Dirac's belt trick, Topology, and Spin ½ particles

ANSWERS TO FREQUENTLY ASKED QUESTIONS:
scholar.harvard.edu/files/noa...
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:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
Largo From Concerto No 5
Exzel Music Publishing (freemusicpublicdomain.com)
Licensed under Creative Commons: By Attribution 3.0
creativecommons.org/licenses/b...

[@HerbertLandei]

Fun fact: All USB ports before USB-C have spin 1/2

[@eivindtriel6107]

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.

[@alicewyan]

Wait, it'd be spin 2/3, right? cause you need 3 180º turns to arrive at the initial position

[@datguy2271]

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

[@zbstof]

USB Plugs were spin 2/3. PBS Space Time did this bit: kzbin.info/www/bejne/mqiUpJihnJuGi7s

[@saturn9199]

Yes definitely, although I got a USB a to micro port that is spin one on the a side like usb-c

[@kiledamgaardasmussen5222]

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.

[@nicolaihaasgiedraitis4082]

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? :)

[@ThomasMeli81]

This is pure genius.

[@kashu7691]

@@nicolaihaasgiedraitis4082 jesus it was disgusting to see

[@abrarkazi2284]

@@nicolaihaasgiedraitis4082 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)

[@EvaTruve]

Too much work, can I hire someone to understand this?

[@LookingGlassUniverse]

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!

[@tricky778]

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.

[@das_it_mane]

@@tricky778 do you remember the title of the video? Sounds interesting

[@tricky778]

@@das_it_mane "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

[@tricky778]

@@das_it_mane it might have been "how electrons make matter possible" on the same channel

[@theopantamis9184]

@@das_it_mane kzbin.info/www/bejne/m4Oci3atjrWbnrc

[@cartlundmonson5164]

"One other thing I wanted to mention is the Hopf Fibration. There. I've mentioned it." My sides. Twin Peaks reference much appreciated too.

[@manamsetty2664]

I explained the belt trick to my class now everybody knows the colour of my underware

[@Samusamu57]

Oh my god the reference to TJ """"Henry"""" Yoshi just killed me. I love you guys

[@JohnSmith-kc6ov]

me too. I can't believe the dude that makes educational videos about the deep inner workings of mario 64 was referenced by the guy that makes educational videos about the deep... hey wait maybe this isn't as big of a crossover as i though. i'm sure there's a pretty big overlap in audience

[@daniellewilson8527]

What reference is that? Also what video was it from? Was this from the one who talked about parallel universes in SM64 via a TAS? I forgot who I watched that did that

[@Samusamu57]

@@daniellewilson8527 yes that one

[@tonyhakston536]

@@daniellewilson8527 the KZbinr you’re looking for is Pannenkoek2012

[@krenv2052]

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!

[@marcobrini]

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.

[@lewdwig]

I think this may be the single greatest video on physics I have ever watched.

[@maxkee9882]

TJ “Henry” Yoshi getting dunked on once again

[@Bruno-el1jl]

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!

[@official-obama]

not knot

[@midnattssol8329]

that's a sharp corner

[@Jesin00]

The color scheme of goldish-yellow and purple for the 2 sheets of the double cover seems like a direct reference to that video.

[@smiley_1000]

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.

[@AlericResident]

OMG thank you for making a youtube that does NOT shy away from formula's, not even from high level math! There are so many youtubes about *interesting* concepts that in the end explain NOTHING because they restrict themselves to what they think every viewer should be able to understand (aka, nothing). This is unfair to the minority of people who CAN understand these concepts in terms of math (if explained well) and in general a disservice to humanity. That being said, after learning that my spacial insight got out of the graph and even off the edge of the paper by the professional trying to measure it; I have spent countless hours trying to imagine 4D space, thinking it has to be just lack of experience that humans "can't" imagine it (and because I suspected that it might be a reason that humanity is stuck with its understanding of physics where we are stuck). Imho it is more insightful to simply imagine a projection from cover spaces to its base space: picture the surface of a sphere as two discs at the same place: one being the projection of the upper half and one being the projection of the lower half, keeping in mind that each disc also has a "distance" (either up or down) to their respective part of the cover space. Putting the discs next to eachother is less insightful (although easier to show in a video). Obviously one then can only move from one disc to the other where this distance is zero: at the edge of the discs. Likewise and 100% equivalent: two spheres in the same place, connected at the edge (surface of the projected spheres) where the extra "up" or "down" distance is zero. Each point inside the two-sphere is then actually two points, where the distance (from the projected point in 3D space) is trivial: sqrt(1 - (distance to the center)^2).

[@SuperMaDBrothers]

yeah I agree. Even this could have done with way more, there was 0 discussion on what a group cover actually is or a rigorous way that shows SU(2)~S3

[@riccardoorlando2262]

@@SuperMaDBrothers Well, if you want all the details, pick up an algebraic topology book and start reading. There's an optimal place to compromise, and this video pushed the boundary but earned the mathy bits with beautiful animations - without which, I'd much rather actually read a book.

[@idontwantahandlethough]

Clicked on your profile pic because I thought it was funny but now I regret it. I gotta say, it's sad to see someone who clearly thinks so highly of their own intelligence and has somehow still fallen prey to blatant xenophobic propaganda. You clearly have the intellectual ability to figure things out, but lack the emotional maturity to see the reality for what it is. I hope you grow up and figure things out someday.

[@thephilosopher7173]

I know this is old, but regarding the part about the viewers knowing nothing: Yea many of us want to understand a concept and don't know much math (or are starting to learn). The reason why you're a minority is because the way these things are taught is backwards. If all of the science in schools today were approached with pure historic reference and first principles methods, then plenty more ppl would be able to understand it, and maybe even more would actually become scientists. Shying away from the hardcore math isn't great in some cases but it isn't bad either.

[@samevans4834]

"Wait, a 360-degree rotation is a 360-degree rotation, you can't say it's only a half!" "Well, Albert """Henry""" Einstein..."

[@Zeus.2459]

One of my favorite references, glad someone else got it :D

[@LordHonkInc]

I can't believe a speedrunning meme got so ubiquitous as to show up in a video proof of a quantum-mechanical phenomenon. I love it

[@mauritz3912]

A man of culture!

[@viliml2763]

It's funny that this is the SECOND time I saw this reference in the #SoME1 playlist

[@mauritz3912]

@@viliml2763 @Owen Maitzen is also a man of culture!

[@oximas]

yeees another 3blue1brown style video, BRO we need more

[@DooDooDiaperShitCunt]

This is absolutely astonishing. Please keep making more mathematics/physics content like this. I have never seen these concepts explained so darn well!

[@simonavarne9635]

Totally agree

[@flmbray]

This has such a 3B1B feeling to it... NICE WORK!

[@tim40gabby25]

Big complement there. Agreed

[@voidisyinyangvoidisyinyang885]

Grant did the audio for the vid clearly.

[@__august__]

you convinced me to watch the entire video with the watch for rolling rocks reference

[@alexbanks9510]

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.

[@williamrhopkins]

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.

[@subhadeepreaditassubhodeep6161]

This gonna blow up. This SHOULD blow up.

[@edvogel56]

Thanks! You are illuminating the the first 300 or so pages of "math primer" in ""The Road To Reality" by Penrose.

[@JaxzanProditor]

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.

[@hydraslair4723]

The intuition that a belt is a path just flipped my world upside down... Only once, so it can never go back!

[@UdarRusskihPudgei]

Thanks for mentioning the Hopf fibration.

[@Osmomorfismo]

This is definetly the best video to explain SU(2) and spin at a fundamental level of all YT

[@natywubet2175]

i came here for party trick and forgot why I came here, true masterpiece.

[@shutosham]

The probability of finding such a video on youtube is 1 in a billion :-) amazing ..

[@AlphaCurveMath]

'A large book' *Serge Lang's 'Algebra' emerges*

[@zlm001]

Once again, one seen this explained in an over simplified way so many times that leaves so much out. I think this is simplified as much as it can be while still giving some useful insights to a non-expert. Thank you.

[@carlkuss]

I keep going back to this video. It shows true pedagogical skill, showing that when the point that you are making is deep enough it deserves tender loving care even if that might seem to those who don´t care it might seem tedious. Go for clarity!

[@ExecutorElassus]

Putting the electron in _that_ room was very chef's-kiss, as they say.

[@EricKolotyluk]

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...

[@AlericResident]

If you try to understand the spin of an electron by looking at it from all angles, you won't get it ;).

[@WorldWaterWars14]

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!!

[@shukius1]

One of the best movies I have ever seen on this complicated topic. Noah you are a true genius. Keep them coing

[@jordanweir7187]

IMO they should first introduce Topology with this sort of clearer material and call it 'Apology' thanks for content bruh

[@MrBebopbob]

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.

[@scepticalchymist]

The haircut in the beginning is a double cover of the one in the end :). Great video!

[@Jop_pop]

It's so cool how you've used manim creatively here, especially with the white theme instead of the dark one

[@Praefectia]

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!

[@filipo4114]

So let's mark the electron with a '+'... xD

[@brooksbryant2478]

was thinking the same thing

[@yourmom-nv9ui]

@@brooksbryant2478 charge is nothing but a ordinary element to distinguish between two different things we can say an electron is positive and a proton is negative it won't change anything

[@yourmom-nv9ui]

Arbitrary **

[@brooksbryant2478]

@@yourmom-nv9ui I know, it's just the opposite of the convention everyone else uses.

[@numbersix8919]

Not about charge.

[@pedroth3]

Truly amazing video! Really enjoyed the philosophic conclusion.

[@jackmcarthur1847]

unparalleled explanation skills, suited for an actually high level audience!

[@zunaidparker]

Came from 3b1b's competition. Great video!👍🏾

[@f-heinze]

That was amazing and my no 1 of the SOME so far! Thanks a lot for taking the time to explain it!

[@daniilzhitov9553]

This is an absolutely brilliant video! I am so glad to find this channel thanks to the SoME1.

[@AlexTrusk91]

i love to listen to ans watch your explanation when I'm sleepy and even more when I'm fully awake.

[@web2wl00p]

This is an amazingly well produced and didactically superbly laid out video! Kudos!

[@siyuanruan3404]

ohh, 3blue1brown style, I love it! You are doing a great job in using manim

[@davidkong2792]

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

[@lazbn90]

Algebraic Topology applied to Quantum Physics: automatic subscription for me. Keep it up!

[@pieterdelaat1463]

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.

[@weinsim3856]

3:42 this is the best form of comedy there is. very well done also great video btw

[@yurisich]

There's another great one right at the end, too.

[@Posesso]

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!

[@irodionzaytsev]

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!

[@stefanosamaxopoulos5285]

Bravo! You made an AMAZING work!

[@frankreashore]

Totally amazing video. Best video I have watched in months.

[@davidkong2792]

I was concerned when he said “for professionals” and he still explained everything he did beautifully

[@Syntax753]

Absolutely fantastic video! Genius presentation - thanks so much for putting this together!

[@nurianav]

This is GREAT, I think I'll go back to watching it every once in a while. It is brilliant. Thank you

[@renaudmathevet9374]

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 :-)

[@aitorgarcia1147]

Amazingly explained, thanks a lot! But I have to say I watched it twice, because the first time I left me with a negative impression ;) Keep doing videos like this one please!

[@athena9098]

with this video, you just stepped up your game Noah!

[@DanielKRui]

I think minutes 0-18:48 should be mandatory viewing on day 1 of a topology class. Starts with a physical phenomenon that's cool; builds up a space that motivates ideas of quotients of topological spaces and manifolds (and identifying antipodal points of spheres ---> real projective plane, Boy's surface, etc.!), then studies loops on that manifold motivating ideas of homotopy theory like contractibility (relative to some fixed endpoints) visualized in different manners, all still grounded in the hands-on real world by Dirac's belt trick. And of course all the topological content afterward: spheres as 2 disks of the same dimension glued along the boundary (i.e. forming the sphere as an adjunction space/categorical pushout of a diagram), covering spaces, the lifting lemma... truly a wealth of content here, all presented in a welcoming way! Even your proof by contradiction was presented in a welcoming way; I'm pleasantly surprised that one can make rigorous arguments with just a 40 minute "picture based" introduction to topology. The summaries were also really nice! Lecturers don't do those often enough in classes, I think.

[@johnitaballmer3966]

Thank you sir, keep continuing such amazing videos and interesting subjects. Please before the spotlight hits you and your channel (which I think is just a matter of time) always strive for quality and thoroughness of your videos over anything else.

[@redaabakhti768]

thanks so much man I'm at 12 min and the video up till now is so much insight. I wanted to investigate paths in spaces of rotations/lie groups as well as quotient topologies for a while but I've been distracted from it, thanks again!

[@IncompleteTheory]

This is really well done, thanks for uploading it.

[@universemaster]

Literally the best math video I have ever watched. Thank you, so much.

[@MasterHigure]

One thing I'd like to mention is that the figure on the cover of Hatcher's Algebraic Topology is the Hopf fibration. Also, many find Hatcher's to not be rigorous enough. There are plenty of more formal treatments, but Hatcher has better examples and exercises than all other AT books combined. To me that's worth it. But to each their own, I guess.

[@DarCMenO]

Wow, this is a masterpiece 😍 I especially like the last part on the Hopf fibration

[@ModusTollendoTollens]

Actually really impresive. Bravo!

[@rohankapoor8289]

This is a phenomenal video, it's so intuitive

[@euclidofalexandria3786]

this was a great vid thank you for your efforts in posting.

[@MarksmanSnir]

Please make more videos like this! It feels like 3Blue1Brown but for physics, which I'm sure for many people is even more interesting. The video was amazing and interesting, so thank you.

[@Benoit-Pierre]

Try minute physics

[@frizzarazz]

Best video I've seen on this topic so far. The tying up loose ends was right on the nose regarding questions I had!

[@SpinjelloLane]

i love that you put the electron in the black lodge

[@GeoffCanyon]

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.

[@amittsabary3907]

This video should be getting a lot more views

[@fredscutler9772]

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!

[@KipIngram]

Superb - absolutely superb video. This laid it all out more clearly than I've ever seen before. Thanks for the great work!

[@johnmccall4035]

That was excellent. Thank you.

[@melkdoria]

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.

[@otv9005]

This is excellent. Well structured and challenging. Understandably this took great effort to put together but I hope there will be more like this.

[@1sanremy]

I feel so stupid when I listen to your lecture...I understand just nothing, but it stimulates my old brain and teaches me humility. Thanx

[@mistertheguy3073]

This was amazing, thank you!

[@spiralofinspiration3653]

Amazing work

[@_srhenry]

This is insane video! You've explained it at the right pace and splendidly! I'm surely gonna recommend your video through all my friends

[@marcoseneto]

I had never seen this so well and thoroughly explained. Thank you

[@ecdavek230]

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.

[@ChrisDjangoConcerts]

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 !

[@kerolosbisheer4843]

Thank you for explaining the basics and the notation, I really needed that.

[@ccamp3175]

Most excruciating, enlightening hour I've spent on KZbin. Excellent presentation, and thank you very much.

[@ARBB1]

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.

[@mikaeels2691]

I recognise that background music from another math animation channel ;)

[@davicruzpestana7329]

An absolutely delightful video, congratulations!

[@diegokent1374]

Absolutly beautifull, i loved It! Keep It coming :)

[@NXTangl]

The connection I am making here is that the 3-sphere being a double cover of rotations is why the unit quaternions (a 3-sphere) are so good at rotation (SO(3)).

[@noahexplainsphysics]

Yes, I think that's right. They give a much more convenient parameterization of rotations at the cost of having an arbitrary sign

[@AA-gl1dr]

Struggling in trig right now and this helped me with the massive dose of perspective I needed to make it click

[@HilbertXVI]

Lmao

[@markos.5539]

You have no idea how entertaining this is for me. Suddenly certain things made for sense. Tbf, you hooked me with topology. Great video, great work!

[@jillianonthehudson1739]

This was so much fun! Advanced math, easy to understand, with your mind blown several times for good measure. Thank you! Definitely earned this new sub!