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Elementary particles are considered point-like. Which makes the concept of spin rather problematic. It can neatly be described by group theoretical considerations, but that offers no explanation what it actually is.

But what if we consider particles to be extended particles, their extension reaching into small extra spatial dimensions? For example, in the one-dimensional case, that the particle is a small circle structure around a thin Planck-sized cylinder, which would have the nice property of a Lorenz invariant Planck length, as the extension is perpendicular to the 1d bulk space. This could be extended to 3d if we consider three small curled up dimension (each to a Planck length circle). In 1d, the circle could rotate in two opposite directions, and spin would correspond to actual rotation.

MatterGauge
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  • Spin is only problematic is one things of it as a spatial degree of freedom, requiring a particle to rotate about a physical axis. But it is known that spin cannot be represented in 3D space (see https://physics.stackexchange.com/a/488754/36194, at least objects with half-integer spin cannot have their spin so represented) – ZeroTheHero May 11 '22 at 23:38
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    @ZeroTheHero "But it is known that spin cannot be represented in 3D space", which is why I suggested extra small dimensions. – MatterGauge May 12 '22 at 00:21
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    so whatcha sayin is that it is a point particle in 3D space… and the extra dimension is where you can fit all the angels on a proverbial headpin… – ZeroTheHero May 12 '22 at 00:36
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    It is not needed, so, no, it is superfluous. – Jon Custer May 12 '22 at 01:34
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    In closed strings there is a circle https://web.physics.ucsb.edu/~strings/superstrings/basics.htm – anna v May 12 '22 at 03:39
  • @ZeroTheHero "so whatcha sayin is that it is a point particle in 3D space… and the extra dimension is where you can fit all the angels on a proverbial headpin…" Nice comparison! Isn't such, in another context, for different properties, done in string theory? (I think properties are just put in an additional math space, the Calabi-Yau structure, with no real existence, as must be the case if particles were a geometrical structure like I think them to be...) :). – MatterGauge May 12 '22 at 10:24

2 Answers2

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Here's an intuitive way to think about spin. Consider a spinning ball. It has an axis with a north and south pole. Now there are two invariant tangent planes on this spinning ball. The ones on the north and south poles. All other tangent planes move.

Now let us draw an arrow on the tangent plane on the north pole. Obviously this will spin along with the spinning ball.

Now let us shrink the ball. The two invariant planes will remain invariant and as we shrink to a point we can imagine both are still there. But they are infinitely close to each other.

So we need to think how the spinning arrow should accomodate this. After all, which plane should it be on? Well, the obvious thing is to say both! And we arrange this by the arrow first spinning on one plane and then on the other.

This gives us a double cover of the rotation group and so we have recovered spin.

Mozibur Ullah
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  • Don't the parallel planes on the north and south move too? They rotate around the axis. Contrary. If they have rotated half they fall together again! – MatterGauge May 12 '22 at 00:12
  • @Felicia: I said those two planes are 'invariant'. Any rotation of the sphere would rotate either of those planes but would leave the plane occupying the same plane. This is what I mean by invariant. Contrarywise, if you look at a plane anywhere else, say he equator, then a rotation of the sphere would also rotate the plane and it would move to an entirely different spot. – Mozibur Ullah May 12 '22 at 10:07
  • Yes. The axis of rotation is perpendicular to the plane at the poles and inside the plane in the equator, while the axis itself rotates around the central axis. So a double rotation! Full turn around the axis which rotates fully around the central axis. Is this the double rotation vs. the single rotation? – MatterGauge May 12 '22 at 10:17
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I agree that intrinsic spin is somewhat unintuitive, but it really is point-like.

Consider a disc spinning with constant angular speed in its plane, with the axis at its centre. This disc isn't made of atoms, it's just an idealised geometrical object, a set of ideal zero-dimensional points. All of the points in the disc except for the centre point have orbital angular motion, but the centre point just has pure intrinsic spin, not orbital.

In our mathematical model of particles, the intrinsic spin of a fundamental particle behaves exactly like that centre point. It's not easy to visualise, orbital angular motion keeps trying to sneak into our visualisation. So don't trust the visualisation, trust the mathematics. ;)

PM 2Ring
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  • Interesting! A bit similar to the other answer. I trust the math but I like to know the physics. What is going on? I don't see how it is a quantum mechanical effect. Is it like the small zero point energy of a qm oscillator? But for angular momentum? Matter always having motion, so also rotation? Is ìntrinsic spin in fact zero point angular momentum? Suppose a Planck-sized hypersphere (like the 1d case of a Planck circle on a thin, seemingly 1d tube, or cylinder) had the charge of an electron, how fast must it spin to deliver the right magnetic moment. – MatterGauge May 12 '22 at 05:00
  • @Felicia In theories with compact dimensions (Kaluza-Klein & its descendants), charge is linear momentum in the compact dimension. But since that dimension is closed (with a length on the order of the Planck length), traveling in that direction brings you back to your starting place. So it's kind of like an orbital motion. – PM 2Ring May 12 '22 at 07:03
  • Yes, I thought about that too! It's a confusing way to define charge though. Like a vibrating string. How can that be charge? Why should a rotation or vibration be charge, which represents the strength of force emanating from it? – MatterGauge May 12 '22 at 07:25
  • @Felicia It's not possible to answer that properly in a comment thread. ;) Kaluza discovered that if you do GR with 5 dimensions (1 time, 4 space) then you get gravity plus Maxwell's laws. Klein had the brilliant idea of making the 4th space dimension compact to explain why our space looks 3 dimensional. – PM 2Ring May 12 '22 at 07:32
  • You think a new question would be appropriate to ask? I mean, it's not really mainstream. :) Though strings are... – MatterGauge May 12 '22 at 07:49
  • @Felicia Kaluza-Klein is ok here. Sure, it's superseded by string theory, but we do have a tag for it, which you should browse before asking a new question https://physics.stackexchange.com/questions/tagged/kaluza-klein – PM 2Ring May 12 '22 at 08:07
  • Tanx, mr. Türing! :) I'll give it a shot. I have the inkling feeking the theory is purely mathematical, like strings. Stashing properties of interaction in an imaginary mathematical space (like the basic interaction symmetries and a corresponding, fitting Calabi-Yau manifold. – MatterGauge May 12 '22 at 08:33