>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 Michael Atiyah mean by this? Was he a brainlet? Spinors are easy as frick to understand.

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Poopin farten, whatchu startin

the algebra is enough

lack of understanding here means "i cant see them as cool 3d objects that spin"

lack of understanding means israeli professors wasting 50 years of funding on string theory

They can be visualized easily if you think of rotations as a pair of reflections. Rotate one mirror around the intersection point by 180 degrees and the object being double reflected rotates by 360 degrees. The object needs to rotate by 720 degrees to get the mirror back to its initial configuration.

The reason people struggle to understand them is because physicists are lazy and often don't bother distinguishing between spinors (purely geometric objects) and fermions (physical objects modeled as spinors). It would be like if we didn't bother distinguishing between vectors (purely geometric objects) and angular momentum (physical objects modeled as vectors). This lack of distinction causes confusion because not only does it pollute our understanding of the geometry with irrelevant physics (especially when quantum weirdness is involved), but also because we often use the "wrong" geometric object to model the physics. This is like how angular momentum actually turns out to be a "pseudovector", meaning it really should've been modeled as a bivector or a rank 2 antisymmetric tensor or something rather than as a vector.

Good answer.

>They can be visualized easily if you think of rotations as a pair of reflections.

DOESNT MATTER

I dont give a frick about the belt trick or other alleged visual effects related to spinors. Watching these things doesnt mean you understand spinors better.

Knowing the algebra is enough, most of math just cant be visualized and theres no point in trying.

The difference is that unlike the belt trick etc, this isn't just a cute trick divorced from the algebra, it's actually a direct geometric translation of the algebra. In Clifford Algebras, transformations are double-sided "sandwich products". Reflecting X about a line:

[math]

LXL^{-1}

[/math]

Rotating X about the intersection point of two lines

[math]

L_{1}L_{2}X(L_{1}L_{2})^{-1}

[/math]

X will always be rotated by twice the angle between the pair of lines that make up the rotor. If you think of the rotor itself as an object that can be factorized into a pair of lines then you have a purely geometric picture of a spinor without the unnecessary physical details, and that geometric picture came directly from the algebra, not from cute tricks.

yeah, but what's the physical interpretation of that? The spinor nature you described implies that a fermion always interacts with another particle exactly twice at a time

>physicists are lazy and often don't bother distinguishing between spinors (purely geometric objects) and fermions (physical objects modeled as spinors).

Why do physicists do this? It makes it harder to understand

To be honest I don't see the problem. A fermion literally is a mathematical model and not a tiny bouncy ball.

I don't really get that perspective. A model of what? Why would you say fermions aren't actual things that exist?

>A model of what?

Nobody knows. We can't directly observe quantum mechanical wave functions pre collapse.

>Why would you say fermions aren't actual things that exist?

Because they are an explanatory abstraction.

Are atoms just a model? Are molecules just a model? Are macroscopic objects just a model? Are basketballs just a model?

>Are atoms just a model? Are molecules just a model

NTA but unironically yes, all are models that just works.

So are basketballs and other macroscopic objects just a model and not "a bouncy ball"?

no, the ball is real.

but if you want to predict the trajectory of the ball, you apply physics which model the ball as a point or a sphere (don't confuse this with the real ball, moron). then you use whatever mathematical physics law needed to predict the trajectory of the ball.

I don't understand why you wouldn't call the actual real object a fermion. The ball(?)-like object the fermion model, models, should be called a fermion..

I mean you would agree that electrons exist, right? Electrons are fermions? So fermions exist? If you wouldn't call electrons existent, what else do you not call existent? Are molecules also not existent?...

>We can't directly observe quantum mechanical wave functions pre collapse.

How do you know the model is modeling that accurately, if you cannot observe it?

You observe the effects. The model is just a tool t predict what your actual observation will be. It isnt natures fault that your sense of vision, smell or other body sense you have cant directly see many things. Heck you cant even see dust mites

Why would you not call the effects you observe a fermion? You call the effect you can have a basketball model for, a basketball, so why not call the effects a fermion model models, a fermion? I don't get this.

Also, you contradicted yourself by saying nobody knows what it's a model of because you can't observe it, and then saying you can observe it. Assuming that was the same person.

spinors are just when you sample at the 1/2 Nyquist frequency

i can say the same about anything

i.e no one fully understands trees, their biology is formally understood but their general significance is mysterious

YOU SPINOR ME RIGHT ROUND BABY RIGHT ROUND LIKE A RECORD BABY RIGHT ROUND ROUND ROUND!!

>You Spin Me Round came out in 1984

help...

physics are mathematical models that were "validated" by emperical data. so many newbies here today.

Either he was a brainlet or one of those poor bastards with aphantasia, and therefore unable to visualize spinors.