The Mystery of Spinors

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:

All support is highly motivating and greatly appreciated! :)

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.

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

[MirzaBicer]

Been patiently waiting for this one. Welcome back Richard. You made up your absence by a literal 70 minute giant, I'm happy.

[AffectiveApe]

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.

[ravani_]

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

[w.o.jackson]

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.

[drakegunter]

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.

[JL-]

I don’t normally comment on posts but this deserves a bump in the algorithm. Well done.

[jordanfarr]

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 more valuable education than I can get from paid sites.

[lunafoxfire]

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.

[grawl]

Am i day-dreaming. This is so ridiculously good. You are a grandmaster educator. Thank you.

[inkdragon]

Guess I found a new channel to binge while I sew. This video was so fascinating.

[aloeparrish]

"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.

[jounik]

"A wiggle is homotopic to an octopus" beats "a donut is topologically a coffee cup" six ways to Sunday. Excellent presentation!

[TheoriesofEverything]

Again, the most pellucid explanation on the topic you cover. Last time complex numbers, and the Dirac equation. This time, spinors. Bravo, Richard. Bravo.

[renzostefanmp]

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.

[sheeesh]

The fact that you provide this wonderful masterpiece of a video for free for everyone to see and study says a lot about your character. Thank you so much for your dedication to science and education!

[remifasollasido]

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

[chri-k]

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.

[ThomasGutierrez]

Amazing work! This is such a great service to the physics community to see this discussed so lucidly and with a friendly tone.

[ΣιγμαΠενισ]

Truly underrated creator. You deserve to be up there with Numberphile, 3Blue1Brown, etc. Loved the video!

[Micha-ngyp]

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!