Geometric Algebra my beloved I love the correspondence between these matrices and the quaternions - it's nice to be able to leverage intuition about one part of math to help with another.
And SU(2) is diffeomorphic to the 3-Sphere--the GLOME! Thanks so much for putting your time and effort into these videos. I really appreciate and look forward to every one. ^_^
This is fun. The derivations are a bit long, but that's OK. I did some more tinkering. Writing the Pauli matrices as sx, sy, sz, Since: sy sz = i sx, etc, one can rewrite the quaternion basis as (using double letters:) ii = -i sx, jj = -i sy, kk = -i sz. And then you can turn a Pauli vector into a pure imaginary quaternion by dividing it by (-i), that is, by multiplying it with i. Also, if you have a Pauli vector U , to make a half turn around it, the rotator (rotor?) is simply R = cos(theta/2) Id_2 + i sin(theta/2) U. That's basically Euler's formula: Exp[ i (theta/2) U] (Id_2 is the 2x2 identity matrix). "Proof:" Use Octave (Matlab) with random examples. 🙂That's how I usually find my math errors.
The correspondence for rotation representations shown is rotation about x -> i in quaternions and -sigma_y*sigma_z in matrices whereas for the vector it is x coordinate -> i in quaternions and sigma_x in matrices This is a a bit un-intuitive. It might be better to introduce the rotations as rotation about x -> jk in quaternions It's identical, but better shows the symmetry between the two systems (except for the minus sign)
The geometric algebra formulation of Clifford algebra is useful here - following hestenes and lazenby - a lot of modern physics has been reworked with geometric algebra
Hi, thanks for the answer in another comment. Your videos are neat as usual, there is just a typo in it's title, in the word "quaterions". Have a good day!
Some thoughts.... 1. the product of quaternion and its conjugate looks suspiciously like the minkowski metric! 2. hmmm i thought physicists introduced matrices in the first place because the quantities didn't follow the rules of numbers. is it possible for a real function coefficient of quaternion to behave like a spinor? take for example the Schrödinger-Pauli equation for spin-1/2 particle in a magnetic field. can we write this equation without pauli matrices, using quaternions instead? 3. quaternions can't possibly enter into measurement result.... but wavefunctions are complex, so maybe quaternions can describe a quantum state somehow.... 4. for rotating vectors with quaternion, we use i,j,k and leave the real part zero. I wonder if the full quaternion (including the real part a+ib+jc+kd) is useful to rotate spacetime 4-vectors instead? 5. Are all of this connected to the "grassmann numbers" used while calculating fermionic observables? 6. Is there any special reason why SO(3) rotation is closely linked to SU(2) instead of SU(3) (having the same dimension)? we know that SU(3) symmetry is used in QCD part of the standard model 7. Are all these group structures responsible for the spin in our 3D space + time taking integer values? All this are just thoughts.... I don't expect the answers to come easily
1. The minkowski metric is more closely related to the "Gamma Matrices"/"Dirac Matrices". I'll cover those eventually. You can watch Sudgylacmoe's video "A swift introduction to spacetime algebra" to learn more. 2. I haven't thought about this too much. I suspect you can write it in the language of Clifford Algebra using Cl(3), which lets you write both the sigmas and Pauli spinors. Quaternions are really like "bivectors" and represent a plane, given by a pair of sigmas 3. I don't think quaternions play any special role in quantum. They are mainly used for 3D rotations. 4. Sudgylacmoe's video I suggested in #1 will explain how to write 4-vectors using a technique called a "spacetime split". The sigmas should be viewed as more fundamental than ijk. An individual sigma is like a vector/arrow. A pair of sigmas is like a bivector/plane, which is why pairs of sigmas give us a rotation in a given plane. 5. Sort of. In Clifford Algebras, some symbols square to -1 (such as ijk), some symbols square to +1 (such as the sigmas), and some symbols square to 0. Symbols that square to zero are "grassman numbers". You can look up the "dual numbers" on wikipedia. They are sort of like the complex numbers a+bi, but dual numbers are written a+bϵ, where ϵ^2 =0. 6. SO(3) is a 3-dimensional space. SU(2) is also a 3-dimensinoal space. SU(3) is an 8-dimensional space, so it wouldn't match up with the 3 possible rotations in SO(3). The reason that there are 8 gluons in QCD is because SU(3) is 8-dimensional. The connection you want to make is the double-cover between SO(3) and Spin(3) = SU(2). More generally, there is a double-cover between SO(n) and Spin(n). 7. I'm not sure why spin only takes integer values. I think that is more related to the rules of quantum mechanics and less about the rules of pure math.
This is a great series and I appreciate this one on quaternions which have always seemed weird to me. I notice that in the double sided rotation the quaternion rotation is applied to the quaternion k resulting in a bi-vector which is a combination of the quaternions i and k. This implies that quaternion rotation is really a bi-vector rotation (not a vector rotation). The thing that lets us apply quaternion rotations to vectors is that the coefficients for a given vector are the same as for bi-vector coefficients for the bi-vector normal to that vector. Is this correct?
Yeah. In the geometric algebra algebra point of view, quaternions are bivectors/planes, not vectors. If you have a pair of sigmas, and you use the double-sided transformation on each of them, the "inner" terms will cancel out, so it's the same as doing a double-sided transformation to the bivector.
Differential forms live in the "exterior algebra", also called the "Grassman Algebra". A Clifford Algebra is like an upgraded version of an Exterior Algebra, so the "dual" operation (also called the "Hodge Star") also exists.
@@eigenchris I had a question regarding the Pauli vectors video (no. 8 in the series) and commented under it. Could you please check it out if you can? Thanks in advance
7:55 by ‘’« they » are isomorphic’’, I assume you mean the four dimensional algebra generated by the linear basis {identity} U {- σ_i σ_ « i+1 » | i = x,y,z } (with standard matrix add and mult) (is isom to the quats)
I’m curious what software you use to make these videos? I have some experience making videos, but making labels and diagrams is so tedious and inefficient with the free tools I have. Not to mention I have no idea to get all the nice math fonts needed.
Yes. This is what we call a "Low dimensional coincidence". In low dimensions, you can come up with several different definitions for various groups, and then find out they are the same. Spin(3) is defined as the "double cover" of SO(3). And SU(2) is defined as the 2x2 unitary matrices with determinant = +1. It turns out these definitions describe the same group. If you watch video #9, you'll hear me talk about the SL(2,C) group, which ends up being the same as the symplectic group Sp(2,C), and also the same as Spin(1,3). As dimensions get higher, these coincidences between different definitions happen less and less often.
is it true that the sigma matrices are equivalent to the standard i multiplied by a quaternionic i, j or k as in Cohl Furey's video? ruclips.net/video/3AAK-4KdGXs/видео.html
Geometric Algebra my beloved
I love the correspondence between these matrices and the quaternions - it's nice to be able to leverage intuition about one part of math to help with another.
Would you consider doing videos that focus on clifford algebra or algebraic topology?
i think its next in the series
It’s the next chapter
@@p.muskett2931 I am talking about a video that only focuses on these concepts, not for spinors.
I'll be going over Clifford Algebra and how spinors show up in them in videos 11-15, roughly. I don't plan on doing algebraic topology.
And SU(2) is diffeomorphic to the 3-Sphere--the GLOME!
Thanks so much for putting your time and effort into these videos. I really appreciate and look forward to every one. ^_^
really nice! I think finding isomorphisms between different mathematical objects is my favourite thing in math, always blows my mind
I'm currently learning geometric algebra and this gives me a strong sense of deja vu 😁
(obviously, as all of these topics are intimately related)
Mind-glowing! Thanks a lot!
This is fun. The derivations are a bit long, but that's OK. I did some more tinkering.
Writing the Pauli matrices as sx, sy, sz, Since: sy sz = i sx, etc,
one can rewrite the quaternion basis as (using double letters:) ii = -i sx, jj = -i sy, kk = -i sz.
And then you can turn a Pauli vector into a pure imaginary quaternion by dividing it by (-i), that is, by multiplying it with i.
Also, if you have a Pauli vector U , to make a half turn around it, the rotator (rotor?) is simply R = cos(theta/2) Id_2 + i sin(theta/2) U.
That's basically Euler's formula: Exp[ i (theta/2) U]
(Id_2 is the 2x2 identity matrix).
"Proof:" Use Octave (Matlab) with random examples. 🙂That's how I usually find my math errors.
The correspondence for rotation representations shown is
rotation about x -> i in quaternions and -sigma_y*sigma_z in matrices
whereas for the vector it is
x coordinate -> i in quaternions and sigma_x in matrices
This is a a bit un-intuitive. It might be better to introduce the rotations as
rotation about x -> jk in quaternions
It's identical, but better shows the symmetry between the two systems (except for the minus sign)
Thanks again! Great explanation and moving towards geometric algebra is very smart. Keep ongoing …
Awesome video. Thanks.
yes, more new stuff!
Awesome, good job chris.
Amazing video 🙏😎
Cohl furey has a nice video on relations between biquaternions and sigma matrices
The geometric algebra formulation of Clifford algebra is useful here - following hestenes and lazenby - a lot of modern physics has been reworked with geometric algebra
Thanks Chris. Good job!
Hi, thanks for the answer in another comment. Your videos are neat as usual, there is just a typo in it's title, in the word "quaterions". Have a good day!
And a typo in your comment... "It's" = "It is."
@@wafikiri_ And a typo in your other comment: Mind-"b"lowing, not mind-"g"lowing.
Thanks for pointing that out. Fixed.
@@viliml2763 It wasn't a typo but intentional. My mind glowed.
@@wafikiri_ "It's" is not a mistake. This is standard English taught in any English textbook.
Some thoughts....
1. the product of quaternion and its conjugate looks suspiciously like the minkowski metric!
2. hmmm i thought physicists introduced matrices in the first place because the quantities didn't follow the rules of numbers. is it possible for a real function coefficient of quaternion to behave like a spinor? take for example the Schrödinger-Pauli equation for spin-1/2 particle in a magnetic field. can we write this equation without pauli matrices, using quaternions instead?
3. quaternions can't possibly enter into measurement result.... but wavefunctions are complex, so maybe quaternions can describe a quantum state somehow....
4. for rotating vectors with quaternion, we use i,j,k and leave the real part zero. I wonder if the full quaternion (including the real part a+ib+jc+kd) is useful to rotate spacetime 4-vectors instead?
5. Are all of this connected to the "grassmann numbers" used while calculating fermionic observables?
6. Is there any special reason why SO(3) rotation is closely linked to SU(2) instead of SU(3) (having the same dimension)? we know that SU(3) symmetry is used in QCD part of the standard model
7. Are all these group structures responsible for the spin in our 3D space + time taking integer values?
All this are just thoughts.... I don't expect the answers to come easily
1. The minkowski metric is more closely related to the "Gamma Matrices"/"Dirac Matrices". I'll cover those eventually. You can watch Sudgylacmoe's video "A swift introduction to spacetime algebra" to learn more.
2. I haven't thought about this too much. I suspect you can write it in the language of Clifford Algebra using Cl(3), which lets you write both the sigmas and Pauli spinors. Quaternions are really like "bivectors" and represent a plane, given by a pair of sigmas
3. I don't think quaternions play any special role in quantum. They are mainly used for 3D rotations.
4. Sudgylacmoe's video I suggested in #1 will explain how to write 4-vectors using a technique called a "spacetime split". The sigmas should be viewed as more fundamental than ijk. An individual sigma is like a vector/arrow. A pair of sigmas is like a bivector/plane, which is why pairs of sigmas give us a rotation in a given plane.
5. Sort of. In Clifford Algebras, some symbols square to -1 (such as ijk), some symbols square to +1 (such as the sigmas), and some symbols square to 0. Symbols that square to zero are "grassman numbers". You can look up the "dual numbers" on wikipedia. They are sort of like the complex numbers a+bi, but dual numbers are written a+bϵ, where ϵ^2 =0.
6. SO(3) is a 3-dimensional space. SU(2) is also a 3-dimensinoal space. SU(3) is an 8-dimensional space, so it wouldn't match up with the 3 possible rotations in SO(3). The reason that there are 8 gluons in QCD is because SU(3) is 8-dimensional. The connection you want to make is the double-cover between SO(3) and Spin(3) = SU(2). More generally, there is a double-cover between SO(n) and Spin(n).
7. I'm not sure why spin only takes integer values. I think that is more related to the rules of quantum mechanics and less about the rules of pure math.
This is a great series and I appreciate this one on quaternions which have always seemed weird to me. I notice that in the double sided rotation the quaternion rotation is applied to the quaternion k resulting in a bi-vector which is a combination of the quaternions i and k. This implies that quaternion rotation is really a bi-vector rotation (not a vector rotation). The thing that lets us apply quaternion rotations to vectors is that the coefficients for a given vector are the same as for bi-vector coefficients for the bi-vector normal to that vector. Is this correct?
Yeah. In the geometric algebra algebra point of view, quaternions are bivectors/planes, not vectors. If you have a pair of sigmas, and you use the double-sided transformation on each of them, the "inner" terms will cancel out, so it's the same as doing a double-sided transformation to the bivector.
Using an axis in one case and a plane in the other for the same rotation seems analogous to duals of differential forms
Differential forms live in the "exterior algebra", also called the "Grassman Algebra". A Clifford Algebra is like an upgraded version of an Exterior Algebra, so the "dual" operation (also called the "Hodge Star") also exists.
@@eigenchris :D that's so cool! Thank you :)
@@eigenchris I had a question regarding the Pauli vectors video (no. 8 in the series) and commented under it. Could you please check it out if you can? Thanks in advance
Nice as I imagined quaternions ijk are planes.
7:55 by ‘’« they » are isomorphic’’, I assume you mean the four dimensional algebra generated by the linear basis {identity} U {- σ_i σ_ « i+1 » | i = x,y,z } (with standard matrix add and mult) (is isom to the quats)
I’m curious what software you use to make these videos? I have some experience making videos, but making labels and diagrams is so tedious and inefficient with the free tools I have. Not to mention I have no idea to get all the nice math fonts needed.
I use Microsoft Powerpoint. It has an equation editor and basic animations, and an "export to video" tool.
@@eigenchrisThanks!
2:10 noooo please make the arrow turn clockwise direction 😢
Spin(3) = SU(2)? So confusing with different names for the same thing.
Yes. This is what we call a "Low dimensional coincidence". In low dimensions, you can come up with several different definitions for various groups, and then find out they are the same. Spin(3) is defined as the "double cover" of SO(3). And SU(2) is defined as the 2x2 unitary matrices with determinant = +1. It turns out these definitions describe the same group. If you watch video #9, you'll hear me talk about the SL(2,C) group, which ends up being the same as the symplectic group Sp(2,C), and also the same as Spin(1,3). As dimensions get higher, these coincidences between different definitions happen less and less often.
What about error correction codes series?
Sorry but I don't plan on finishing it. I've lost interest.
@@eigenchris OK, Thanks a lot that you share it at all :)
How can we contact you by email?
I'd generally prefer questions be asked here in the comments.
dagger
is it true that the sigma matrices are equivalent to the standard i multiplied by a quaternionic i, j or k as in Cohl Furey's video? ruclips.net/video/3AAK-4KdGXs/видео.html
I believe so. Off the top of my head, taking 𝓲 as the complex unit: k ≈ -σx*σy = - 𝓲 *σz. So therefore, 𝓲 k ≈ σz.
@@eigenchris thanks!
I am Just wandering what is the mathematics for non inertial frame... I mean, for a accelarating observer , the light does not travel at c, right?