Spinors for Beginners 7: Square Root of a Vector (factoring vector into spinors)

Sources:

0:00 Introduction
1:46 Factoring 2x2 Matrices
5:45 Factoring Pauli Vector
11:31 Relating back to spinors in videos 2-5
15:10 Factoring matrices with non-zero determinant
17:50 Summary

[pelegsap]

Love your videos. Also as a German I appreciate the fact that you pronounce Stern-Gerlach correctly :)

[prosimulate]

If youtube was a 1000 years old your video here on this topic would be A1. What a gift you have in explaining something I’ll never use in my boring job but am so interested in your topics. Bless you Chris with great health, intelligence and happiness always👏📕🏆

[aidenwinter]

Chris is the kind of guy who comes across a vector and square roots it into spinors, and says it's more complicated than that and proceeds to make a brilliant video on that

[cloudyyau]

I wish you could get more attention. Your videos are indeed instructive. We appretiate your efforts.

[edwardlulofs]

Very good explanation, thanks. In all my years, I have never seen the square root of a spinor.

[Duskull]

Thank you for this wonderful series! It was immensely helpful for studying for my Quantum Mechanics II test :)

to me it seems that you have very profound knowledge and understanding of both physics and geometry, do you mind answering what path (degrees, courses, books etc) you took in order to reach such level?

I'm a 3rd year undergraduate physics student with very little mathematical background beyond what is required for a physics degree by my university.. but after getting into classical field theory and introduction to general relativity i was really mesmerized by all the geometry involved!

now I am very curious to learn more about the mathematical machinery behind all these beautiful ideas in general relativity, quantum field theory and string theory, but when i tried looking it up i was very overwhelmed by the shear amount of advanced mathematical topics that stand behind these fields..

(topology, differential geometry and deferential forms, Riemannian geometry, semplectic geometry, spectral geometry, algebraic geometry, algebraic topology, differentiable manifolds and exterior calculus, calculus of variation, tensor calculus, functional analysis, harmonic analysis, spectral analysis, measure theory, stochastic processes, ergodic theory, gauge theory and hodge theory, abstract algebra, lie groups and lie algebra, category theory(?) and the list goes on and on..)

it would take a full course just to study either one of these subjects (some of them will probably take 2 perhaps more) and at the end of the day i still want to focus on physics and not forsake everything for the sake of mathematics..

it would make me very happy if you could share how you managed to balance all these physics and mathematics that stands behind it.

[GiordanoGaudio]

Great video! I’ve been trying to figure out how to define spinors in an arbitrary Clifford algebra, in such a way that I make a connection with what you did here. Is it possible to take an arbitrary abstract Pauli vector (where the Pauli matrices are symbols not matrices) which is not necessarily isotopic and express it with a spinor decomposition? How would you do this? Can’t wait for the rest of the series!

[Neuroszima]

Funny how the most interesting thing that i learned from this video is the one you didn't mention. Got me curious and then fell into a "AHA!" moment when i realised you sneakly can actually materialize "i^2" as "1 = (-1)*(-1)" and then use one of these in your own calculations, only then to obtain "b = -c", which becomes trivial. Sorry i graduated as a chem student and didn't do much maths, so this was new to me that you didn't need to "divide by i" out of a sudden and wondered for a minute where did it go.
The moment in question is 9:08

[GeekyNeil]

I love your thorough and methodical style, paying attention to all the details. I think the misleading part about spinors being the square-root of a vector is that it only applies to null vectors. The normal description makes it seem like you can somehow take the square-root of (3, 4, 5). Thanks for explaining the caveats and how the actual decomposition works. I'm wondering if it's necessary to use as many as four pairs of spinors to decompose a non-null vector. It seems like the Pauli vector ((a, b), (c, d)) could be decomposed as ((a, b), (0, 0))+((0, 0), (c, d)), but that's not very symmetrical. Perhaps there's a more symmetric (but more complicated) way to decompose an arbitrary Pauli vector using only two pairs of spinors.

[jaspreetsingh-gdgi]

hey man great videos I am a graduate student in physics can you suggest books with a simple explanation to study more about these topics as you know self-study is important

[pannegoleyn]

Another excellent video; my only complaint is that I'm left wanting more! (I shall have to start on the GR course ;)
What was Pauli's direct motivation for this analysis?

[Cosmalano]

Excited to see what happens to null spacetime vectors! I wonder how the metric signature allows for different things to happen, since the components are all real (or would 1-3 of the components (depending on east or west coast metric respectively) of a null spacetime vector be complex, given the fact that their squares are what have the minus sign; do we have to use “ict” to talk about spacetime spinors?)

[SergioGarridoPaco]

Sorry, I can't find the solutions for the proposed problem in this particular video. Is it supposed to be in the github? By the way, the course is amazing and helpful, you're an insta subscribe

[stefanosvasileiadis]

If I remember correctly, you followed the exact same procedure in your playlist about tensors to decompose (1, 1)-tensors into a sum of products of the basis vectors and the basis covectors (L^i_j e_i ε^j). Is this the same thing with spinors or is there a difference?

[linuxp]

Back when studying linear algebra, in my first semester in physics course. I didn't understand why would anyone even consider matrices as bases and call it an isomorphism to the vector spaces of row/column vectors, but after your course on tensors and now spinors, that seems more reasonable.

On another note, I even thought you wouldn't metion that matrix decomposition trick, but seeing to how much calculation that leads, I got why you left it to the end.

I was only thinking if it would be best to break it as a sum of two matrices: one with the first row/col and the second with only zeros, the other with the second row/col and the first with only zeros. Since the determinant of both will still be zero.

[cmilkau]

I'd say rank 1 matrices is a more useful description than zero-determinant 2x2 matrices, particularly if you already know that nonzero determinant means full-rank.

[TheOneMaddin]

So here is what I don't get: why the complicated first half of the video if it only works if x, y, z are complex, which they are not? You then split the Pauli vector it into four summands where the factoring is trivial. Why not like that from the start?

[cmilkau]

2x2 Hermitian matrices can always be represented as a sum of two rank-1 matrices. Four seems a bit excessive. In general, diagonalizable nxn matrices (of which Hermitians are an example) can be represented as sum of n rank-1 matrices.

[ravikantpatil]

I also so astonished that mathetical power is very powerful.chris explained so nice in Hindu scriptures vishwRup darshanam❤❤❤

[M.athematech]

@8.50 you are distributing square roots over complex factors ... which is the sort of thing that leads to -1 = sqrt(-1)*sqrt(-1) = sqrt(-1 * -1) = sqrt(1) = 1