Spinors for Beginners 3: Polarizations and SU(2) Matrices [and O(3), SO(3), U(2)]

0:00 Introduction
0:57 Waveplates
2:25 D-to-L rotation
4:43 H-to-L rotation
7:32 Ignoring overall phase shifts
8:52 H-to-A rotation and the Poincaré Sphere
10:08 U(2) Matrices
10:49 O(3) Matrices for Real vectors
13:25 SO(3) Matrices for 3D Rotations
15:12 Hermitian Conjugates
17:00 U(2) Matrices for Complex vectors
18:33 SU(2) Matrices for Rotating Jones Vectors
20:53 Why Jones vectors are Spinors (angle-doubling)
23:31 Next Video

[lourencoentrudo]

babe wake up first eigenchris of the year just dropped

[evilotis]

wow. this is the first video i've seen that's really explained what O(x), U(x), SO(x) and SU(x) actually ARE. thank you. ❤

[tomgraupner]

Whenever Physics sounds too complicated to be understood and the wikis are full of spooky symbols, it is worth to see eigenchris' video on that topic. It simply shows that Physics is smart and easy, if the teacher also is. Thanks Chris. It's always a pleasure to watch your excellent work.

[chattava]

I like the new 'suggested exercise' sections, very helpful.

[soranuareane]

This brings me back to group theory in college. As a CS major, group theory wouldn't qualify as an elective for my major, but I _wanted_ to learn it. So I took it pass/fail as a "for fun" course. I'm so glad I did.

[niosacyswiatlo]

Great videos! I work with electricity, low voltage. I know basics and I love to have my mind blown, spinors are not new thing for me, but I never even tried to understand them. It's pretty easy to understand if you have fundamental physics knowledge.

[cmilkau]

It is important to note that v† U† Uv = v† v only implies U† U = I because you can pick any v you want. In particular, if v is an Eigenvector of U† U, U† Uv = λv, we have v† U† Uv = λ v† v, so λ = 1. That means 1 is the only Eigenvalue of U† U. Matrices of the form A† A always have an Eigenvalue decomposition, so we can conclude that U† U is the identity.

OTOH, consider some unit length v and the rank-1 matrix A = v v†, which clearly isn't the identity. But v† Av = v† v v† v = 1 v† v.

[eamonnsiocain]

I watched the first two videos without realising they were more than 15 minutes long. It felt like 5 minutes each. Very good writing and presentation.

[ytKuna]

This is the best explanation of spinors, SU(2) matrices, and their relation to the Jones vectors. I will for sure watch this video several times.

[meenakshiiyer]

BRO you are god!!!! the best person I have seen on youtube. the way you explain these concepts are phenomenal. the clarity and simplicity along with the unique examples you provide are always beyond expectations. I have been really grateful to check out your channel. please make more videos on such topics, that would be great help. can you mention, how do you study these things and what kind of patterns do you follow and which books you prefer? thank you very much for your help. bless you.

[FormalSymmetry]

Incredible work. You've answered so many questions I didn't know I had. I'm sharing this with my classmates. Thank you!

[michkrumskas]

What A wonderful clear description of the things that are described in a too complicated manners by the books. thanks and wish the best for you. finally I hope you prepare some videos on the how Heizenberg made his quantum mechanics version of poission bracket. It's described at the end of his book and shankar's book but I'm confused why the others left this matter behind and they have mostly used the Schrodinger method.
best regards.

[phygeniuxYTcreations]

The topic and delivery is amazing. I thing I have noticed is the wave passing through the medium at 1:53 should have shorter wavelength than the outside if n2 > n1. Please keep on working on this series. Thank you!

[mattkerle]

Wow thank you! Finally a decent explanation of the Hermitian dagger notation! I might have to watch this video a few times to really absorb the rest of it though. When you finish this series a good epilogue might be going back and linking the SU(2) and unitary groups link to the standard model. But thank you for this awesome series!

[vacuumisallinone]

sir,
after read the series you've posted on the channel,
I know you are my great teacher i ever met,
thanks!

[afammadudaniel]

You have demystified quantum computation for me. I am emotional right now because I have struggled with it for two years.

[thuvalusmandratis]

"Wake up babe, Eigenchris uploaded a new video"

[ytpah]

🎯 Key Takeaways for quick navigation:

00:27 🌈 Jones vectors represent polarizations of electromagnetic waves with magnitude and phase components, allowing for various polarizations.
01:47 🔍 Wave plates can rotate one polarization into another by introducing phase delays, making them valuable optical devices.
04:18 🔄 Special matrices, SU(2) matrices, enable rotation between various polarizations, revealing an angle-doubling relationship between physical space and wave polarizations.
06:00 icon
08:56 🤔 Total phase shifts don't affect the polarization of a wave, allowing the simplification of Jones vectors by ignoring these phase factors.
12:00 icon
15:18 🔀 U2 matrices represent rotations in polarization space without changing vector lengths, making them suitable for manipulating polarizations.
19:26 🌀 The special unitary group SU(2) contains matrices with determinant +1, providing unique rotations in polarization space, removing redundancy.
21:02 🔄 Jones vectors are considered spinners because they require two full rotations in polarization space to return to their original state.

[alejrandom]

I really recomend watching mathemaniac's video on "rotating in higher dimensions". It gives strong intuitions that complement this video quite nicely.

[VincentBILLET]

Awesome video, thank you very much. It explains something I didn't understood : Transposition from 2D space to 3D. Thank you !