Spinors for Beginners 6: Pauli Vectors and Pauli Matrices

Sources:

0:00 Introduction
1:02 Pauli Matrix Properties
5:48 Pauli Vector Reflections
11:11 Pauli Vector Rotations
17:24 Proof of double-sided SU(2) formula
23:28 Form of SU(2) matrices, parameter counting
25:32 Summary

[orktv]

I like how you right off the bat show that Pauli matrices are like unit vectors endowed with the special property that you can actually multiply them (and that therefore squaring a Pauli vector amounts to finding the magnitude squared). Maybe it's just me but I think it's very important that the Pauli matrices are introduced as initially abstract objects, that can be cast in a more familiar representation of matrices, lest I start to look way too deep into what it means that 2×2 matrices are now our vectors.

[abhijithcpreej]

11:53 this blew my mind beyond repair; not that rotation can be viewed as two reflections, but the angle doubling relationship here that connects back to the behaviour of spinors

[ditch]

I've struggled for so long to understand it all, but thanks to you it's all falling into place now. Thank you so much for the videos.

[nice]

I'm loving how comprehensive this series is

[winterturtle]

I love how approachable this series is. I have an undergraduate degree in mathematics, and feel like I have more than enough background to fully understand what you are doing without any experience with the physics.

[nm-com]

in 20 years i have not found a better explanation on spinors than this video series. you break it down to the most crucial properties, and from what i get you leave all the irrelevant rest. at least i really have the feeling i understand step by step why the spin is formulated the way it is. i come back at some of your videos and watch them again. i wish you many many more subscribers. there are very very few channels as didactic as this.

[shafiulhossain]

I'm in great debt of yours. Your videos helped me a lot. Can't Thank you enough.

[kikivoorburg]

I discovered for myself that the Pauli Matrices are not as 'arbitrary' as they may seem when I ended up rederiving them from scratch.

I had been exploring geometric algebras (clifford algebras) after being introduced to them by sudgylacmoe's great video "a swift introduction to geometric algebra", but found remembering the entirely new algebraic rules a little confusing. However, I had remembered that the imaginary unit 'i' can be represented by a 2x2 matrix (and obviously the identity matrix acts like the real unit '1'), so wondered whether or not there would be 2x2 matrices that acted like 2D G.A.'s 'x' and 'y' vectors. I reasoned there could be, since a 2x2 matrix has four degrees of freedom -- only two of which were used by '1' and 'i'. Indeed, after solving the relevant systems of equations (xy = -yx and x^2 = y^2 = 1), I found a pair that worked! (These turned out to be -σ_z and σ_x later.)

I kept exploring past that, trying to find a basis for 3D G.A., which at first led me down the futile path of 2x2x2 matrices (which have the same issue as vectors, in that multiplication either bumps you up a dimension (outer product) or down one (inner product). However, 3D G.A. has twice the basis elements as 2D G.A. (1, x, y, z, xy, yx, xz, & xyz compared to just 1, x, y, & xy), so if I could somehow "augment" the 2x2 matrices I found for 2D G.A. with a 2nd degree of freedom for each term, that would work! Since complex numbers are an algebra that is 2D but also 'plays nicely' (often more nicely than the reals), I checked if 2x2 complex matrices worked and indeed they did! I noticed that they looked similar to the Pauli Matrices, and they turned out to be equivalent!

(I had labelled x = -σ_z, y = σ_x, and z = -σ_y)

I think it's amazing how these seemingly 'arbitrary' matrices actually appear quite naturally when exploring the maths involved in 3D space. I am by no means a professional mathematician, so the fact I could stumble across them entirely on my own shows how fundamental they actually are!

P.s. This series has been really awesome so far! I've learnt a lot, and my exploration of other subjects (such as G.A., mentioned here) has been helped a lot by the added context and detail you've presented! Looking forward to what comes next (especially when you get to the Clifford Algebra perspective, as that will probably inspire my explorations even further)!

Edits: Small typos and clarifying the basis I found.

[GeekyNeil]

Thanks for the videos! I've been interested in spinors for a while but haven't managed to get my head around them previously. Now with this series they are starting to make sense. I love your attention to detail.

[jordanchen]

Thank you much for making such clear video, I can't wait to see the next one. This series of videos solved the doubt I have when I was in physics department as a undergraduate student.

[EzraAChen]

Very clear interpretation with tone posed and pitched with clarity and pace moderate for voice auditory lobe reception and hippocampus formation spatial short and long term memory transition. All well practiced. And the content is logical. Recommend more Conceptualization. Since computation must lead to a Concept. A La Alex Grothendieck. Du boir Marie.

[uff]

saved my day! thank you! Straight direct to the point. actually explaining something. and fast, my brain stays focused and gets happy 😁😊

[TheMultifun]

Babe wake up a new Eugenchris video just dropped

[scottcarothers]

So many things here are reminding me of geometric algebra. Especially that sandwich product for reflections!

[twoonesixsixonetwo]

thank you for teaching everyone the negative sandwich operation we will be forever grateful

[marshallsweatherhiking]

Thank you! The fact that you don’t leave out any details makes this much more satisfying than anything else on the internet, yet still easier to follow than a typical text book. I feel like I need a refresher on linear algebra though. I know that 2 and 3 dimensional stuff can mostly be found by just plugging and chugging with algebra, but I know there are way more elegant proofs for arbitrary numbers of dimensions. I just never got to the later half of the course covering complex valued matrices. I don’t know if I can even find my old book. :(.

[frozenturtl]

my brain is not big enough to comprehend this fully. i've been watching this video and for the past two weeks trying to understand, but alas i am not this comfortable with linear algebra and complex numbers to throw them together so casually. gotta study up, i hope i can understand this video and the rest of the series soon.

[AMADEOSAM]

Thanks for all your work! All very educative.

[angeldude]

I disagree that the calculations get tedious if you only work with the abstract vectors rather than the matrices. After all, it just ends up as exp(θ/2 σ_x σ_y), since it's possible to show, using the algebraic properties covered, that (σ_x σ_y)² = -𝟙. It is also just very satisfying watching the limit definition of the exponential, when applied to _these objects, _ slowly increase the number of iterations.

Regardless, this was a pretty good thinly veiled introduction to Clifford Algebra. I have some personal gripes regarding "reflecting along a vector" rather than "reflecting across a plane, " but they're just different geometric interpretations of the same algebra. The latter extends better when you have mirrors that don't pass through the origin or aren't flat, but for now that doesn't matter. It also more clearly shows that reflecting across a line is the same operation as rotating 180°, since the product of two planes is a rotation around their intersection, but if they're perpendicular, then that intersection is just a pure line.

[Bldravnz]

Thank you for the video, I have a question: at 10:40 we note that Vperp U = -U Vperp. I didn't have an immediate intuition, but after writing it down, I assume we define two vectors U and V to be perpendicular in the usual sense with the standard dot product, so that ux vx + uy vy + uz vz = 0. With that in mind, expanding U Vperp will only leave us with the antisymmetric mixed terms, which will pick up a minus sign when we swap the order of multiplication. Hence U Vperp = -Vperp U. Is this correct, or is there an easier way to see this?
Edit: Oh sorry, you show it right after it!