
Inside the Knotty World of ‘Anyon’ Particles - bainsfather
https://www.quantamagazine.org/20170228-anyons-a-new-kind-of-quantum-particle/
======
zmgehlke
So, I've been curious: having a "worldline" seems to imply that the objects in
question are a point object -- adding one dimension (time) to zero gives you
one, a line. Is this actually a reasonable model, though?

If we think of particles as little spheres (or loops), it should be possible
to knot them -- you can knot objects 2 dimensions less than your space.
(Hence, there are 1-knots in 3D and 2-knots in 4D, where an N-knot is a
N-sphere embedded.)

Is the question of 3+1D "anyons" actually settled, or has no one performed the
analysis on the wave equation(s) looking for 2-knots? (My reading of MS papers
implied the second, but I'm hardly an expert.)

Ed: As an aside, Im super happy Wilzcek is covering this. I've always found
his writing to be rather accessible.

~~~
danbruc
_If we think of particles as little spheres (or loops), it should be possible
to knot them [...]_

As far as we can tell elementary particles are point particles. They are
however surrounded by a cloud of other particles due to vacuum polarization.

 _[...] you can knot objects 2 dimensions less than your space. (Hence, there
are 1-knots in 3D and 2-knots in 4D, where an N-knot is a N-sphere embedded.)_

Could you actually knot world lines if particles were solid spheres? That's
certainly above my ability to visualize and I know practically nothing about
knot theory but naively I would think that solid spheres would not be
different from points by imagining the radius shrinking towards zero. But my
intuition may of course be misleading.

~~~
zmgehlke
> elementary particles are point particles

This might sound dumb, but how does a point have a wavelength?

(I actually think QM is a conceptual mess of hacked together math, but that's
a rant for another day. Here I am being sincere, because maybe (probably,
almost certainly) I just don't understand what you mean.)

> I would think that solid spheres

Sorry, I think I was unclear.

I meant sphere in the topological sense of just the "shell" part (the
surface), as opposed to a ball, which contains the interior. Think bubble.

You can vizualize a 2-knot fairly easily: tie a knot in a piece of string,
hold each end, and spin it. The "sphere" you get by identifying the start and
end of a cycle is knotted.

~~~
danbruc
_This might sound dumb, but how does a point have a wavelength?_

This turns out to be possibly surprisingly complicated. I thought I knew the
answer, that all photons are the same and have no wavelength by themselves and
that the wavelength is in the wave function. Now I just wanted to check that I
am not mistaken in order to not spread false information and of course failed
to verify what I thought. It may be correct, I may be incorrect, it may be an
approximation, I can not tell, that will probably require a day of reading to
understand.

Because photons are massless you have to use quantum field theory, simple
quantum mechanics does not apply. This means there is no wave function as in
quantum mechanics. The classical electromagnetic field seem not to be well-
defined for single photons due to the uncertainty principle. It matters
whether or not you take absorption and emission into account. Just google
photon wavelength, there is a lot to read. All of this may obviously be wrong,
mostly just bits and piece I just picked up while skimming articles.

I will certainly try to figure this one out, such an obvious question and
something I thought to understand at least in broad strokes. But not today,
its late enough. This paper [1] might be useful nut I did not yet read it.

I figured you might refer to a sphere but I think a ball would be the more
likely thing if elementary particles were not points. Was I correct thinking
you can untangle world lines of balls?

[1]
[http://www.cft.edu.pl/~birula/publ/APPPwf.pdf](http://www.cft.edu.pl/~birula/publ/APPPwf.pdf)

~~~
zmgehlke
I appreciate the reply!

Yes, if you put little R^4 balls around points of the line, you can still
untangle it. It's basically still a line. (They might have to be _really_
little balls, but the definitions all use neighborhoods for defining the
legality of knot moves, so you can't "shrink" the knot out of existence.)

My actual thought was slightly more complicated, in that particles dont need
to necessarily not be a point _all_ the time, since a sphere you're slicing
with a plane can appear as a point -- see sibling comment for what inspired
the idea.

~~~
danbruc
Final thought for tonight. It is certainly an extremely weak argument because
it is root in intuition from our macroscopic world, but based on this it makes
sense that elementary particles have to be points. At least for me it is
somewhat hard to imagine how something could be spatially extended and
elementary, i.e. not subdividable. If it has a length, an area, or a volume, I
should be able to break it into pieces. This of course immediately conflicts
with string theory with which I am sympathetic, sometimes more, sometimes
less. But as said, it is an extremely weak argument rooted in a world many
orders of magnitudes from where I want to apply it.

------
mrcactu5
These diagrams may be helpful in discussing the entanglement of the anyon
worldlines

[https://news.ycombinator.com/item?id=13787283](https://news.ycombinator.com/item?id=13787283)

