4 ms·

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,

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.

IIAOPSW 10y ago
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 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 10y ago
  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 10y ago
    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 10y ago
      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.
danbruc 10y ago
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 10y ago
  > 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 10y ago
    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

IIAOPSW 10y ago

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 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 10y ago
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 10y ago
  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 10y ago
    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.

danbruc 10y ago

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 10y ago
> 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 10y ago
  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