To understand spin it's good to consider a gyroscope. when it has a lot of angular momentum there are two stable states for it in a gravitational field: aligned or anti-aligned with gravitation, up or down. in all other cases the gyroscope precesses. a spinor doesn't spin quite like a gyroscope but it is spinning in a sense (after all spin is angular momentum). but just like the gyroscope, you can think of it as having two stable states: in alignment with a magnetic field, or in anti-alignment. and because the magnetic field is a measure of some kind of rotation it can add to or subtract from the angular momentum of a spinor. this difference is "felt" as a negative or positive potential difference. this you can think of as two opposite forces on the spinor that split it apart into up and down components. the interesting thing is that a spinor with an arbitrary axis can always be written as the sum/superposition of an up and down spinor for some chosen direction. turns out quaternions have precisely the properties that you need to model this. i hope this was intelligible, it's a bit hard to put the geometry into words.

Sounds interesting - would you have a good intro book on spinors you could recommend?

Unfortunately i can't. I've been thinking about spinors and related concepts as a hobby for maybe two years now. But i never found an explanation like mine above anywhere. Good geometric intuition about these things does not seem to be highly valued in physics education so people don't seek it. It turns out you can represent a dirac spinor (as the sum of left and right weyl spinors) extremely well just with your two hands. the dirac matrices simply tell you how to rotate your hands or mirror them (which you can do because you have two!). I should make a video about this actually...

Woah. Ok, that model totally blew my mind. I never understood what angular momentum had anything to do with it, but the gyroscope analogy makes total sense. Thank you!

Yes, Quantum Field Theory can be explained through Lie groups. SU(2) is isomorphic to the quaternions of norm 1, and SU(2) is important if you want to understand the Lorentz group and Poincare group, which represent the symmetries of spacetime and special relativity. Check out the text book Physics From Symmetry by Jakob Schwichtenberg if you would like an approach that derives modern physics primarily from algebra

OMG thank you. Purchasing right now!

You DO NOT understand how happy I am right now. Truely!

I did general physics for a year at uni as part of my Computer Engineering course, then switching to Computer Science where I picked up a year of quantum mechanics. Since then whenever I lay in bed and thought about physics I would end up awake for hours. So damn interesting but the maths always held me back, so sadly gave up.

I don’t know what’s changed (maybe maturity or maybe Vyvanse lol) but I’m slowly putting the pieces together. It’s always been in my outer periphery but still out of reach. Your confirmation has and will change my life. Maybe not career wise or life altering seen from the outside, but hot damn you have at least cleared my constant nagging guilt for not perusing maths and physics because you’ve just made it slightly closer within reach. Can’t wait for the book to arrive. Thank you!!!

I just want to say I'm rooting for you, and hope you enjoy the book and learn a lot from it.

I had a bad experience with complex analysis as a teen (took a grad class that was a bit over my head). Many years later, I got Tristan Needham's "Visual Complex Analysis" and the whole thing clicked for me - I'm a visual person and do a lot of geometry. I hope your experience is similar.

haha thank you!

Awesome :) Yes, I saw that book this morning on my Amazon travels... though I'll put it in my wishlist because I think this morning blew out my yearly book allocation lol.

Quantum spin is an intrinsic angular momentum of a particle. It's angular momentum that is 'just there' as a key component of the particle.

In early days it was hypothesised that particles were spinning about their own axes, but this isn't accurate.

All the interesting stuff of Spin from its quantizable nature, the non-commutatability of spin measurements along orthogonal directions, the very different fundamental behavior of particles with half-integer spin (Fermions, eg Electrons, Protons) vs integer spin (Bosons eg Photons), how Spins interact (eg spins of say two electrons with half-integer spin interacting as a Spin-0 Boson in a Cooper Pair of a superconductor), or spin interacting with orbital angular momentum eg electron spin interacting with it's orbit around proton in an atom.

At the end of the day Spin isn't a terrible name for it.

I believe Spinor vectors are merely named after the eigenvectors used to represent spin itself, not the other way around as you suggested.

I think we maybe saying the same thing (unless I'm not reading that right) - Spin is not named because it's physically spinning (ok intrinsically perhaps, but that still isn't intuitive to me) but because the way we measure its interactions it's easier to describe using Spinors?

> Oh… dont tell me Quantum Spin just called Spin because it’s a Spinor rather than something actually metaphorically spinning?!

That's a great way to understand them if you are already comfortable with the algebra of spinors and spin groups, but it doesn't short-circuit the history—spinors were so called after quantum spin (https://en.wikipedia.org/wiki/Spinor#History), and I believe that was so called because, yes, it was envisioned as something at least conceptually spinning.

Interesting.

I've tried many times to go through Modern Algebra texts, and so I on-the-surface get that it's an algebra of sorts.

When I have enough time I'm going to finally go through mathacademy.com, because I think it really does suck not knowing advanced maths but want to do the hard sciences