Spinors for Beginners 10: SU(2) double covers SO(3) [ SL(2,C) double covers SO+(1,3) ]

0:00 - Introduction
3:05 - Real projective spaces RP^n
7:29 - SU(2) double-covers SO(3)
11:02 - Simply Connected spaces
14:34 - SL(2,C) double-covers SO+(1,3)
20:34 - Mobius Transformations
23:10 - Spin Groups

[stevelamprou]

Let me express it once more, you are a gift to the community. Thank you.

[MaxxTosh]

God I can’t thank you enough for all the hard work and time you put into these videos, they’re truly a gift

[makespace]

Your buildup of the double cover has given me a much clearer understanding of it. Thanks!

[monadic_monastic]

I know this is a series about spinors, but man do I wish I had your vids back in early 2020 when I was taking an algebraic geometry course! (Particularly the parts that required projective geometry). It was a class intended for undergrads funnily enough, but it was pretty challenging.

[.]

Episode 10 releasing after episode 11 is like when the 3d orbital is higher energy than the 4s orbital.

[AkamiChannel]

You are an absolute giant. I've been working on this stuff/stuff adjacent to it lately and this is extremely helpful.

[HelloWorld-lvwe]

This video is simply amazing! Very well done sir. Thank you

[rustybolts]

I WAS, getting depressed because every theory I had considered for years, were all crashing in flames and all I knew is something big was missing. NOW the sun is shining in my mind again, thanks to your infinitely enlightened videos. Now I know that God and infinity are the same entity. Thank you more than words can say.

[sietsebuijsman]

Hey, I would like to say that I really appreciate your videos and the work you put in! I have been watching the Tensor Calculus playlist for a while now and you make everything so incredibly clear, much clearer than what I encountered in university. It really helps me build an intuition for what spinors are and how they relate to regular vectors. Hope you can and will continue with your videos! have a nice day.

[angeldude]

U and -U are _not_ the same _rotation._ They _are_ the same _orientation._ Rotation matrices don't actually encode rotations, but rather orientations. SU(2) matrices, quaternions, and rotors however encode _rotations._ For every orientation other than 0 and 360°, there are 2 distinct rotations that lead to it. In 2D, These are usually called "clockwise" and "counter/anti-clockwise" (or "widershins").

There's also the fact that 180 degrees around the y axis is a completely different orientation than 180° around the x axis. (This is trivial to show with any physical object.) The rotations around each axis therefore can't collapse into a scalar until having already rotated 360°. In addition, said scalar must be -1 in order for the rotation to be a traditional spherical rotation rather than a hyperbolic rotation.

The contradiction with Born's rule doesn't seem too hard to resolve. Just redefine orthogonality. Two vectors are orthogonal if their inner product with respect to a given metric is 0. In the Euclidean metric, this is equivalent to vectors being at right angles, but in a Minkowski metric, vectors that don't appear to be orthogonal can still be such, and vectors that appear to be at right angles might not be orthogonal.

[CarlosRodriguez-mxxy]

For Dante, Virgilio.
For us, Eigenchris.

[mesterfriend]

Could you please tell us Which books you follow? This is indeed great effort from a passionate person

[meahoola]

So many concepts I have seen here and there in physics, and only now they become parts of a bigger picture... 👍

[Tstopmotion]

Very nice! Thank you for these videos!

[MGoebel-ce]

Thanks for this overview. There is one thing which confused(s) me: you say multiple times across the series that “you can express a 3D vector with a Pauli matrix”. That is certainly true - what troubled me however is why a 2*2 complex matrix is NEEDED for both 3D and 4D vectors (as in spacetime).

The answer could be the following: Not ONLY a Pauli matrix can express a 3 D vector. ANY (incl real-valued matrix) can do that. Right? Actually not sure…) A Pauli matrix is only needed for a special subset of 3D vectors - those relevant for Pauli:) So these vectors are likely better thought of as some sort of 4D vectors which suppress a dimension. If this is so it would help a lot of you make that wording clearer. This is all about navigating between nD-spaces after all. Thanks for considering!

[hydraslair]

Ecstatic to see where the next videos will take us.

What I'm really looking to learn in this video series is a mathematical explanation of why "rotating a half-integer spin particle once on itself will result in the spinor picking up a minus sign", possibly within the context of Dirac's equation.

I understand that this probably has to do with double covers. It would make sense if "running around SU(2) once" was equivalent to "running around SO(3) twice", because it would mean thst running around SO(3) once (i.e. full 360 rotation) is equivalent to running around SU(2) half-way and stopping at the negative.

Sadly it still doesn't include an explanation of integer spin particles.

[GeoffryGifari]

i'm trying to visualize how to map SU(2) matrices to a 3-sphere and SO(3) matrices to projective 3 space (just a volume?)

its not clear to me how matrices can correspond to points on a surface, so please correct me if i'm wrong:

- SU(2) transformation can be parametrized by three angles multiplied by the basis (pauli matrices)
M = θx σx + θy σy + θz σz
by selecting one value for each angle and rotating from the origin, we can locate where that matrix is in the space of SU(2) matrices (the entire 3-sphere)
it transforms a pauli vector V by
V' = M V M^(-1)

- for a 3-vector

|x|
|y|
|z|

we can rotate around the z, y, and x axes respectively with the matrices

| cos α -sin α |
| sin α cos α |
| 1 |

| cos β sin β|
| 1 |
| -sin β cos β |

| 1 |
| cos γ -sin γ |
| sin γ cos γ |

we can make the angles themselves as a vector ( α β γ ) that corresponds to a point in a 3-space (volume)

[GeoffryGifari]

interesting to note that the boosts can become rotations if we (somehow) make the boost angle imaginary
φ -> iφ
as seen using exponential form of trig and hyperbolic trig functions

[monadic_monastic]

Would it also be accurate to say that Clifford/Geometric Algebras basically provide a way to *summarize* all of our work we've been building up to (using seemingly different tools)?

[willemesterhuyse]

They say Special Relativity treats time and space the same. This isn't true: there is a +1 for time and -1 for space signature.