Introduction to the complex quaternions (Video 3/14).
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- čas přidán 15. 10. 2017
- 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
damtp.cam.ac.uk/people/nf252/
arxiv.org/pdf/1611.09182.pdf
arxiv.org/abs/1603.04078
arxiv.org/abs/1405.4601
www.furey.space - Věda a technologie
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...
I almost understood some of that!
Same
I was an (ik) rotation away from putting it all together.
Then Quanterion conjugates for the win...
jenga!
Same. I am just learning quaternions today.
Glad to see people smarter than me.
Excellent lecture series!!! :)
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 :-)
nominative determinism certainly
of course, hopefully you work in a Bank too
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
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.
Very clear!
very clear explanation. Thanks
I purchased "Group Theory in a Nutshell for Physicists" and I'm finally understanding these videos.
Can S be any real linear combination of the six elements, or is it a specific one?
Cohl may need to teach millions of students at a time in the coming years in VR
i commutes with everything. That's important!
Nope. Only true if your calculations are in the plane of the vector. At first I didn't get her making a distinction between complex i and quaternion i but I get it now since you brought up the commute thing.
Mam...
Please give some example for triple representation quaternion hermitian matrix
Had to go do some reading about quaternions, even wrote a Python class implementing them, but I’m with you so far...
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?
No she didn't. See 2:41 where there is a factor of i in the exponential. Physicists usually set things up in this way so that eigenvalues of spin operators come out real.
@@cohlfurey8766 Holy crap thank you for replying. Trying to teach myself physics online so I really appreciate the outreach!
@@SirTravelMuffin My pleasure - it sounds like you think about these things as do mathematicians. In the end, theirs is the best way.
Leptons changing flavour
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?
Yes, of course.
@@cohlfurey8766 Thank you for the rapid response! I am just strolling around CZcams, checking the videos which explain subjects that Geometric Algebra reframes and "explains". GA and GC are still completely unknown to most scientists, and absent from most curricula. In my opinion this is a major obstacle in developing physics' theories and understanding the universe.
@@AndreaCalaon73 I'm interested, what are you talking about?
@@iro4201 I replied twice, but the reply keeps disappearing.
@@iro4201 Look for David Hestenes on Google
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.)
SL(2,ℂ) is actually a group not an algebra (sl(2,ℂ) would be the corresponding algebra). To properly understand that you have to study Lie theory. The exponential function roughly speaking relates the Lie algebra with its associated Lie group ( en.wikipedia.org/wiki/Exponential_map_(Lie_theory) ).
For physics, at the end of the day it's all Euler's formula where e^(i theta). For quaternions :
e^(ipi)=-cos theta+isin theta=ii=jj=kk=ijk=-1. For physics this is all you need.
Apparently S is an alternative expression of the radian angle theta. Euler's formula is THE wave equation. There is nothing in physics that can't be expressed in terms of Euler's formula.
This is the most complex question I ever saw in the comment seciton
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.
Jeez she writes fast
Doodelay Explains, I have never noticed that. I have been paying attention to maths, while forgetting her writing speed. You can write extra fast, if you practise cursive handwriting and Gregg shorthand. Try speed drawing too.
Fury please speak of the theory of everything and your math perspective.
The theory of everything is Euler's formula.
Is there a reason you exponentiate i*s instead of just s? (Both are real linear combinations of elements of the lorentz algebra.)
Such a great explanation! Where were you when I was studying algebra?
... Same age as I am, so I guess you were still in school too.
FYI, I'm totally going to use that triangular octonian relational chart for non-scientific purposes.
so basically we can get much of QM and special relativity from quaternions. So how come it's not popular to teach undergrads this?
News flash. The whole of physics is nothing but the study of the interaction of the units of mass, charge, length, and time expressed as the 4 quaternions i,j,k, and -1. This is all concisely expressed within Euler's formula which is THE wave equation.
So that's how she spends her time after Kill Bill trilogy. 👍🏽👍🏽
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.
Are you shure, that she knows something about geometric algebra? It would be intresting to see the relations to geometric algebra expecially where to find the octoniens in geometric algebra?
P.S The presentation is terrible. I don't like the chalk tok tok. She expects that you know every thing she talks about. Nothing is deduced from first principles.
@@hermannwacker1902 Yes. The basic idea of multiplying directions flipping sign when you commute them. Clifford Algebra is GA; with a different focus and notational differences.
Even my 10th grade daughter knows the very basics of GA. You don't need to talk about wedge or dot, or any of that stuff at all. Change the axes to be "(right up)"... makes sense. She did 2D rotations with normal matrices last year. All the crazy complicated stuff that tries to avoid ever using coordinates? No. That's not really the essence of it anyway. You can reproduce all of trig by explaining what "(right up)" means.
I see GA all over this presentation, even though the notation is kind of obscure compared to typical GA notation.
What is Boost
It's a chocolate bar
en.wikipedia.org/wiki/Lorentz_transformation#Boost_in_the_x-direction
It's a file system
transformation between uniformly moving reference frames
A rotation-free Lorentz transformation
COHL
why is so much missing?
probably because its a 4 minute video in a series of short videos
wait what.. her name spells quaternion??
I saw it too, on the left side of the blackboard "C (x) H l" ;-) Indeed a striking coincidence for a woman who is passionate about octonions.
Comments in other episodes of this lecture also talk about this BTW, and even in more detail ;-)
Baila Hie Cool. Good to know. Thank you!
This is the hottest professor I have even seen especially at this age and status. Great diet control and physique control. Mad respect!!!
i dont know, im just here because of The 100
Makepeace
Okay I was with you until you started introducing sigma. I think I'm just not literate enough in mathematics.
mommy
I had to stop watching because I couldn't stand the sound of the chalk on the board...