[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

[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.

[raattgift]

> Because photons are massless you have to use quantum field theory, simple quantum mechanics does not apply

Photons themselves are quantum objects; they do not appear in classical theories of light.

They have several physical attributes which are quantized: spin (+/- 1), helicity (+/- \hbar), chirality (+/- \hbar), charge (0). The masslessness is what makes chirality == helicity as massless particles spin in the same direction along the axis of motion for any observer seeing such an axis.

Frequency is not quantized. A monochromatic light source at frequency f will emit individual photons with energy hf, but the frequency is observer dependent (e.g. one observer may decide that the photons are all hf_{lower} and another may decide that the photons are all hf_{higher}, and an accelerated observer may see the photons arriving from the source with different frequencies at different times; and in all cases each photon will have some hf_{observer} energy).

Photon number is also not conserved -- they can be freely emitted or absorbed. They also always move relativistically (even "slow light" in media does). These features conspire such that the wave-function of a photon only makes sense as long as the photon is neither emitted nor absorbed, and only in a fully relativistic quantum theory. That's what drives your "... you have to use [relativistic] quantum field theory". [0]

You can however devise an (effective) single-photon wave function, and this is done for single-photon experiments, e.g. in studies of single-photon spatial structure, as in the Bialynicki-Birula paper you link to (see eqn 42, and note the point about non-renormalizability).

World-lines are attached to all objects in a relativistic theory (i.e., one where there is Poincaré invariance in the infinitesimal neighbourhood around every point in spacetime), whether those objects are classical or quantum.

Extended objects technically have worldlines attached to each of their microscopic components, but one can usually consider such objects as having a single worldline in some limit; as in fully classical Newtonian mechanics, this would usually be attached to the centre of mass. (In special relativity, which is the theory of flat spacetime, you just use the Schrödinger equation. In very weakly curved spacetime you'd use the Schrödinger-Newton equation, and progress from there as necessary).

[0] It's not strictly the masslessness. Neutrinos have a tiny positive mass and almost always move relativistically. (The exceptions form the cosmic neutrino background, the elements of which have lost momentum in the expansion of space).

[6nf]

> As far as we can tell elementary particles are point particles.

When elementary particles interact with each other, they appear to do so as a point particle. However when they move they do so as a wavefunction which might be spread out over a large volume in space.

[IIAOPSW]

Knotting worldlines of anyons is the basis of topological quantum computing. Yes it can be done in theory, yes there is active research into doing it in practice.

https://en.wikipedia.org/wiki/Topological_quantum_computer

The best research into this right now is being done at Caltech

edit:

ok so tqc is mentioned in the article. my bad. I kind of skimmed the article the first time since I already know about this stuff.

[zmgehlke]

Yes, sponsored by MS at Station Q.

I believe the question there is if non-Abelian braiding statistics can be found to enable universal quantum computation, as Abelian anyons don't enable a univeral quantum computer. (Ideally a 12/5 FQHE, I believe.)

But my reading of the MS paper suggested that they were working with traditional anyons in 2+1D and the question of 3+1D analogs was unresolved. Here Wilzcek seems to suggest that there are no 3+1D analogs.

My question was if the treatment of objects as points, hence having a worldline rather than worldsurface or worldsolid wasn't the cause of that -- you can't knot a line in 4D, but you can knot a surface.

So my question was if the point object model was accurate or a simplification we need to move beyond.

[IIAOPSW]

ah

I'm fairly sure you cannot have a "lineicle" instead of a "particle". I guess maybe you could construct a weird albeit complete set of basis states based on lines through position space. However even if in some crazy interpretation your linicles had the properties of 2+1d anyon's, observing in this wicked basis is likely much harder than just making anyon's.

More simply at the end of the day the experimentalist observes a particle not a string. Thus there is a world line not a world surface.

disclaimer: this is partly my intuition. Maybe your idea has more merit than I give it credit for. My main research is not in anyons

[zmgehlke]

Well, the inspirtation in my mind was the interference from the double slit experiment.

If we think about a particle going from A to B, over t from 0 to 1, then the common interpretation is that it takes all paths and that those paths can interfere with eachother.

If we think about t=0.5, then the "particle" is not in a definite place, but sort of "smeared" out across its possible paths. The thought was if those possible paths, together, could form an embedded sphere in 3+1D, and in some sense could have similar computational knotting behavior. (And moreso that form shells of equal probability in a solid in 3+1+1D, with the extra dimension coming from the probability associated with each point.)

Im not sure how you'd "read" that, though possibly by "weakly" interacting with it in flight or it might contribute to the "random" outcomes of measured values. Since knotting is low-energy in general, we might not notice that behavior under normal conditions because of thermal interference adding a lot of computation in flight, hence making the values seem random. Similarly, heavily controlled experiments likely don't permit enough freedom to be anything but the trivial knot.

I expect that there is a reason that doesn't work, but figuring out where my hunch fails will help learn more about QM.