Spinors for Beginners 3: Polarizations and SU(2) Matrices [and O(3), SO(3), U(2)]
HTML-код
- Опубликовано: 9 июл 2024
- Full spinors playlist: • Spinors for Beginners
Leave me a tip: ko-fi.com/eigenchris
Powerpoint slide files + Exercise answers: github.com/eigenchris/MathNot...
MinutePhysics video on polarizers: • Bell's Theorem: The Qu...
3Blue1Brown video on polarizers: • Some light quantum mec...
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
babe wake up first eigenchris of the year just dropped
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. ❤
I would say there are many that tried :-) but those left me as dumb as I was before - yep - so, I fully agree with you: here I get explanation instead of confusion
Maybe try reading a book once in a while
Indeed, usually SU(2) is just mentioned, without being introduced.
@@HilbertXVI a book, you say? wow, i don't know why i never thought of that! you are a brilliant genius and not at all a snide comment section cunt
are you only watching physics videos? because group theory is possibly the best-documented area of math there is lol
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.
Maybe the key ingredient is that the teacher first struggled to fully understand the topic, then struggled again to find the simple way to explain it. In both cases, that's a huge sum of work. Smartness certainly helps, but work is also critical.
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.
I like the new 'suggested exercise' sections, very helpful.
Thanks. A few people were asking for exercises. They're not terribly hard, but they should be just enough to make you think about what you're watching.
@@eigenchris Yes, that's the point. Exercises do fix knowledge, especially when recently acquired
"Wake up babe, Eigenchris uploaded a new video"
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.
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.
that was cool :o really clear and easy to follow, thanks for your effort! I can't wait for future episodes
An absolutely outstanding explanation!
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.
Incredible work. You've answered so many questions I didn't know I had. I'm sharing this with my classmates. Thank you!
sir,
after read the series you've posted on the channel,
I know you are my great teacher i ever met,
thanks!
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!
Awesome video, thank you very much. It explains something I didn't understood : Transposition from 2D space to 3D. Thank you !
Thanks for your explanation of poincare and unitary matrices.
It is so cool to see how that whole polarisation business corresponds nicely to spin / quantum rotations!
Of course it should - spinors they are! :)
Also all of those remind me about qubits and quantum computations - one more application!
Looking forward to the next videos!
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.
Very good! Thank you.
🎯 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.
Great video thanks a lot!
You have demystified quantum computation for me. I am emotional right now because I have struggled with it for two years.
So anyway, here's group theory.
You are awesome 🥺
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.
Thanks for the compliments. I don't generally read any one single source. I tend to read a lot in "parallel". I do a lot of googling on a given topic and look at many textbooks, PDFs, videos, and forum posts in parallel. I try to take the best parts from each source and combine them together. Spinors I found very confusing though, and it took me several years for them to make sense. So my technique is basically "lots of parallel sources" + "waiting/thinking time for things to fit together and make sense". Unfortunately this isn't a great strategy for undergrad students, because they always have tons of assignments to do.
There's actually a "rough draft" of this video series in the description links of this video: ruclips.net/video/O12Y0DkLDf8/видео.html . You can see some of the explanations I had an understanding of 2 years ago, but still some things were not clear, and I even make some mistakes. Back then I don't think I understood Jones vectors were spinors.
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!
Gyro, it's time to spin again!
I should have seen a steel ball run pun coming...
@@takenspark546 lesson 5, johnny
Good lecture
wow, nice topic
u r great man, u r so great i want curse, I love you my bro
The ideas are clear. If I spent three weeks working on the math, I might be able to do it. Back to the old linear algebra notes.
Hopefully the mini-exercises in the videos help.
@@eigenchris I should also say that when first introduced to linear algebra, I thought matrices were magic.
The crystal should reduce the wavelength, while keeping the frequency, thus reduces the light speed in it.
eres un condenado crack
Seeing this in real life. Purchase a Doorscope. It is a peep-hole for the door. What is interesting is that as you turn the scope, the image on the other side is 1/2. Years ago I let physicist Sylvester Gates borrow mine. (should have just gave it to him.).
Hello Chris. I have a question about problem sets. But first, big kudos! Your video's are my absolute favorite for truly understanding the meat. I get both mathematical utility and insight. I watch each video several times, try to copy the work from memory, and ask myself questions. Here's my question for you though. Where can I find problem sets online to practice what you're teaching? I think the better part of learning comes from solving problems.
I don't really have a specific place to look for more problems. You can start with the exercises in the videos. If you want more, you'll have to google the topics in the video and search for more. Or try making up your own questions.
If determine be is how much you scale areas and volumes, what does it mean to scale an area and a volume by a complex non-real number?
I'm honestly not sure. A lot of the geometric intuition we have for real vectors/matrices doesn't work out so well for complex vectors/matrices. But when the determinant has a magnitude of 1, the magnitude of Jones vectors is unchanged after matrix multiplication.
From the beginning of the series I felt that there is a double covering situation going on somewhere - and it finally showed its face here as I believe SU(2) is a double cover of SO(3). Your video gives a nice intuition why following a path in SO(3) (where every point represents a rotation in 3 dimensional physical space) is locally the same as following a path in SU(2) (where every point is a unitary transformation in polarization space). Am I right?
Yes, you're exactly correct. It's impressive you were able to see that. SU(2) is the double-cover of SO(3). SU(2) is also equivalent the set of quaternions of unit length (for any quaternion "q", the quaternion "-q" will do the exact same rotation on a vector, so there are 2 quaternions for each SO(3) matrix, hence they are a double-cover). Another name for SU(2) is "Spin(3)". Every SO(n) group has a double cover called Spin(n). This will become relevant later when we want to build spinors in any dimension.
@@eigenchris Couldn't you claim that -q _isn't_ the same _rotation_ as q, but rather the same _orientation?_ In terms of rotations, it's the same, but in the opposite direction. If q is clockwise, then -q is counterclockwise, but both end up at the same place. From what I can tell SO(3) describes orientations, while SU(2) describes rotations.
@@angeldude101 The formula for rotating vectors "v' using a quaterion "q" involves a double-sided transformation: "qvq*". If we replace "q" with "-q", we get: "(-q)v(-q)* = (-1)(-1)qvq* = qvq*", which is the same as the original formula. So both "q" and "-q" do the same transformation on vectors.
@@eigenchris Yes, they both send the vector to the same orientation, but they do so with rotations in opposite directions. With quaternions, 90° clockwise ≠ 270° counterclockwise.
Is the voice generated by speech synthesis from text entry, perhaps using samples from your own voice? The delivery is essentially perfect with no hesitations or extraneous sounds. If so, it’s not a criticism, but actually a very effective way of delivering the information.
It's my natural voice, but I do a lot of very careful editing to remove stutters. I often misspeak and make mistakes in the original recording, but those get removed or fixed while I make the video.
What does the “physical space” angle represent (at 21:00)? Is it the position of the waveplate or something else?
It's the direction the light wave is oscillating. V is a vertically polarized wave and H is a horizontally polarized wave.
Thanks 👍
Bonjour existe-t-il un développement en série du logarithme en base b ?
Je ne vais pas faire un série sur les logarithmes.
I am at 21:23. Does the 180 deg in physical space vs 360 deg in polarization space simply boil down to the fact that concept of vertical polarization does not distinguish between up vertical and down vertical? A 180 deg rotation in physical space takes us from up to down, but it's still vertical polarization. Anyhow, great stuff and thought provoking.--- p.s. OK, a few seconds later you said the same thing!!!
I think it's not "up vertical" vs "down vertical", more "up down up down up down etc vertical" vs "down up down up down up etc vertical"
😀
7:52 Isn't that a right hand rotation?
You need to imagine the wave tracing out a dot in the XY plane as it passes through. If you follow that dot, it will go +y -> +x -> -y -> -x -> +y, which follows the fingers of your left hand.
Your picture at 1:58 shows the wavelength as getting a lot longer in a glass or crystal. This is the opposite from the usual. In effect you have made the light in the crystal or glass move faster than the speed of light in the external medium. If that external medium is air or vacuum, your light violates special relativity.
Excuse me,Sir! I can't see your new video that you have just posted.
It will be up tomorrow. It has some errors I need to fix.
@@eigenchris Thanks!
At 1:56 you show wave length INCREASING in the denser medium. WRONG.
Frequency remains constant and speed decreases, so wave length DECREASES when light enters a refracting medium.
fans coming
I'm so unsatisfied with so many stuff... I don't mean the first thing you can understand from it. I mean I'm being uncapable of imagining how things being ignored like that can't be relevant.
I understood that in terms of what we're experiencing it's all the same but if for vertically polarized you have more than one option then how would you convince me that a wave function colapse loses informarion?
If it just looks like there's some information missing fron the start, how could you actually measure there were loss?
Also if multiple SU matrices that could be used to interpret the same transformation can end up having imaginary determinant and the determinants are a form of "size" of the matrix, what's the problem exactly in allowing imaginary or negative sizes? Which ends up being one more information that just wasn't there.
My feelings are telling me that we're not losing information, but the information never existed to start with... and... why am I wrong??? (Like seriously I'd love an explanation)
You're right that we're "losing information", in a sense. In video #5, I'm going to explain why spinors belong to a space called "the complex projective line". The "projection" causes multiple objects to merge into the same object (similar to how if you project the 2D plane downward onto the x-axis, multiple points get projected to the same point on the x-axis). In the complex projective line, we project the vector's overall phase away, so objects with different phases all merge into the same object.
I feel a bit uneasy with how Jones vector is presented as a spinor here. Let's step back to 1/2 spin fermions for which the notion of spinor was originally established. For them, the turn for 180 degrees in the physical space is corresponding to a 90 degrees turn in the state space in which the quantum states live.
Now per photons we do not anymore consider quantum state space, but rather introduce some "polarization" space in which we have 180 degree angle between vertical and horizontal polarization. This is clearly not a physical space, nor a quantum states space. Now, where does Jones vector live? Definitely not in the polarization space as in that space the components of this vector are not orthogonal. Yes, we can use polarization space if it is convenient to represent Jones vectors, but 90 degrees between Jones vector components (H and V) are seen only in the regular space.
In other words, it looks to me that "polarization" space has no mathematical or physical meaning, it is just a "diagram" and that in this space (in which Jones vector does not live) a 180 degree rotation corresponds to 90 degree rotation in the physical space does not make a Jones vector a spinor.
Now, even if to follow this video idea that Jones vectors live in the polarization space, I see a discrepancy:
electron's spinor: lives in the abstract state space in which 90 degrees correspond to 180 degrees in physical space
Jones vector: lives in "polarization" pace in which 180 degrees correspond to 90 degrees in physical space.
Obviously, this is not the same proportion as it goes in opposite direction relatively to the space in which a vector lives. So to fix that, we would need to write:
Jones vector: lives in (complexification) of a real space in which 90 degrees correspond to 180 degrees in polarization space. Yet, then question is asked again like why, first of all, did we need to introduce some "polarization" space which is not physical, nor a space of quantum states and why it may define what Jones vector (not leaving in that space) is. Remember that for electron the situation was much easier as the second space in the pair of spaces was a physical space in which we seem to live, not some arbitrarily introduced "polarization" space.