Spinors for Beginners 14: Minimal Left Ideals (and Pacwoman Property)

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

[spitsmuis]

I still believe this series should be called "Spinors for beginors"

[NotThatTallAtAll]

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 ❤❤❤.

[AtmosMr]

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.

[pan]

These videos are designed to make someone love physics. Many thanks

[AlfredDiffer]

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.

[ebog]

ty so much for this vid. its truly my fav vid on this channel SO FAR

[shivammahajan]

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

[bartlomiej]

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.

[justingerber]

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 \rfloor}. 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.

[sensorer]

39:44
Here we see that for CL(3, 0) our projector is 1/2(1+ixy)1/2(1+z). Sanity check: this should be the same as 1/2(1+z) (the projector we established gives us a minimal left ideal for CL(3, 0). But isn't 1/2(1+ixy)1/2(1+z) a projector to the {0} ideal? My reasoning: if we replace i with the pseudoscalar, 1/2(1+ixy) becomes 1/2(1-z) which is the orthogonal projector to 1/2(1+z). If we check with the matrix representation of the sigmas, we also get the zero matrix. 1/2(1-ixy)1/2(1+z) would give the right result here.
The vector and bivector commuting is not enough to construct the projector for a minimal ideal because there are two situations where projectors commute. 1. If they are parallel 2. If they are orthogonal.
Edit: in the book you reference the theorem explicitly states the idempotents need to be mutually non-annihilating

[AdrienLegendre]

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.

[dariushimani]

Thank you so much, these are super useful! Keep them coming!

[quantumphysics]

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!

[AlfredDiffer]

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.

[anirbanmukhopadhyay]

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).

[swalscha]

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!

[BakedAlaska]

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.

[stodent-mgbp]

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.

[metallicarocks]

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!

[xieziqian]

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?