The biggest misconception about spin 1/2

“If you rotate a spin 1/2 particle by 360 degrees, it doesn’t go back to its original state, rather you need 720 degrees”. This is only technically correct if you interpret the words “rotate” and “state” in the way it’s intended, and here is the video on what they really mean, and what this sentence is saying.

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Many people have talked about spin 1/2 before, but I do want to chime in because there are a few things that I myself am not satisfied with all the explanations.

(1) I personally don’t like **any** “demonstration” of spin 1/2, aimed at demonstrating in some physical situations you need two full rotations to get back to where you started, like the Dirac belt trick / spinning your hand trick. This is simply because physically, you literally can’t tell the difference before and after rotation - even in principle. You can only tell the difference when you have a superposition. And these demonstrations, to me, give the false perception that physically there is something different. This is, I think, an extremely important point that people miss out when talking about spin 1/2 and how “weird” this is.

(2) From U(2) to SU(2): the “usual” explanation for the choice of determinant is that the phase factor does not matter, so in passing from U(2) to SU(2), we sort of “remove the redundancies'' in the description of our transformation. But why did we leave the ±1 factor redundancy untouched? Either you remove **every** redundancy, or **none**. It doesn’t make sense (at least to me) that you don’t also remove that sign redundancy as well.

This is actually because the projective representations (obtained using Lie algebra methods) of SO(3) **must** have determinant 1, and this is known prior to constructing such a (projective) representation. This might be briefly explained if I decide to make a video on Lie algebras / groups / representations. I think that using the Lie algebra method means that we are imposing the constraint about analyticity and genuine representation in the neighbourhood of the identity, but I can’t be sure about this.

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In a similar vein, people often think that spin-s particles are just particles that magically goes back to its original state after 2pi / s rotations. Even Wikipedia says this! Except this is not even technically correct - it is just plain wrong. I’ll talk a bit about that, again if I decide to make a video about higher spins / Lie groups / algebras / representations.

Hopf fibration / quaternions:

Video chapters:
00:00 Introduction
00:47 Chapter 1: "State"
07:42 Chapter 2: "Rotate"
17:46 Chapter 3: The construction
25:41 Chapter 4: The "spin-1/2 property"

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[mathemaniac]

Sorry that I only had about 5 days to make the video, and there are a few glitches in it before I have to put out this video, but hopefully they are minor. Please read the description on some of the thoughts I have around the topic, because something that I mentioned / did not mention might be a deliberate editorial / pedagogical choice.

[enterprisesoftwarearchitect]

THANK YOU! when I tell people that it’s a rotation in SU(2) spin space, not SO(3) actual space, they think I am a lunatic.

[KieranORourke]

Geometric Algebra simplifies so many of the concepts here. Matrix algebra is marvelous but, by definition, it only works on vectors so representing other geometrical objects means sprinkling difficult-to-interpret matrices ~(like the Pauli Matrices) all over a representation.

[lexinwonderland]

YES!! PLEASE ANOTHER VIDEO ON THIS TOPIC!! great work as always!!

[narfwhals]

Often these explanations include a statement like "In statespace (1, 0) and (0, 1) are orthogonal, so 90° apart, but on the Bloch sphere they are on opposite poles, so 180° apart." Tadaa 90° corresponds to 180°! But i never quite understood why I'm supposed to care about this correspondence in the first place. This video helped me with that.

[davidhand]

The whole geometric description of a spinor thing prevented me from understanding them for years. You aren't rotating a geometric object in 3D space, and the moment someone showed me what they were actually rotating, I got it immediately. I caution everyone who wants to understand spinors against watching any explanation of "how to visualize spinors", the belt trick, the coffee cup trick, etc. Obviously there is no object that must be physically rotated twice before looking the same.

[quadrannilator]

I was in my head thinking of quaternions ever since the video started and was also delighted they got their due mention at the end. Also was suprised seeing Hopf Fibration but intuitively got it why its there in relation to the projection type presentation. Keep up the amazing videos coming.

[ghassanehajji]

Amazing explanation. Clearest discussion of the subject I've encountered so far. Thank you for your efforts, and I'm really looking forward to be watching your next videos.

[SmogandBlack]

When I started to read about the Standard Model I didn't think that I would like to go into its Math: I felt happy just to know about the existence of Fermions and Bosons. But time passes and the brain is an unresting structure that wants satisfying answers to its questions... so thank you very much for this video.

[tanchienhao]

You explained the notion of “intrinsic property” much better than any other sources out there! Awesome video

[fangjiunnewe]

I'm cackling in my room near midnight because of the ChatGPT segue 😂😂

[japanada]

Fantastic video, very nice derivation of the projective representation of SU(2)! Just wanted to leave a note regarding the comment in the description on spin 1/2 "demonstrations": while I agree that they can easily be misleading if presented incorrectly (i.e. as a description of what physically happens to a spin 1/2 particle during rotation), as they almost always are, I respectfully disagree that they necessarily misrepresent the nature of spin 1/2 particles. If explained properly, I would argue they describe the nature of spin 1/2 particles _extremely_ well, to the extent that you can actually use arm-twisting to perform accurate calculations involving the behavior of entangled states.

At their core, all these demonstrations are exhibiting the behavior of _paths_ in SO(3): whether you use a belt (fixing one endpoint and allowing the other to rotate) or your arm (fixing your shoulder position and allowing your hand to rotate), in every case there is an extended object (belt/arm) that parametrizes a continuous trajectory from the trivial rotation to some other rotation in 3d space. When you rotate the belt end or your hand 360 degrees, the belt/arm now parametrizes a _loop_ in SO(3). What the demonstrations show is that the fundamental group of SO(3) has an element of order 2: there is a loop that cannot be contracted to the identity path, but if you repeat that exact same path twice, then there _is_ a contraction to the identity. Now paths from a fixed base point (up to homotopies that fix both endpoints) are in 1-to-1 correspondence with the universal cover. So one way to interpret these demonstrations is just that they are _performing calculations in SU(2)._ Every matrix multiplication in SU(2) can be perfectly replicated, with no information loss, as a sequence of rotations of the belt/arm! This already gives a nice explanation for the purpose of the demonstrations: even if you don't want to attach any physical significance to them, at the very least they perfectly explain the _mathematical structure_ that is required for performing calculations on spin states.

But I would go further and argue that the demonstrations actually _do_ have a very nice physical analogy. Most explanations make the mistake of comparing the particle to the entire extended object (belt/arm), saying that the 360 degree rotation has "twisted" the particle somehow. In a more accurate interpretation, the arm is a _timeline_ of the spin state of a particle; that is, it describes what quantum operators were applied to the spin state to put it into its current state. Only the very end of the belt, or the hand at the end of the arm, represents the current state of the particle. When it comes time to make a measurement of the hand, we have absolutely no way of telling whether a 0 or 360 degree rotation was applied, because all we have access to is the current position of the hand, which has no memory; the history of the particle (the shape of the arm) is lost to the depths of time.

However, this all changes once entanglement enters the picture, because this allows us to keep track of histories _between_ particles! In particular, the twisted/untwisted arm exactly conveys the difference between the "twisted" (a1-a2) and "untwisted" (a1+a2) entangled states. Instead of treating shoulder=past and hand=present, we now interpret shoulder=entangled particle #1 and hand=entangled particle #2 (initialized in the exact same state a1=a2, giving entangled state a1+a2). By applying a 360 degree rotation operator to the hand (i.e. to the spin state of one of the entangled particles), we've encoded into the wave function (arm) the fact that one must follow a nontrivial loop in SO(3) (ie apply the quantum "360 degree rotation on spin state" operator) to get from the state of one particle to the state of the other particle. If you just look at my shoulder, or just my hand, you would have no way of knowing that I did anything to either of them - but the twisting of my arm in the middle is evidence that I performed some odd number of 360 degree rotations to the hand, putting me in state a1-a2. This "twist in the arm" can be measured experimentally _only_ because we've set up an entangled system: even though the particles have both ended up in the exact same state (up to projective equivalence), the twist allows us to detect the fact that at some point different operators were applied to each particle. And again, while this is just an analogy, it's a very precise one, in the sense that SU(2) acts in the exact same way on the arm as it does on the entangled pair! That is, if you want to know which sequence of operators will recover the a1+a2 state, all you need to do is figure out which rotations of the hand bring the arm to the untwisted state.

tl;dr: the arm (belt) demonstrations captures the behavior of an _entangled pair_ of spin 1/2 particles; the hand (end of the belt) alone captures the much less interesting behavior of individual spin 1/2 particles.

[martine]

Finally a video that goes into the math of spin.

I wish I hadn't forgotten all the math I studied 15 years ago, ha!

[friendly_hologram]

Thank you! Explanations of spin are often disappointing, but this one was great, I feel like I have a much better understanding now. I study pure maths and I was a little disillusioned with physics (I used to study it but swapped to maths recently), but mathematical explanations like this are making me regain some of my appreciation for the subject :) (especially the mention of the hopf fibration!)

[NoNameAtAll]

eigenchris is doing video series about spinors, explaining SU2 symmetry and everything

[bboysil]

Love how this ties into Lie theory, which I happened to learn last year, needed for work :D. Excelent video!

[ChrT]

Great video. Just one concern: you call two spin states differing just by a phase "physically equivalent". That is true unless the phase plays a role in the experiment.
Consider the Bohm-Aharonov-effect. Here, the wavefunction of a spin-1/2-particle is first split into two parts, then one of the two parts is sent into a magnetic field that "rotates" it (by precession) 360°, and then the two parts of the wavefunction are again combined, allowing and forcing them to interfere. And we see destructive interference!, showing that by the 360^-rotation, the spinor phase has been multiplied by -1.
So the phase is not really irrelevant, and two spin states differing just by a phase can result in a different physical measurement.

[profdc]

This also ties into the geometric phase (Berry phase or Pancharatnam phase) which generally for a spin x particle is going to be 2 pi x, for example, for the spin 1/2 particle, rotating it through 360 degrees added a pi phase shift.

Of course, this leads into all kinds of directions, such as adiabatically varying systems (Foucault's pendulum, etc.), adiabatic and diabatic transitions in molecules, etc.

[MarceloRobertoJimenez]

Congratulatios for the video. Definitively, we need a video on Lie groups and Lie algebras.

[tablettorrensabellan]

Thanks for the ordered way to present the issue. I d just gone through various series of really nice and good YouTube videos, noah explains physics, eigenchris, and your video has given me the final nail I. The coffin to understand this complex issue. Thanks a lot. And the order in your presentations are a key asset!!