Spinors for Beginners 14: Minimal Left Ideals (and Pacwoman Property)
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- Опубликовано: 9 июл 2024
- Full spinors playlist: • Spinors for Beginners
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Powerpoint slide files + Exercise answers: github.com/eigenchris/MathNot...
Thesis by Crystal-Ann McKenzie: scholar.uwindsor.ca/etd/5652/
0:00 - Introduction
2:01 - Review of Cl(3,0)
4:28 - Fitting Spinors into Cl(3,0)
8:09 - Minimal Left Ideals
11:39 - Projectors
12:55 - Pacwoman Property
14:17 - Calculating Minimal Left Ideal in Cl(3,0)
17:45 - Spin Operators in Cl(3,0)
19:35 - Dual Spinors and Inner Product
24:40 - Spinor Outer Product
27:20 - Hestenes Definition of Spinors
30:18 - Generalizing to Cl(1,3)
37:10 - Generalizing to Cl(p,q)
41:00 - Faithful Matrix Representations
I still believe this series should be called "Spinors for beginors"
These videos are designed to make someone love physics. Many thanks
Hmm very interesting. Even though your video's are extremely good I noticed something you looked over.
The series should have been called "Spinors for Beginnors".
Still, I love your video's. I dont think anyone could explain anything better than you do ❤❤❤.
I'm quite drunk tonight but I have been following these beautiful videos for some time now.. When I'm not drunk I have followed slowly and done videos 1 to 13. I am not a physicist or a mathematician but I wish I was. Thank you for going step by step. I'm so in awe of the beauty of this subject. Thank you for this time.
I've only posted cos i'm blasted. sorry. x
ty so much for this vid. its truly my fav vid on this channel SO FAR
Keep up with this amazing work.
Great video! I noted that physics use the simplest mathematical terms to describe universe. For example Standard Model is based on SU(3)xSU(2)xU(1) symmetries that they are simplest. There is SU(2) instead U(2). Spinors describe spin 1/2 particles such as electrons, quarks. Spinor which are minimal left ideal means that spinor is most elemental object. Complex number can be written by using real numbers in 2x2 matrix where 1 is identity matrix and i is matrix where has -1 on the up right and 1 on down left. If is gives 2x2 matrix with four complex numbers and each complex number will be written as 2x2 matrix with real numbers then we get 4x4 matrix with real element. I noted that taking hermitian conjugate of 2x2 matrix and substituting each complex number with 2x2 matrix with real numbers will get the same result if we take transposition of 4x4 matrix with real elements got from primary complex matrix. So can pauli matrices be written by using 4x4 matrices with real elements and spinor by column with 4 real elements? Going further dirac matrices would be have size 8x8 and dirac spinor column would be a column with 8 real numbers.
Thank you so much, these are super useful! Keep them coming!
this is the best vid so far hehe
Thank you Chris!
Thank you for doing this one. I finally made it through to the end of the video.
I remember my prof grousing at us every time we wanted to write out a matrix representation. He said those hid the geometry from our intuitions. I've still got some of my notes with his red ink on them complaining about resolving into a basis too early as well. For example, the projector in Cl(1,3) with the time-like basis vector is one he used to write as 1/2(1+p) [unit 4-momentum of a particle] because we wouldn't be scared away from imagining how boosts and rotations would work on it. The full projector was similarly abstracted to a particle property that commuted with the momentum. From THERE he expected our physical intuitions to kick in.
I'm honestly having a hard time understanding the 2nd half of your comment. I see what you could take the momentum "p" vector to just be gamma_t in some other reference frame. But I don't really understand how this helps visualize boosts or rotations on spinors. I still don't find that very intuitive.
Oops. I found us. They filed Bill, Forrest, and myself under Glen Erickson. Ken passed away in the summer of '90, so that makes sense.
My understanding is that Lounesto calls the Hestenes spinor a spinor operator. Thus 3 kinds of spinors: column vector, left minimal ideal (algebraic spinor) and spinor operator.
What about an operation that does the inverse of a projector in that it takes an element of the ring/algebra and forms a new ring/algebra with the original left as a subset? In my head it seems like an uninteresting idea because there are potentially infinitely many ways to "open up" a given set definition.
I wanted to how would you construct something similar for a dirac spinor? I also want to add that your lectures on diffrential geometry helped me a lot with general relativity, I can't express my gratitude to you
I cover Dirac spinors around 30:00-37:00. Is there something else you wanted to see?
Glad my other videos have helped.
Hello, very nice video. I'm not 100% certain about the correspondence you establish between Hestenes "elements of even-grade subalgebra" spinors and standard "minimal left ideal spinors". I believe minimal left ideal spinors correspond to "standard" Dirac spinors which are also characterized as elements of the irreducible representation of Clifford algebra. Apparently, for a vector space with dimension d the complex dimensionality of the Dirac spinors is 2^{\lfoor d/2
floor}. However, I think the Hestenes spinors have complex dimension 2^{d-2}. For d=3 and d=4, the examples you work out, these two dimensionality agree. But for larger dimensions we see there are more Hestenes spinors than Dirac spinors. I think the correspondence you establish in the section comparing to Hestenes spinors is one-to-many instead of one-to-one in the case d>4.
I haven't investigsted in higher dimensions honestly so you might be correct. I've seen people argue about which definition is "best". I wanted to show for the standard Pauli spinor case they end up being equivalent.
@@eigenchris This is something that has tripped me up as well, when previously I have tried to understand spinors and geometric algebra. I think @justingerber9531 is exactly right: "the" even-graded subalgebra of a Clifford Algebra will *not* in general have the same number of degrees of freedom as the minimum left ideal.
I think this comes down to the number of simple projection operators needed to go from the full algebra to the minimum ideal. If there is only one projector P, of the form (1/2)(1+e), so that that reduces the algebra from G to GP -- then in this case I think GP does corresponds to "the" even-graded subalgebra, equivalent to an algebra of rotations acting on that projector.
But if then we can apply a second projector P2, projecting us further from GP to (GP)P2, that it seems to me will make the algebra smaller again, corresponding I think to a *smaller* even-graded subalgebra, equivalent to a *smaller* algebra of rotations acting on P(P2). The projection will have projected some of the previous transformations down to identity transformations -- the change they made to P no longer changes P(P2).
So, continuing, there will ultimately be *an* even-graded subalgebra of the Clifford Algebra that relates to the minimum left ideal and transformations within it, but it will not be *the* even-graded subalgebra of the Clifford Algebra (or at least not a 1-to-1 faithful copy of it); equivalently, most even-graded subalgebras of Clifford Algebra will *not* faithfully correspond to the transformations of a minimum left ideal.
The emphasis on minimal left ideals in defining spinors is useful I think, because (i) that is how spinors have been defined since the 1930s; and (ii) it ties in closely with irreducible representations, and Wigner's theory in group theory that elementary states of a quantum system will correspond to irreducible representations of its group of symmetries.
On the other hand, Hestenes gives a useful reminder, I think, that we can happily switch backwards and forwards between thinking about a spinor and thinking about the transformation needed to boost to it from a particular reference spinor -- and that the latter can very much help our geometric intution as to what the spinor may represent.
I was left feeling there was scope for another video here in the Clifford Algebras and spinors part of your staircase - SfB 14a perhaps - taking this point further. It seems you have spent several videos building up a geometric intuition for Clifford Algebras in the form of geometric algebras; now, by way of pay-off, it would be useful to discuss how spinors can encode geometric transformations that can be interpreted with geometric algebra, and review this for the different spinors met so far; and perhaps present (though not necessarily endorse) Hestenes's geometric interpretation of the Dirac equation.
How about it? Any chance of such a 'bonus video', that might slot here into the series?
"Spinors are members of minimal left ideals in Clifford algebras." In the video you find all spinors in a single minimal left ideal. How many minimal left ideals are there in an algebra? In a Clifford algebra? Does it matter which minimal left ideal we select as the space of spinors? Are all the minimal left ideals isomorphic?
I like the pacman analogy, but when we were first learning this stuff in the 80's my advisor referred to the projection operators in terms of shark movies. JAWS. They'd eat the geometry multiplied against them leaving the result inside the relevant ideal.
We spent some time looking at various algebras asking what their natural minimal ideals were because multivector current densities in an ideal essentially remained in the ideal after being operated upon. The ideal doubled up as a way to identify the kind of particle one might be able to describe because particle identity got conserved that way.
I've heard Cohl Furey say ideals are a bit like particles. Do you know of any books or papers where I can read up on this more?
@@eigenchris My prof's name was Ken Greider from UC Davis. He didn't publish much on this, but what he did was from the early 80's. He got five or six of us through to our PhD's as well (depends how you count), so there are a few dissertations from that era too.
The point of it all was that current densities for E&M have scalar charges, but what about charges with geometry? What would a field theory look like if momentum were treated as a 'charge'? Current densities that were in ideals couldn't be bumped out, so multivector charges were the thing to examine.
I have a question that’s been gnawing at me for awhile:
After finding the z+ projector (P+) in Cl(3,0) would it not be more obvious to write the basis spinors as simply P+ and P-, where P- is defined as the orthogonal version of P+? In other words, would it be more straightforward to use P- to describe the “down state” rather than using another multiple of P+ to describe the “down state”?
Likewise for Cl(3,1) or Cl(1,3) (I use the alternative metric signature (-,+,+,+). For this, we need two projectors in a similar style to Cl(3,0) to form a minimal left ideal if we won’t use the pseudo scalar term. Since we need two of them, each of those two has an orthogonal pair, so we can imply that the basis required for spinors in this algebra are formed from these four projectors. Is there any advantage or disadvantage to simply defining the basis this way, compared to multiplying the algebra on the left and finding the basis spinors to be multiples of one or two projectors?
I hope that question makes sense 😅, because it’s a lot of words when I can’t just simply point to my scrap paper!
We need the 2 spinor components to be able to "rotate into each other" using the standard SU(2) matrices / Spin(3) operations we've been covering throughout this series. You can't ever rotate P+ into P- because the P+ component lives in the 1st column (aka ideal) and the P- component lives in the 2nd column (ideal). Meanwhile, P+ and σxP+ both live in the same column (ideal).
We are looking forward to making new series videos on physics topics.
Hy
From Pakistan
Sir how much you take time to prepare one lecture on this software..
It seems we want to find a one to one correspondence between a matrix U and a vector Uv in a set of matrices. We need to find a vector that is not an eigenvector with eigenvalue of 1 of the difference of any pair of matrices. Is this the same as finding the minimal ideal?
I'm not sure I can follow you're reasoning. It's been a while since I've thought about eigenvalues and what they imply. Can you explain a bit more?
At 25:16 and the following slide, why do you need a distinction between xi and zeta in the names of the ket and bra spinors, but when expressed in terms of components, you use xi's again?
It's not actually the dual of xi (that would just be xi's components complex conjugate), so I needed to give it another name. I nust happened to choose zeta.
At -18.11, the Inner Product, 😊we get that this is Pz+. Is that because at -19.13 the calculation appears only in the top left slot? I don’t see any other way to arrive at that description.
Yeah, the inner product needs to follow the formula I show at 24:20, unfortunately. I'm not sure of a more simple way to make it work.
Interesting, I wonder how this relates to the definition where spinors are defined to be vectors in irreducible representations Cl(V)_0. Also, if there are any resources on the answers to the questions given at the end of the video I would appreciate if anyone shared!
The book by Lounesto has some answers but I'm still working through it.
I love your videos. It makes me wonder if the question of why are there more particles than anti-particles, or the chiral symmetry breaking, isn't just a mathematical awkwardness that would lead nature to prefer minimal left ideals instead of the right one.
Thanks for sharing your knowledge, it's highly appreciated!
I think the left/right choice is just a convention. You could easily swap them.
@@eigenchris yes, I agree that it is just a convention for the mathematical object. I meant, since in nature we have more particle than anti-particle, was it because there are natural objects that act as projectors to "push" fermions in a particular minimal ideal than an other. That's what I'm wondering while seeing your video 🤷🏻♂️😁
Spinors without matrix representations woo! This one will take some time to digest...
👍👍👍👍
please discuss Majorana spinors to
Will you create lecture series on Quantum mechanics and Quantum field theory after finishing this spinor series ? You teach the subject in great detail and I would like to be taught by you ( regarding QM and QFT).
Sorry, but I won't. QM just isn't a subject I like that much, and so my passion to make videos for it isn't there. QFT, even moreso, is just too big and complicated, and I simply don't understand it. Usually an graduate intro to QFT is broken into 3 courses, and I'm only about 1 course into it knowledge-wise. I hope you can find someone else to each it to you.
Are all bivectors also spinors, if they are even graded elements of the algebra? If so, why can't we construct a basis for spinors in the clifford algebra out of just the bivectors basis (oxoy, oxoz, oyoz) or otherwise out of all the even graded elements? Why do we need to use projections to create another set of elements of the algebra to call spinors? I understand the dimensions of the basis won't be equivalent to the spin basis but not why?
According to the Hestenes definition, bivectors are spinors. According to the definition I use, bivectors are not spinors. You need to use the projector to make sure they match up with matrix representations whwre only the 1st column is nonzero.
any referenceable books for this?
There's the thesis I linked in the description. I think it's appendix "B" that covers the pacwoman property. A lot of the stuff in this video is extremely niche and not well-documented in most places. You can try looking at various Geometric Algebra books to see if they cover this, but I don't have any specific advice.
Fun fact: Hugh Jackman's daughter is not, it turns out, named Hughina Jackwoman.
its interesting how Cl(1,3,R) is not enough to represent the dirac spinor, but i feel like Cl(2,3,R) where the fifth basis would be gamma 5 is more natural than Cl(1,3,C) because there, the pseudoscalar behaves like i.
Yum! Brain food😋
I think the interesting thing about spinors is how often they appear in nature. Mathematically, you start with first principles that define rules, and then you build a bunch of properties while staying in the confines of those rules. What makes spinors remarkable is how often those first principles pop up in nature.
Spinros have been fascinating me since I heard about them (like tensors and differential forms). Unfortunately they elude me :( Tensors turned out to be very easy, differential forms - relatively easy. But spinors turned out to be surprisingly complex or maybe elusive. I still need to learn Clifford algebras and representation theory before I comprehend them.
I don't think you necessarily need to learn the Clifford Algebra approach in order to understand the Pauli and Weyl spinors used in 99% of physics applications. For those, the first 10 videos are probably enough. But the Clifford Algebra approach is used in more abstract generalizations to higher dimensions. It's up to you if you want to keep learning at that level.
@@eigenchris I think I must. I don't believe you understand some topic if you only understand some particular instance.
Maybe I just haven't tried hard enough, but no matter how many times I try to learn about algebraic ideals I feel that the whole left-vs-right distinction gets in the way of my understanding.
What we're really trying to capture is a sense of the order of application. But we're mixing that concept with arbitrary teminology based on our notation system.
It feels like someone is trying to explain the concepts of 'inside' and 'outside' to me but they're using terms such as 'hot' and 'cold'. My brain just gives up after a certain point.
I'm not really sure what to suggest. Have you watched the previous video (#13) in this series? It goes over how left ideals often look like matrix columns and right ideals often look like matrix rows.
@@eigenchris Ah thanks for the reply.
Yeh I'm not sure what the solution is either, but thanks for offering a suggestion. I actually get the intent behind it and the relation to matrices.
Something just feels deeply clumsy about the notation to me though, and I find myself having to fight through that clumsiness where others might just let it pass. Might be a personal thing... Or maybe there is some good reason for the discomfort.
Anyway just a rant haha. Not a criticism of your excellent videos ;-)
@@eigenchris can we use terms like covariant, contravariant, dual, or co-ideal to explain it?
yup definitely did a double take about that name
This is the closest I've come to a clickbait thumbnail in a serious math video.
@@eigenchriswill you ever do a video on the cox-zucker machine? (clickbait but it's a serious video)