Spinors for Beginners 8: Are the Pauli Matrices also Vectors? (Intro to Spinor Spaces)
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- Опубликовано: 9 июл 2024
- Full spinors playlist: • Spinors for Beginners
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My Tensors for Beginners playlist:
• Tensors for Beginners
0:00 Introduction
3:07 Vectors
5:08 Dual Vectors
8:34 Tensor Product
11:22 Spinor Spaces
15:00 Sigma as a linear map
19:45 Doubling indices; rank 1/2
20:45 Change of Spinor basis
23:00 Summary
i'm so lucky you chose the topic of spinors to make a whole series about. i'm to take a final exam on a massive quantum mechanics course in a couple of months, and spinors is probably the most counterintuitive and difficult thing throughout the whole thing. thanks for the amazing work you do man
I'll probably only be able to make 3-4 more videos in the span of the next 2 months. Next video will be on Weyl and Dirac spinors (used in special relativity), though, so hopefully that helps.
I'd curious to know what (if any) stuff this series has helped you with so far.
@@eigenchris the series mainly just helped me with intuition. the best way to understand a mathematical concept is to see it expressed and used in different contexts, and you did a great job at that with even the first couple of videos in this series. for a while in my quantum mechanics course i just had to take spinors as a weird algebraic thing that only made sense algebraically and for some reason led to accurate physical predictions. your vids helped me create some kind of picture of what a spinor is in my head, and that makes it much easier to interpret and work with equations
Best description I've seen in the past 60 years...
After completing this video, I re-did your Tensors for Beginners 15. Double thumbs-up.
Thanks, I was waiting for your video!
Amazing series, please keep them coming! Can't wait.
Literally couldn't sleep last night thinking about how the restriction to null vectors by the factorisation is resolved! 😂😭 Bring on pt8.
Great video as always!
I love your explanations.
If any viewers have made it this far in the spinor series and haven't already seen eigenchris' video series on tensor algebra, I would highly recommend. It is probably the best introduction to tensor algebra you will find anywhere. Also, he has a series on general relativity which is also amazing, if that's your thing.
Great video, connecting many concepts from linear algebra and tensor algebra, to spinors and representations.
Great series, looking forward to next video!
Excellent insight! Thank you.
Would be nice if you could release a video tomorrow about the many real-world applications that topology has to offer.
If you put this series for sale I would definitely buy it. Great work.
Always to the point, thank you.
Excellent video
Thank you! You finally helped me understand what the 1-forms are about. I worked with them often but never really felt in my bones what they were about. Just saying 'linear map from a vector space to R' really makes it a bit abstract.
i did a previous work on higher-spin objects (p.s. its very complicated). if a spinor can be thought of the square root of a vector ("spin-1") and a 3-vector can be mapped into the spinor (x) dual spinor space with pauli matrices,
is it possible to "start" at spinors and then construct from it higher half-integer spin objects (3/2, 5/2, etc). this thing is used in calculating the properties of fermionic excited states
Great video as always, though I'm slightly uneasy about the fact that we've gone a whole lecture without referencing the projective nature of spinors, which is their defining characteristic and probably has some interesting relevance when it comes to inner products.
This is my first introduction to inner products over spinors and I was confused when I also realised the (apparent) non-uniqueness you seem to be hinting at here. Would you (or @eigenchris) mind explaining how this is resolved? Do we only use specific representatives of classes of equivalent spinors in these products? Thanks in advance.
Awesome I’ve been studying susy and I didn’t know those spacetime sigmas were called infeld van der waerden symbols! Thanks :)
@eigenchris I am very thankful and appreciate your work,this channel is a gem for education your work on Tensor Calculus, and for beginners series helped me a lot in studying diff.geometry and General Relativity
And this spinor series has made my understanding really clear and I can't appreciate it enough. Side note: why do you sound so dead inside all the time or is it just me?
I have an issue I can't solve. The eigenstate |x+>, for example, is 1/sqrt(2)^*[1,1], although if we use the formula to obtain x,y,z from ξ1 and ξ2 I don't obtain the vector (1,0,0). Are the ξ1 and ξ2 Pauli vectors components the components of a spinor? Is there anything I don't get about spinor eigenstates or bloch sphere? Thanks a lot for the help
Could you please answer this question?
I know how to find the covariant components of a vector by utilizing dual basis. Using that what is the next step that connects the covariant and contravariant components indices of a tensor? I understand that the upstairs indices are contra variant and the downstairs and this is covariant but what is the geometric connection between these indices and the graphical representation of covariant components based on dual basis vectors.
No one has ever come close to answering this question, and I’ve asked many mathematicians
Sir Which books are proper to study these topics?
Great videos series. I really enjoy your work!! A point that feels really important tome but wasn't put forward in the video is that 1 3D vector indice matches 2 2D spinoff indices. I don't know why (yet?), but making extra explicit the fact that switching from vectors to spinors reduces the range of the indices feels quite important to me. Kind of like a foreshadowing of some sort... maybe the relationship can be expanded upon in higher dimensions (4D vectors to 3D spinors to 2D 2-spinors?)
can you tell why only pauli matrices are used as basis any reason
So tensor product = outer product?right
In 19:34, “not every elements in spinor-pair spaces corresponds to Pauli vectors” Is this statement related with SO(3) - SU(2) homomorphism? Not isomorphism.
I’m a Korean who really like your video. thanks for upload!!
Excellent, now I know the reason the components have upper indices is that the arrows point in the opposite direction. 😂
I wonder how the bivectors can be represented as a tensor. Would it be the tensor product of σ_i^a_b with itself? I feel like that would have too many components though, given that (in 3D) the dimension of the bivectors is the same as the dimension of the vectors... so if σ_i^a_b has 3 x 2 x 2 = 12 components, I feel like if bivectors could be represented as a tensor, then they would also need 12 components.
I'd normally think of σ_x times σ_y as a bivector. Written as arrays, the multiplication is easy. In index notation, I think it would be: (sum over b) σ_x ^a _b σ_y ^b _c = σ_xy ^a _c.
Written that way, you get an extra tensor index, but I think that makes sense, it would be the output of the map of σ(e_x∧e_y).
I'm not sure if there's a way to "distribute" σ over the wedge product... as in σ(e_x∧e_y) = σ(e_x)∧σ(e_y)... I'll have to think about that.
@@eigenchris Ah, I see. So the "extra" tensor index leads to more components, but they should either be "kronecker delta" for σ_xx^a _c, or signs should be flipped when xy are flipped. Interesting...
In APS (Algebra of Physical Space, aka VGA/Vanilla Geometric Algebra or Cl(3), the sigma matrices are vectors and their (GA) outer products are bivectors. In STA (SpaceTime Algebra, aka Cl(1,3), the sigma matrices are bivectors, but the products of each basis are still the same bivectors from APS, even if not the (GA) outer product.
This relationship between their representation in Cl(3) vs Cl(1,3) is actually an important one in Clifford Algebra, where every (ℝeal, finite) Clifford Algebra both has an even subalgebra that is itself a Clifford Algebra a dimension lower, and _is_ the even subalgebra for another Clifford Algebra a dimension higher. It is also related to the idea of "reflecting" a 2D image on a sheet of paper by flipping the paper in 3D.
@@IronLotus15 "I wonder how the bivectors can be represented as a tensor."
Aren't they basically just anti-symmetric (rank-2) tensors? Or am I misunderstanding your question?
@@seneca983 Basically I'm wondering how to express them as a tensor with a similar framework as this video, how to transform the bivector bases to 2x2 matrices. So a rank 2 tensor is the desired output of the tensor I'm looking for, just like how the rank 3 tensor in this video maps vectors to rank 2 tensors.
21:07 you say dual basis spinors are contra variant. But what you were describing at 6:52 makes me think they should be co-variant. Why is it different with the spinors? @eigenchris
Dual spinor COMPONENTS are covariant. Dual BASIS spinors are contravariant, as shown at 7:20.
Let's write it this way: v1 => 1v, we put the indices at the top before the vector!!!
RUclips does not allow to write formulas.
Can you share your reference books?
You can try googling "Infeld Van Der Waerden Symbols" to learn about the equivalent map in special relativity. I don't have any sources for 3D space as I've done in this video. I basically looked at the IVDW symbols and "simplified" them for the case of 3D space. en.wikipedia.org/wiki/Infeld%E2%80%93Van_der_Waerden_symbols
At 15:13 I'm not sure how you got the coefficients of the basis elements independent of simply already knowing what σ_x is supposed to look like. You've said that the coefficients are exactly the entries of σ_x, which makes me think you've somehow derived the coefficients and are verifying them against what we know σ_x should be. Apologies in advance if I've missed anything.
You're right that I'm just taking the coefficients from the sigma matrices, which I already knew ahead of time. I'm just re-writing the coefficients in a new matrix form people may not have seen before, to highlight how vectors can be mapped to a pair of spinors.
@@eigenchris Oh okay. Are there derivations or motivations for these forms of the sigma matrices that you could direct me to?
@@toaj868 If you start with the following requirements:
1. Matrices that square to the identity
2. Matrices that anti-commute with each other
...you get the sigma matrices (or some linear combination of them). The above 2 properties are loosely what clifford algebras are about.
When studying spin in quantum mechanics (see video #4), the sigma matrices are operators for measuring the spin of a quantum state in various directions.
Brother I want to ask you does this all topics comes under jee advanced?
While I don't know anything about the JEE, my guess is "almost certainly no". This video covers an extremely specific topic even most university textbooks don't cover, so I wouldn't worry about it.
so are pauli vector vector or linear transformation ?
The sigma with 3 indices is a linear map. The 3 Pauli matrices with 2 spinor indices each can be thought of a vectors in the space S⊗S-dual.
are pauli vector/Matrix same thing as corresponding vector ie can i think pauli vector as arrows in same space as origan vector space is that so what are columns of pauli matrix physically says about vector ?
This was such a great video. But something's bugging me. This all seems a little... artificial? For example, what is the difference between saying sigma is a linear map from V to spinor tensor dual spinor and saying sigma is a linear map from V and the space of 2x2 matrices such that etc etc. Why do we need spinors, or, how do spinors really come into play? Couldn't we obtain those 2x2 matrices with an appropriate choice of tensor product of vector covector spaces?
Also, additional question: normally the rank of a tensor reflects what kind of multilinear map it is. In that framework, what does a spinor having rank 1/2 mean? Somehow 1/2 of a linear map? Like it is linear in some part of its argument but not the other one? Only thing I can think of tbh; so, what kind of multilinear maps would spinors be? Keep up the great work, can't wait for the next video!
To my understanding there is a single linear map that transports a S X (S dual) tensor product (vector) , as there are two of them (spinors) they can be thought of as having half linear maps hence a rank half (I am not an expert, I am just a 15 year old on the internet so please take it with a grain of salt .)
I guess you're right. In theory, you could re-write most of this video without using the word "spinor" and just talk about this mildly interesting linear map from 3D vectors to 2x2 matrices. But this just so happens to be the linear map which leads to a space which contains the "Pauli spinors" we see in physics, which transform with SU(2) under rotations.
The "rank 1/2" is more of a metaphor than a formal definition. Normally a tensor "rank" tells you how many indices its components will have. So the metric tensor g_ij is "rank-2" because it has 2 indices. A spinor is metaphorically "rank-1/2" because it has "half a tensor index". What I really mean is that it has 1 spinor index, and since 2 spinor indices are worth 1 tensor index, the 1 spinor index is like "1/2" a tensor index. Physicists would call it "spin-1/2", just as they would call a vector "spin-1" and a scalar "spin-0". The hypothetical gravitons are spin-2 because they are described by the metric, which is a rank-2 tensor.
I have a similar question. I agree that this video (in isolation) does not illuminate the usefulness of spinors, but it does clarify / show another perspective of the nature of the sigma matrices. But I'm sure not all tensors with 2 lower indices and 1 upper index act as a map from "vectors" to "spinor pairs", so that can't be the key property of spinors. There has to be something else that makes the space of "spinors" or "spinor pairs" useful. Probably just that spinors are rotated with SU(2) rather than SO(3). I have a feeling that this will become more clear when Clifford algebras are formally introduced...
@@eigenchris What would a rank-3/2 mathematical object or spin-3/2 physical quantity look like?
@@viliml2763 There are no fundamental spin-3/2 particles, but you can get composite spin-3/2 particles (the Δ++ baryon is an example). I've never dealt with them mathematically, but I would assume they have 3 spinor indices, nd I assume they would transform with three SU(2) transformations.
😀
ok so spinnors and cospinnors, nobody can't do anything to change my mind
How many videos going to come
My guess is there will be 20-25 total.
0:15 erm what the sigma
I love the mispronouncing of pronunciation when talking about the pronunciation of Pauli. It comes off as a really punk response to a petty complaint about some mispronunciation that doesn't really detract from the content of the previous videos. super cool.
When I am benevolent dictator for life, I’m gonna make it illegal for any non-Greek scientist or mathematician to ever write or say the litter xi again.
A dual vector is just a linear map of a vector of the vector space to a scalar.
Try changing your voice tonality once in a while so as to not sound robotic. It' s like you're just reading off a script without imparting much understanding.