There is a LOT more to it, so consider this a start. See Chapter 8 in “Geometric Algebra for Physicists” by Doran, that is the source of the material. And thank you for your kind comments, I appreciate your thoughts,
First let me join the others in thanking you for your return! (I am looking forward to your lectures on the Space Time Calculus which I had trouble with when I first tried to work through Doran et al.) A question about eq'n 3.62. Although there is no reference to it here, 3.62 implies that F can include complex bivectors as in the spacetime split (eq'n 3.50). I'm not familiar with spinors so asking if this extends to spinors as well?
Love your work. I've watched for several years and been a supporter on Patreon for a year or more. I may have a problem with Patreon. While they seem to have no problem taking my $, they don't recognize me as your supporter. I'm wondering if my dollars are finding their way to you or if they are just sticking to Patreon. The temptation is to cancel the support and try again from scratch. Thoughts? Again, love what you do. Thanks, Andrew Riley
@@XylyXylyX in a way yes, and undoubtedly your connection between normalized spinors and rotors is a fun demonstration of it. But I disagree with the authors choice to call something a spinor which does not transform under some representation with half integer spin. It seems pedagogically foolish. Spinors are confusing enough to students and we already have a name for the even subalgebra of the clifford algebra, it's called the spin group. Furthermore the relationship is surely muddled when we leave the Bloch sphere, as is the case with Dirac and Weyl spinors, since their normalization is no longer a probability density which need be normed to one.
@@TrailersReheard But it does transform under half-integer spin. In fact that is one of the things that makes this interesting in STA: it is completely natural to find spin 1/2 mathematics immersed in the pure geometry of STA. The “phi/2” of the spinor seamlessly appears in the STA’s representation of spinors. However, you do bring up an interesting point: How do we represent spin-1, spin- 3/2, …. and higher spinors in the STA?
@@XylyXylyXI believe i "get it" now, based on additional lectures by Cohl Furey and EigenChris. You can identify with every spinor, an element of the algebra - and if we transform not by conjugation, but by left or right multiplication, we see that of course the algebra element is rotated by half an angle, not a whole angle like a vector. My lack in understanding then is why do we throw away transformation by conjugation? It seems disastrous to have our vectors and spinors be made out of literally the same objects. How do I know how to transform which one, and when? Am I still misunderstanding?
Great video! Every Xylyxylyx video makes me want to live a bit more.
Thank you for your kind comment!
REEEEEAAALLLY APPRECIATE THESE GA VIDEOS bcuz u break it down to da bitty gritty ta fully interpret da papers which is time consuming!
49 :34 tracking differences. You rock. Thx
Great episode! I've been looking forward to the STA treatment of spinors for a long time, and you've summed it up perfectly, as always.
There is a LOT more to it, so consider this a start. See Chapter 8 in “Geometric Algebra for Physicists” by Doran, that is the source of the material. And thank you for your kind comments, I appreciate your thoughts,
Awesome video as always! I feel way too blessed that you make this subject so easy to follow.
Thank you for your kind comment
First let me join the others in thanking you for your return! (I am looking forward to your lectures on the Space Time Calculus which I had trouble with when I first tried to work through Doran et al.) A question about eq'n 3.62. Although there is no reference to it here, 3.62 implies that F can include complex bivectors as in the spacetime split (eq'n 3.50). I'm not familiar with spinors so asking if this extends to spinors as well?
You okay, Mr Xylyxylyx? It's been a while. Hope all is well.
Will you continue the series? :(
Yes. Thanks for your patience. I do want to finish this paper…..!
At around the 4 minute mark it should be the hyperbolic sine and cosine since sigma3 squares to 1.
Yep! I noticed this also….I’ll mention it in the errata of the next lesson. Thank you for noticing!
Love your work. I've watched for several years and been a supporter on Patreon for a year or more. I may have a problem with Patreon. While they seem to have no problem taking my $, they don't recognize me as your supporter. I'm wondering if my dollars are finding their way to you or if they are just sticking to Patreon. The temptation is to cancel the support and try again from scratch. Thoughts? Again, love what you do. Thanks, Andrew Riley
Thank u for your kind comment and support. I am earning a little from Patreon. I will check your name to put you at ease (or confirm your suspicion 🤨)
Seems more than foolish to call something a spinor when it does not transform under a projective representation of the lorentz group
Wasn’t this entire lesson a demonstration of a projective representation?
@@XylyXylyX in a way yes, and undoubtedly your connection between normalized spinors and rotors is a fun demonstration of it. But I disagree with the authors choice to call something a spinor which does not transform under some representation with half integer spin. It seems pedagogically foolish. Spinors are confusing enough to students and we already have a name for the even subalgebra of the clifford algebra, it's called the spin group.
Furthermore the relationship is surely muddled when we leave the Bloch sphere, as is the case with Dirac and Weyl spinors, since their normalization is no longer a probability density which need be normed to one.
@@TrailersReheard But it does transform under half-integer spin. In fact that is one of the things that makes this interesting in STA: it is completely natural to find spin 1/2 mathematics immersed in the pure geometry of STA. The “phi/2” of the spinor seamlessly appears in the STA’s representation of spinors. However, you do bring up an interesting point: How do we represent spin-1, spin- 3/2, …. and higher spinors in the STA?
@@XylyXylyXI believe i "get it" now, based on additional lectures by Cohl Furey and EigenChris. You can identify with every spinor, an element of the algebra - and if we transform not by conjugation, but by left or right multiplication, we see that of course the algebra element is rotated by half an angle, not a whole angle like a vector. My lack in understanding then is why do we throw away transformation by conjugation? It seems disastrous to have our vectors and spinors be made out of literally the same objects. How do I know how to transform which one, and when? Am I still misunderstanding?
He said he knew column that is not exactly in some cases necessary