Spinors for Beginners 8: Are the Pauli Matrices also Vectors? (Intro to Spinor Spaces)

My Tensors for Beginners playlist:

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

[ivanklimov]

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

[lanimulrepus]

Best description I've seen in the past 60 years...

[BakedAlaska]

After completing this video, I re-did your Tensors for Beginners 15. Double thumbs-up.

[stephenmcateer]

Amazing series, please keep them coming! Can't wait.

[richardneifeld]

Great video, connecting many concepts from linear algebra and tensor algebra, to spinors and representations.

[MW-lyzu]

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.

[kevinsellers]

If you put this series for sale I would definitely buy it. Great work.

[Abon]

Thanks, I was waiting for your video!

[PM-]

Would be nice if you could release a video tomorrow about the many real-world applications that topology has to offer.

[SerbAtheist]

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.

[jardelcestari]

Great series, looking forward to next video!

[shivammahajan]

@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?

[GeoffryGifari]

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

[mathieulemoine]

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?)

[orktv]

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.

[tanchienhao]

Awesome I’ve been studying susy and I didn’t know those spacetime sigmas were called infeld van der waerden symbols! Thanks :)

[changethiswhenyouareok]

So tensor product = outer product?right

[dr.rahulgupta]

Sir Which books are proper to study these topics?

[thevegg]

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

[toaj]

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.