Spinors for Beginners 17: The spin 1/2 representations of SU(2) and SL(2,C)
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- Опубликовано: 9 июл 2024
- Full spinors playlist: • Spinors for Beginners
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Powerpoint slide files + Exercise answers: github.com/eigenchris/MathNot...
Physics of the Lorentz Group PDF (hints at how to re-write tensor product representation): iopscience.iop.org/book/mono/...
Previous videos:
SfB #6.1 on Bivector-Quaternion equivalence: • Spinors for Beginners ...
SfB #9 on Weyl Spinors: • Spinors for Beginners ...
SfB #10 Double Cover of SO(3): • Spinors for Beginners ...
SfB #12 Spin Groups: • Spinors for Beginners ...
0:00 - Introduction + Review of SO(3)
1:39 - Spin-0 representation of SO(3)
4:16 - No spin-1/2 rep of SO(3)
5:04 - SU(2) is the double cover of SO(3)
7:10 - su(2) Lie Algebra
10:03 - su(2) commutation relations
12:30 - Exponentiating su(2) generators
15:50 - Rotations of 720 degrees
18:40 - Lorentz Group SO+(1,3)
20:44 - SL(2,C) and sl(2,C)
25:26 - Left and Right SL(2,C) representations
30:27 - (1/2, 1/2) tensor product representation
34:39 - Complexification of su(2) and sl(2,C)
![Skilla Baby - Misfit (Feat. Polo G) [Official Video]](http://i.ytimg.com/vi/IjsylSnNMyw/mqdefault.jpg)
ERRORS:
1:26 - the 3x3 so(3) generator should NOT have a 1 in the lower-right corner... should be zero.
22:20 - the M^dagger = -M is not required for sl(2,C)... this was a copy+paste mistake from the su(2) case.
I love you
@@hyperduality2838why did you feel the need to copy paste this everywhere
@@hyperduality2838 please stop spamming this. If you want to have some sort of philosophical discussion, just info-dumping your beliefs in random comments isn’t the way to go about it.
Edit: apparently this comment differs slightly and doesn’t mention the bosons and fermions. I’ve edited it out and replied to the comment with the mistake instead. Here’s what I said:
Also your first line contains a demonstrable mistake. You say “(fermions, particles)” have spin 1/2 and “(bosons, waves)” have spin 1 but _fermions and bosons are both particles_ and those particles are _described by wavefunctions_ so they’re also waves. If you stick to just mentioning “fermions” and “bosons” it would at least be a bit better.
Even then their duality is rather uncertain. Supersymmetry suggests they’re related, but that theory is unproven.
23:23 in the bottom line on right you introduced a k turning ij into ki. So the commutator [K_ti, K_tj] = -J_ij as indicated in the middle right of the slide is correct and the line at the bottom right of the proof where it shows -J_ki is incorrect.
BABE WAKE UP NEW EIGENCHRIS SPINOR VIDEO JUST DROPPED
Accurate
Another banger from the man himself
Thanks for providing all these lessons, wondering how many more you plan to bless us with.
About two weeks ago, RUclips re-recommended me one of the earlier videos in this series. However, when I saw it in my feed I only saw the top of the thumbnail and at first I thought it was a new video in the series and got very excited because your last one took Lie Algebras from “confusing and abstract” to “logical and concrete” in my mind instantly. I was quite sad when I realised it wasn’t a new one.
Today my excitement _is_ warranted. It’s going to be a good day!
P.s. your video series on tensors was clear and detailed enough I was able to independently reconcile the Tensor-form of Maxwell’s laws (mentioned in passing in an E&M course I took) with the Geometric Algebra form (mentioned in sudgylacmoe’s great intro to GA). E&M feels much less strange and complicated now. Thanks for the amazing work!
Thank you for not getting tired of posting 🙏
Note the spin 1 counterpart of the pion is the rho meson, which is also an internucleon force carrier. They only stick around for a few yoctoseconds.
Clarifies preceding lessons especially 9. Exceptional explanation of challenging subject. Thank you 👏👏
Fantastic series! Many thanks for sharing this with us.
I am only at Lesson 2 but man I am so excited to look at this stuff. You are a blessing thankyou so much!!!!
Simply: Excellent - thank you !
Your video is great and helpful🙏.
Thanks Chris! ♥
Amazing! Thanks so much!
YOU DESERVE A PRIZE.
In the direct product matrix at 33:17, the beta star and gamma star should reverse places.
Ah finally i understand all this chirality stuff
Thank you so much 🙏
Thank you 🙏
You know I almost have an easier time understanding repersentation theory in higher dimensions because in low dimensions 2,3, and 4 there are so many exceptional isomorphism between groups that it's hard to keep track. Where as in higher dimensions the group structers diverge and differentiate themselves.
I'm not very familiar with higher-dimensional rep theory right now. Are you talking about stuff like how SU(2) = Spin(3)?
@@eigenchris Like that and how it’s equal too Sp(1) which is why you get the quaternionic representation. Or how Spin(4) = SU(2)xSU(2) and then is also related to the SL(2,C) which is then related to SO(1,3). It’s just a lot to keep as a mental model. Past dimension 6 their are no more relationships between the spin groups and Special Unitary groups and Symplectic Groups just it’s double cover of the Special Orthogonal group remains. This seems to be because there are no commutative division algebras past 4 and the Alternating group past 4 is non abelian.
In Cl(1,3) we like gamma sub t on the right. Kudos for getting the z x bivector in the right order, though. It seems to really bother physicists who want i < j in variable sub i j.
I really wish the title was "Spinors for Beginnors"
He knows.
ruclips.net/video/KjzO_pcdTOw/видео.htmlsi=mjgu_eGcz0t7oNuN
He made a video about this exact comment, it’s called “my greatest mistake” (or something very close to that, I might have misremembered slightly)
@@hyperduality2838 Your first line contains a mistake. You say “(fermions, particles)” have spin 1/2 and “(bosons, waves)” have spin 1 but _fermions and bosons are both particles_ and those particles are _described by wavefunctions_ so they’re also waves. If you stick to just mentioning “fermions” and “bosons” it would at least be a bit better.
Even then their duality is rather uncertain. Supersymmetry suggests they’re related, but that theory is unproven.
Also, as I had said in a different reply, please stop spamming this. Nobody wants to engage in a philosophical discussion when you just bombard them with random unexplained info.
@@kikivoorburg omg are you serious???
@@hyperduality2838 Yes. Not to mention...Dielectric Materials belong to physics while Dialectical Materialism belongs to sociology. 🤔
I think there is mistake at 24:35 in the video. The three boost matrices
should be the same as the boost matrices at 22:33 ...
Is su(2) o+ i su(2) equivalent to su(2) ox C?
Since taking the tensor product there would basically allow for “complex coefficients”, right?
(o+ and ox are direct sum and tensor product if that wasn’t clear)
Yes. The "complexification" C subscript can also be written as ⊗C. They are different notations for the same thing.
How was the matrix decomposition from 22:09 to 22:22 done? I got 6, 2x2 matrices. 2 of them were the vector sigma_z and the bivector sigma_xy, but the other 4 have only one non-zero element and I have failed to convert them into the other vectors and bivectors of cl(3,0). For the decomposition given, multiplying the 6 scalars into the vectors and adding does give a traceless matrix since the 2 main-diagonal elements use 2 of the scalars to form a complex number and its negative but the 2 minor-diagonal elements each have all of the other 4 scalars.
It's a 6-real-dimensional vector space, and there are an infinite number of different choices of basis. For the 4 parameters you're talking about, you could take the follow combo:
A -> [0 1]
[0 0]
B -> [0 i]
[0 0]
C -> [0 0]
[1 0]
D -> [0 0]
[ i 0]
In this case we could change basis to get:
a = B + D
b = A - C
d = A + C
e = -B + D
@@eigenchris Understood. I would also like to ask whether there is any care to be taken in much of the operations involving matrices. Most involve multiplication of several matrices in one term and you would wonder if the order of multiplication is to be considered.
Is the Hermitian operation relation of the bivector at 15:20 correct? It was equal to the negative of the bivector in video 6.1
The order of i and j is flipped, which is the same thing as putting a negative sig in front.
@@eigenchris I just realized that, thank you nonetheless
Is the so(3) xy-generator at 1:26 written incorrectly? Should it not be a 0 in the 3,3 entry?
Yup. My bad. :')
So there can't be an isomorphism between SU(2) and SO(3) since the correspondence is 2 to 1?
That's right. It's a homomorphism but not an isomorphism.
Can somebody tell me why on 30:44 there are no equation to relate z from pauli vector form to pauli spinor form? My background is computer science and after some searching on google, it seems like I can only factor a pauli vector into spinor pair if it is an isotopic vector where x²+y²+z²=0? Does it work for arbitrary x,y,z value? And doesn't the outer product of a column/row vector pair (in this case the spinor pair) can only at most be a rank 1 matrix?
Yes, you're completely correct. You can only factor Pauli / Weyl vectors if they are isotropic. I covered this in video #7 in this series if you want to get more info.
Thanks👍 I will check it
@eigenchris, I have checked the video #7, and now a new question pop up in my head 🙏, does this "isotropic vector only" constraint also applicable to the Clifford algebra with projectors method in the other videos of this series?
@@hellfirebb I believe so. I cover this in video #14, which is a long video.
Thanks for the explanation 👍
Could you please refer some literature
related to spinor ?
33:12 to 33:22. How does that decomposition of the 4x4 matrix work? It doesn't seem to be correct since the y* and b* are reversed
I did it wrong here, unfortunately. It's fixed in my most recent video, #20, in the section on the vector representation.
@@eigenchris I've seen the correction. Thank you
At 22:20 I think the M^dagger = -M is not requiered for \frak{sl}(2, C) matrices
oops. that's a typo.
If I can do something for the world in the future, the world must owe you because I learned a lot from you. Is it possible to upload the PowerPoint file of part 15, 16 and 17?
I feel like the introduction of the left/right Weyl spinors should have been shown in the reverse direction: start with the Dirac spinor and show that in Minkowski space it can be decomposed into a direct sum of left/right spinors. But maybe I'm wrong in my understanding.
I view Weyl spinors as the simpler building blocks, since they are from the smallest non-trivial representations and can be combined with direct sums/tensor products to make all other SL(2,C) representations.
@@eigenchris It's a good introduction to get a feel of them as representations. But imo the reason different representations are interesting is because larger representations can be reduced to smaller ones. For instance, 3×3 rotation matrices are interesting (they are certainly "natural"), but the reason we're also interested in different representations is because functions defined on a sphere can be transformed using irreducible transformations, by expressing them as a sum of spherical harmonics. This is also why trivial reresentations aren't just a ridiculously simple curiosity, but actually worth mentioning: they are still part of the expansion.
The fact that Dirac spinors have 4 entries is inherently tied to their living in 4D space, and the fact that they can be decomposed is an accidental property of the Lorentz group... if I'm not mistaken.
I might just not understand the Lorentz group very well yet. Personally I didn't understand what a dirac spinor was under I say it broken up into left and right parts. Although in the Clifford Algebra approach, Dirac spinors pop out naturally using the "minimal left ideal" definition, and you have to break them up into smaller pieces afterward.
31:08
37:57 This needs additional explanation, as it can cause misunderstanding without it: J/K Lie algebra elements and their real sums have physical meaning in that exponentiating them gives (a combination of) rotations/boosts. By complexifying them we move to a larger algebraic space, which no longer makes complete physical sense (what is e to the power of i times a generator??), but which lays bare additional structure. How this structure can be exploited without losing touch of physical reality, I do not know, even though it's very important that this is properly understood.
Pretty sure you can just use the Taylor series definition with i*generator, just as you normally would. You just get a series with extra factors of +1, +i, -1, -1,... repeating. Actually most physics classes already do this, because they use the Hermitian convention instead of the anti-Hermitian convention for their operators. As for the direct physical meaning, I'm not sure.
@@eigenchris Oh I have no problem with exponentiating a generator, just with the fact that the inclusion of the extra imaginary unit would give your rotated object complex-valued physical coordinates.
@@hyperduality2838 Go away crank
the uwu jumpscare hit me like a truck
UWU† jumpscare
Can you do a "What is Energy (joke)" video
no new videos?
I found some errors in #18. I'll post it later this week.
@@eigenchris thanks..
7:44 no, this argument does not transfer to complex numbers. There are infinitely many complex numbers whose exponential is one, explicitly 2πiℤ
That's an interesting observation, but since the property must work for any real number θ, you can't guarantee that using an integer multiple of 2πi will be an integer multiple of 2πi after multiplied by an arbitrary real θ.