Spinors for Beginners 12: How the Spin Group Generalizes Quaternions to any Dimension

0:00 - Introduction
2:45 - Terminology overview
4:00 - Reflections in 3D space
9:00 - Reflections in 4D spacetime
13:20 - Rotations in 3D space
22:07- Exponentials
26:37 - Rotations + Boosts in 4D spacetime
32:07 - Galilean Boosts
32:42 - Spin(n) Groups
40:27 - Grade Involution
42:07 - Spin(p,q) Groups
44:15 - Transforming Multi-vectors
46:34 - Hestenes Definition of "spinor"

[anassbellachehab]

Sir this is the best video series on spin that I found on the internet. Thanks for your hard work and dedication.

[metrictensor]

Chris. I love your site. As a math and physics geek from the 70s, I often ask "where was eigen Chris back then". I love your point of view. Thanks. MetricTensor!

[lame_lexem]

the pin group being derived from crossing s in spin group is just perfect
we must have more tomfoolery like that

[naidoeshacks]

I've been thinking about this video dropping for weeks, so glad it finally did!

[AlfredDiffer]

Thank you. I'm preferring your take on the terminology. We all headed straight into confusion a few decades ago when we couldn't agree on what 'spinor' meant to those of us using clifford algebras in physics. Hestenes had his way, but I'm come to prefer 'rotor' for the defining element of 'rotation' operations. Spinors should be what we need them them to be for quantum and polarization. 8)

[superk]

11:23 The hyperbolic angle is defined as the half area according to the arc length along the hyperbolic curve for my knowledge. So I think that the end point of Vreflected should be positioned such that the reflection axis (cyan line) divide the area of the diagram composed of V, Vreflected and hyperbolic curve exactly in half (or divide the length of arc from V to Vreflected along the hyperbolic curve exactly in half).

[ericbischoff]

I find it a bit disturbing to use "length" for the number of vectors in a multivector, as it has little to do with the usual concept of length. "multiplicity" would have been better IMHO.
As usual, thanks for the great contents.

[teodoroalves]

Great video, as always! Thank you :) When you say that V is outside of the plane in 19:22, does it mean perpendicular to the plane?

[Narutonokia]

Great video as always, keep the good work, we appreciate it!

[Juergen-bu]

Great Video! As always! Between 31:26 and 32:06 should it be Gamma_x instead of Gamma_z in front of sinh?

[ffs]

Performing the inverse of the flim-flam operator is what always trips me up, help! Kind request to cover that next video!

[erikstephens]

"I'm in pain without the a"
"Wait, you're in pin?"

[longsarith]

Have you been focusing on Quantum field ?

[pelegsap]

Amazing as always.
I wonder, are Cl(m, n) and Cl(n, m) homomorphic for any natural n and m? I know that it's true for 1, 3 since the n special relativity the difference between spaces with signatures +--- vs. -+++ is only up to a sign.

[davidharris]

Hi Chris, part way through another good video. I couldn't understand the formula for tau at 25:29 in the last line (is tau here a unit vector?), or where the alpha squared comes from in the equation.

[cmilkau]

I find it very confusing that when talking about just space, we use Cl(3, 0), while when taking about spacetime, we use Cl(1, 3), so the σ_x, σ_y, σ_z change behaviour. That seems like a quite unnecessary cognitive load

[RussellCreek-bg]

I don't get the diagram at 32:00. Reflecting the mauve line (in -t direction) across the mirror (pale red) would give a line near the -x/+t lightline but the diagram shows the purple line near the +x/+t lightline. What am I missing or is this a mistake?

[DavidSartor]

3:01
This only gives k-blades, not k-vectors generally.

x1^x2+x3^x4 is a 2-vector, but not a 2-blade, so it can't be made by wedging vectors. Right?

[cmilkau]

23:41 to complete the proof you also need to prove absolute convergence (so you can reorder the terms), which amounts to proving e^θ is finite

[Duskull]

Its a bit unrelated to this point in the series, but how does ths spinor(n) spaces relate to the spin ½, 1, 1½...? If i remember correctly all those spin systems like the spin ½ system are the unitary representations of SO(3) in various dimensions. But now i am a bit confused about how all of this relates to the physicists and chemists point of view. Im guess you are going to go over it in the section about lie groups