Spinor Puns

I wonder if we could group tensor and spinor rank notations in a meaningful way. It could be very useful in my opinion.

Current notations

• (n,m) tensors are n covariant and m contravariant
• (i,j) spinors are the members of the corresponding Lorentz group representation

Motivating relations

1. N-rank tensors are 2N-rank spinors
2. spin-j space has dimension 2j+1
3. tensors transform using the normal and inverse matrices while spinors use normal and complex conjugate.

I am wondering how good is it at closing the gap between Numerical Linear Algebra topics like the SVD and the FFT with the work of David Hestenes (Clifford Algebras, Grassmann Algebras, and Gibbs-Heaviside's Vector Calculus).

So I’ve just been introduced into spinors in QM, and I’m quite confused, which from googling seems to be a common thing.

I’m struggling to understand what one is, and Wikipedia goes mostly above my head.

So the way In which spinors we’re introduced to me is considering some spin state that has a well defined spin in some direction n. And I can represent this state in the basis of Sz such that for (for spin up) my state is [cos(θ/2), e^(iφ)sin(θ/2)]

This makes sense to me, but then I’m just told that this is a spinor and any general spin state may be represented in lz basis as

Ket(ψ) = [ψ1, ψ2]. where ψ1 is the component of my state in the spin up in the Lz.

This still makes some sense, but I was hoping someone could provide a bit of added clarification, because this just seems a lot like what I’ve done a lot which is representing some state as the linear superposition of some other states, but now we’ve called it a special name as a spinor.

If (1, 0)^T is spin-up and (0, 1)^T is spin down, what is -(1, 0)^T?

To my understanding, spinors are used to describe 1/2 spin particles like electrons in qm. I am familiar with the concept that the spinor requires a 720º rotation to return to its original state, but what does it mean when a spinor "changes sign" from a 360º rotation? What is the sign of a spinor?

A quote by Michael Atiyah

"“No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the 'square root' of geometry and, just as understanding the square root of -1 took centuries, the same might be true of spinors.”

What did he mean by this, is there a problem to Spinors we don't understand?

Hello! I am trying to find the irreducible spinor decomposition of W and Z bosons and how/why they differ from photons.

Im thinking its just

(1/2,1/2) = (0, 0) +(0,0) +(1,0) + (0,1)

where the (0,0) + (0,0) portion would describe the W and Z massive vector bosons, and the (1,0)+(0,1) describe the antisymmetric Maxwell tensor? but im confused here because wouldn't (0,0)+(0,0) be a scalar?

Im thinking they are different because the photon only has two polarization states while the W and Z have four, due to the fact that they have mass while a photon is massless. Do I have this right? Everywhere I look it says the W boson has 3 polarization states and does not mention much about the Z. Why are they hard to learn about in comparison to EM?

Thanks for any advice!!

I've been casually trying to understand spinors and spin structures for a while now. I've sort of accepted the definition of, they are just a bunch of elements in the Clifford algebra built out of an even number of tensor products of unit vectors. But what the hell does that mean? What do they "do?" do they act on stuff? When I think of tensors, another topic I had difficulty understanding once, I now have the snappy line "(p,q) tensors are elements of a vector space of maps that eat p vectors and spit out q vectors in a way such that they are linear in every argument", this immediately tells me what they are and what they do. I haven't got a similar sort of intuition for spinors yet.

Another thing that confuses me is globalising this construction. Most things from differential geometry turned out like, we have this construction on a vector space V, now replace V with TM_x and use a vector bundle to get smoothly varying versions of what you just built (i.e. forms, vector fields). When I see lecture notes about spin structures instead of saying "a spinor field is a section of the spinor bundle" which is what I'd have expected I've seen

"A spin structure on a principal SO(n) bundle Q -> X is given by a principal Spin(n) bundle P -> X"

I won't lie I genuinely have no idea what this means, is it like the case of "a tensor transforms like a tensor" where it sounds useless until you already understood the concept and then becomes helpful?

Is there some kind of baby cases that people keep in mind when they are reading abstract constructions? Or some intuition to remember?

Hello everyone,
I realized there are different "systems" of multidimensional objects which behave specifically under coordinate change.
There are:

• tensors from tensor algebra
• spinors from Lorentz groups
• pseudovectors and multivectors from exterior algebra

I wonder if there are links between those different systems, especially:

• Is a a pseudovector a (0, n-1)-tensor?
• Is a spinor a (0, ½)-tensor ?
• Is the difference between left and right Weyl spinors the same as the one between vector and covector ?
• Is a bispinor a multivector ?

I guess I'm trying to make the topic more general, so I welcome all ideas of an intuitive explanation. Imagine you have a clever kid and you need to teach that kid spinor, how would you explain it? Age is up to you.

The usual ideas are the plate tricks, belt tricks or Mobius strip, but they barely scratch the surface. I was thinking if anyone know some interactive games or something that could help?

So why am I asking? I actually made a post a few weeks ago but was auto-modded, but recent post reminded me of it again. I am babysitting this kid occasionally. She's good at math and physics, and want to do physics (though it's a long way from now). So I had been teaching her some stuff that I think would be pretty unintuitive if taught too late. Spinor is one such topic, but I had a hard time explaining it to her. She got vectors just fine though, but spinor is just too hard. So I am asking for some ideas, as I want to revisit that topic some time later.

I learned that spinors are elements of [;\mathbb C^2;] when it is regarded as a representation space for [;SU(2);].

The "when it is regarded as" part makes me wonder: where do the spinors live, really? Do they live in [;\mathbb C^2;]? But then suppose the [;SU(2);] disappears, would that magically make the spinors turn into vectors? How can the nature of a mathematical object depend on something external?

The problem comes with trying to understand tensors as well. A rank-2 tensor on [;\mathbb R^3;] is an element of [;\mathbb R^9;] regarded as a representation space for [;SO(3);]. But why can't 2-tensors suddenly turn into 1-tensors (aka plain vectors), if we suddenly use the [;\mathbb R^9;] to represent [;O(9);]?

In 2D for example, when we have a spinor (a,b) describing a spin direction, where a and b are complex numbers, the thing has 4 degrees of freedom since it takes 2 real numbers to describe each complex number. Fine.

I understand that the "overall" phase (and not relative phase between components) does not make a difference to its magnitude, since the product of a complex number with its conjugate cancels out the phase.

How exactly does this reduce the degrees of freedom from 4 to 3?

Thanks!

Hey I am trying to do spinor calculus by myself for now, can someone tell me what it is and some for resources to study it

I understand that there is some important connection between spin and the groups SU(2) and SO(3). It's often pointed out that SU(2) double covers SO(3), and that the Pauli matrices form a basis for the Lie algebra su(2) (up to a factor of i). But I'm having trouble understanding the connection.

As I understand, SU(2) is the symmetry group of spinors (as diagrams like this one point out - you need a double-covering of SO(3) in order to cover all spinor-space symmetries). This is why this group is so important in the theory of spin. A question that just occurred to me is why the Pauli matrices happen to form a representation of the Lie algebra su(2); does the algebra of observables of a system always have to be some representation of the system's symmetry group Lie algebra? But we can put that question to one side.

I'm especially confused because the theory of spin in quantum mechanics is usually presented as (a generalised version of) the theory of orbital angular momentum. The commutation relations between spin observables are derived in terms of the canonical commutation relations of linear position and momentum.

So then, could someone connect this intro QM exposition of spin to the groups SU(2) and SO(3)?

I'm also vaguely aware of Clifford algebra representations of spin. As far as I can tell spin states are represented by rotors which happen to double-cover SO(3). But that just kicks the can down the road: why should rotors represent angular momentum?

Apologies if this question is poorly formed or seems scattered - that's because it is. It's hard to form the question exactly because I don't fully understand its subject matter yet. So any answer, even partial, would be greatly appreciated.

I understand why if we divide our Dirac spinor into two Weyl Spinors, we have the distinction between the right and left handed Weyl Spinors because they transform opposite to each other when it comes to boosts. However, if we want to write the Dirac equation in terms of the left and right handed Weyl Spinors, how does the handedness affect the sign of the 3D gradient? Peskin and Schroeder, Equation 3.39

I'm a physics student and today I came across spinors while studying QM out of Shankar's "Principles of Quantum Mechanics". There was relatively little said about these mysteries "spinors" so I decided to google it, and I found that I totally have no clue what they actually are. In fact, there's an awful lot of math in Shankar's text that I don't truly "understand" and I'm wondering where I can go to get a grasp of these topics in a purely mathematical context.

The Wikipedia for spinors has so much information, it's quiet overwhelming. It does mention lie algebra and clifford algebra, so maybe the best place to start is with an abstract algebra text?

Bonus Question: If somebody could give me a brief overview of what a spinor "is", I'd greatly appreciate it!

I understand what vectors and tensors are, but have never understood what spinors are. Could someone please help me!!!

So...I am trying to understand particle spin and "half-spin" etc.... I have heard it mentioned that this isn't physical spin as we know it a such but I am having a hard time understanding it... I guess it makes sense on one level since these aren't really "solid balls" as such, but then why is it called spin?

I have no idea really, just thought I'd see what anyone can offer me in terms of either the official standard model version or possibly what Nassim's version might say about it...

It did occur to me that the fermion half-spin could have something to do with the inversion through a torus, like a mobius-strip has a half-twist in it....but I'm just guessing. When I looked up "spinors" in wiki, it does display an image of a mobius-strip...

It also seems to be explained using Euclidean and Minkowski space, which, as far as I understand, uses the concept of "flat space" - infinitely thin 2D manifolds, which I don't understand since 2D flat space simply doesn't exist in reality except as a illusionary and incomplete perspective of surface areas...

Can anyone offer any insights here? Cheers. :)

I'm a sophomore electrical engineering student and I recently just finished vector calculus. It has opened up a world of physics and applied math that I never knew existed.

That being said, I recently stumbled upon the concept of Tensors and Spinors. I have a very basic understanding of these mathematical objects, but I'd like to have some real physical examples to help me understand better.

If someone could succinctly explain what Tensors/Spinors are, and give some simple examples, that would be awesome.