Introduction to the complex quaternions (Video 3/14).

Video 3 of 14

Series: Division algebras and the standard model

Some short videos filmed by Vincent Lavigne

Seminar by C. Furey,
Walter Grant Scott Research Fellow in Physics
Trinity Hall, University of Cambridge

[roberthuber]

Wow! Just stumbled across your work Dr. Furey... love it. Looking forward to viewing your lectures while taking notes alongside. This is the kind of stuff I'm interested in - the intersection of physics and mathematics. I'm interested in studying this stuff for the rest of my life...

[crocshock]

If you look at what she wrote down at the very beginning of this video, it *almost* looks like it spells out "Cohl".
I don't know if that was on purpose, but it's perfect :-)

[jonathanjackson]

Great videos. But I think this works out in a much more transparent way if you use a Cl(1, 3) algebra.without invoking quarterions ie. Spacetime algebra.

[AkamiChannel]

I purchased "Group Theory in a Nutshell for Physicists" and I'm finally understanding these videos.

[letmeoffendyou]

What's the link between SL(2, C) and the standard model ? I thought that SL(2, C) was the gauge group used in *relativity*.
Interesting video, I didn't know about CxH ~ sl(2, C), can we do the same with other Lie Algebras s.t. su(n) ? what about all the other (simple) Lie algebras ?
(stupid question: If sigma's are rotations and i, j, k are boosts how interpret the complex i in the context of the Lorentz gauge group ?)
Thanks again

[SuperDeadparrot]

i commutes with everything. That's important!

[Eric-xv]

Cohl may need to teach millions of students at a time in the coming years in VR

[perilousgourd]

Can anyone tell me what the significance is of multiplying S by i and exponentiating to form an element of SL(2, C) (an element of the Lorentz algebra?)? What are the physical consequences of this step? Why is that algebra a useful/interesting one to have and **how does the exponentiation and i cause this usefulness**?

(I'm a physics undergraduate with no knowledge in this area yet, so a layperson answer would be acceptable/appropriate.)

[peterboyajian]

Can S be any real linear combination of the six elements, or is it a specific one?

[joshuamowdy]

Hello.

U(1)
Heisenberg Euler in abelien guage theory with parity violation

The dark And the light makes it matter. The expectation values designed to offset the singular spin interests me. Is it possible to single out a small direct set of octonians within a prime construct?
I'm just sayin.


Good luck.

[AndreaCalaon]

Dear Dr. Furey, with all due respect, do you know about Geometric Algebra, Space Time Algebra, Geometric Calculus and the reformulation of the Dirac equation with real spinors?

[SirTravelMuffin]

At 2:27 did she mix these up? If you exponentiate the quaternions you get rotations, and the pauli matrices (clifford vectors) you get boosts? Or when you say "generator" does that imply and extra imaginary element in the exponent?

[jfuller]

wait what.. her name spells quaternion??

[TheWyrdSmythe]

Had to go do some reading about quaternions, even wrote a Python class implementing them, but I’m with you so far...

[dr.m.rahamathunisha]

Mam...
Please give some example for triple representation quaternion hermitian matrix

[theman]

So that's how she spends her time after Kill Bill trilogy. 👍🏽👍🏽

[zhin]

This is the hottest professor I have even seen especially at this age and status. Great diet control and physique control. Mad respect!!!

[Evan]

Is there a reason you exponentiate i*s instead of just s? (Both are real linear combinations of elements of the lorentz algebra.)

[Rachelebanham]

so basically we can get much of QM and special relativity from quaternions. So how come it's not popular to teach undergrads this?

[rrrbb]

this is Geometric Algebra in disguise. e_1, e_2, e_3 are mutually perpendicular directions. (e_1 e2) is a bivector... that rotates from e_1 to e_2. "i", usually written I, is (e_1 e_2 e_3). Each direction squares to 1: e_1 e_1 = 1, and (e_1 e_2) is anirreducable rotation, and squares to -1. Every pair of mutually perpendicular unit vectors squares to -1, ie: (e_1 e_2)(e_2 e_2) = -(e_2 e_1)(e_1 e_2) = -(e_2 (e_1 e_1) e_2) = -(e_2 e_2) = -1. The pseudo-scalar (e_1 e_2 e_3) also squares to -1.