundaries identified like RP2.
Great stuff!
Could you do a video on the Hopf fibration and maybe even the Bohm-Aharonov effect? I think you could make a great visualization.
This is an amazingly well produced and didactically superbly laid out video! Kudos!
I loved this video! The correspondence between unitary matrices and rotations was easily the part of my quantum computing class that I struggled with the most
Great video! The first three chapters alone already make it interesting and self-contained, so I'll definitely direct my students (and other people of good wil) here.
By the way, I guess that there's one minor hiccup with the definition of a group at 20:38. The standard definition involves a two-sided identity and a two-sided inverse. As far as I know, your definition (with left-identity and left-inverse) is equivalent, but it's a difficult (albeit googlable) exercise to verify that.
Also, it might be nice to see the proof 46:20 animated. The "positive formulation" of it - that homotopy preserves the end of the lifted curve - should be more manageable. But I appreciate the pun, so please don't think I'm complaining.
Thanks for the content!
I am always fascinated by physics experiments and studies, but the notation system and the formulas engage me about as much as music notation.
Having said that, I believe this is valuable information, and I’m wondering if you ever considered making available some example models in Blender scenes.
In this way, with limits pre-defined by nodes, one could experiment with the rotations, and make additional observations by getting creative with some other principles of physics.
Just a thought . . .
Thank you for explaining the basics and the notation, I really needed that.
Thank you very much for this video! It was great, and it made me want to do algebraic topology. I had an introduction on it where I didn't have all the prerequisites, so I didn't understand much, but I found it very beautiful nonetheless. Yours was much easier to follow without tons of mathematical background.
This was a brilliant video. I was completely lost with all the matrix calculations, but the explanation with the covering groups made perfect sense
wcpgw - 2021-10-11
Fun fact: All USB ports before USB-C have spin 1/2
Eivind Triel - 2021-10-18
I concur. First you try and it dos not fit. Then you rotate Pi and it still dos not fit. Then you rotate it Pi again and now it fits.
Alice Wyan - 2021-10-19
Wait, it'd be spin 2/3, right? cause you need 3 180º turns to arrive at the initial position
DatGuy - 2021-10-24
Also when you don't observe it it goes into superposition, and when observed it always collapses into the state where you need to rotate twice
Stas - 2021-10-28
USB Plugs were spin 2/3. PBS Space Time did this bit: https://youtu.be/dw1sekg6SUY?t=1032
Saturn - 2021-10-31
Yes definitely, although I got a USB a to micro port that is spin one on the a side like usb-c