3 ms·

One absolutely astounding aspect of the path integral is that it ascribes amplitudes to trajectories that don't obey the (classical) equations of motion. If yo

by evanb 4y ago

One absolutely astounding aspect of the path integral is that it ascribes amplitudes to trajectories that don't obey the (classical) equations of motion. If you like: the classical paths are at turning points of the action (loss function) but all other paths have some associated imaginary loss. In the path formalism we sum every possible path, weighted by the imaginary loss. Since the classical paths are at turning points, small deviations around that path tend to sum coherently, while large deviations tend to have neighboring paths that contribute with opposite phases. If you write the amplitude for a photon going from your flashlight to your wall, the path integral tells you to sum up all possible paths, not just the classical ones. So yes, the path integral formalism really does say that a photon goes from the flashlight, around Jupiter, and to the wall. But not in 48 minutes! In time = (distance to wall)/c! It goes every which way without caring about the classical law. The "insane" ones cancel. Well, if all the insane ones cancel, what's the point of all this? Just forget about them, they're a huge surplus of nonsense in the theory! However, it's perfectly possible to construct experiments where it's clear that photons really do "smell out" alternatives. Diffraction gratings, the double-slit experiment, etc., all show this quantum weirdness.

CyberDildonics 4y ago
Are there any other household appliances that emit photons that circle around any other planets?
- evanb 4y ago
  “Emit” usually means as radiation. The photons that take every path are not on-shell. When your refrigerator is open the light you see contains an amplitude of the photon going around Uranus.
  - CyberDildonics 4y ago
    Flashlight -> Jupiter Refrigerator -> Uranus What planet does the oven radiation go around?
    - raattgift 4y ago
      > What planet does the oven radiation go around? All of them, including planets outside our observable universe. Same with the flashlight and refrigerator. Each photon also goes into and through all the planets, and near each of their electrons in an infinite number of ways. They also go through all the stars, all the vacuum, etc. And with all the speeds, all the accelerations, ... Most of these infinite possibilities are a negligible contribution to the final probability that photon-detected-at-B was photon-emitted-at-A. The non-negligible contributions come from the (still infinite) trajectories allowed by relativity and mostly from those near the lightlike geodesic.
      - CyberDildonics 4y ago
        Which part of the lightlike geodesic's infinite relativity trajectories does my toaster glow come from?
        raattgift 4y ago
        The whole glow? Treat it classically, it's mostly a solid-state physics problem. One individual photon, treated quantum mechanically? From a single inelastic interaction between a free electron and a metal cation. That origin marks one end of the photon's worldline. Repeating for large numbers of photons will make you quickly bored, so you'll probably want to look into the Stefan-Boltzmann law (which still includes the constant c and Planck's constant) and the concepts in Bohr's correspondence principle and effective theory. More at <https://en.wikipedia.org/wiki/Joule_heating https://en.wikipedia.org/wiki/Joule_heating> if you're curious.
- raattgift 4y ago
  There are two different "circle around distant bodies" here that are probably causing some confusion. Classically, if near your appliance you can see stars and Mars through the window on a cloudless night, a flash of light from your appliance can in principle be seen (e.g. by a really good telescope orbiting one of those bodies). Indeed, some of the light from a laser pointer shining out your window will go to infinity, and has some astronomical chance of being stuck the photon ring around a black hole. It works just like you being able to see through your window starlight or (in principle) a bright flash on the surface of Mars. However, this is not the sort of "goes around Jupiter" that we are talking about; we are talking about something like the probability of one microscopic component of your appliance's flash of light scattering off or being absorbed by a microscopic component of said window (for example, in part of the spectrum in which the window is not transparent), where calculating that probability considers an infinity of possible paths. Indeed, an infinity of the path-integral trajectories of one such photon (that classically is definitely intercepted by the window) will go out to infinity, an infinity of those trajectories will go out to Jupiter, an infinity of them will go to somewhere very near the window pane, an infinity of them will speed up and slow down along the way rather than travel at the speed of light at all times, etc. We then want to assign a probability to each of these paths. The idea is that trajectories that are not like a classical path between appliance and window are suppressed because (somewhat technically) they [a] have the same magnitude of probability amplitude, but [b] have different phases, [c] different changes in trajectory have different changes in phase, and [d] for "longer" paths compared to the classical one, a small change of trajectory makes a big change in the action compared to Planck's constant, so [e] for sets of small changes in "longer" paths the phases change rapidly and thus are highly likely to cancel out. So most of those infinities are effectively irrelevant and we can focus on the smaller infinities that make a marginal-to-relevant change to the total probability amplitude and figure out a way to calculate it from them (e.g. by sampling).
raattgift 4y ago
> a huge surplus of nonsense in the theory It gets even worse if one starts thinking about the flashlight's filament or semiconductor, the wall, stratified Jupiter with its optical depth, the flashlight-wielder's visual system, and everything in between all being (sets of) quantum objects. Maybe the thing to do in this example is to assume correspondence and think classically, and insist on using a fewer-quantum-number example if one wants to think quantumly. However I tend to admire detailed wrestling with everything-is-quantum in an example like this. Maybe start by remembering that the spot-on-wall-to-retina is relevant, not just the flashlight-to-wall.

by evanb 4y ago

CyberDildonics 4y ago

Are there any other household appliances that emit photons that circle around any other planets?

evanb 4y ago
“Emit” usually means as radiation. The photons that take every path are not on-shell. When your refrigerator is open the light you see contains an amplitude of the photon going around Uranus.
- CyberDildonics 4y ago
  Flashlight -> Jupiter Refrigerator -> Uranus What planet does the oven radiation go around?
  - raattgift 4y ago
    > What planet does the oven radiation go around? All of them, including planets outside our observable universe. Same with the flashlight and refrigerator. Each photon also goes into and through all the planets, and near each of their electrons in an infinite number of ways. They also go through all the stars, all the vacuum, etc. And with all the speeds, all the accelerations, ... Most of these infinite possibilities are a negligible contribution to the final probability that photon-detected-at-B was photon-emitted-at-A. The non-negligible contributions come from the (still infinite) trajectories allowed by relativity and mostly from those near the lightlike geodesic.
    - CyberDildonics 4y ago
      Which part of the lightlike geodesic's infinite relativity trajectories does my toaster glow come from?
      - raattgift 4y ago
        The whole glow? Treat it classically, it's mostly a solid-state physics problem. One individual photon, treated quantum mechanically? From a single inelastic interaction between a free electron and a metal cation. That origin marks one end of the photon's worldline. Repeating for large numbers of photons will make you quickly bored, so you'll probably want to look into the Stefan-Boltzmann law (which still includes the constant c and Planck's constant) and the concepts in Bohr's correspondence principle and effective theory. More at <https://en.wikipedia.org/wiki/Joule_heating https://en.wikipedia.org/wiki/Joule_heating> if you're curious.
raattgift 4y ago
There are two different "circle around distant bodies" here that are probably causing some confusion. Classically, if near your appliance you can see stars and Mars through the window on a cloudless night, a flash of light from your appliance can in principle be seen (e.g. by a really good telescope orbiting one of those bodies). Indeed, some of the light from a laser pointer shining out your window will go to infinity, and has some astronomical chance of being stuck the photon ring around a black hole. It works just like you being able to see through your window starlight or (in principle) a bright flash on the surface of Mars. However, this is not the sort of "goes around Jupiter" that we are talking about; we are talking about something like the probability of one microscopic component of your appliance's flash of light scattering off or being absorbed by a microscopic component of said window (for example, in part of the spectrum in which the window is not transparent), where calculating that probability considers an infinity of possible paths. Indeed, an infinity of the path-integral trajectories of one such photon (that classically is definitely intercepted by the window) will go out to infinity, an infinity of those trajectories will go out to Jupiter, an infinity of them will go to somewhere very near the window pane, an infinity of them will speed up and slow down along the way rather than travel at the speed of light at all times, etc. We then want to assign a probability to each of these paths. The idea is that trajectories that are not like a classical path between appliance and window are suppressed because (somewhat technically) they [a] have the same magnitude of probability amplitude, but [b] have different phases, [c] different changes in trajectory have different changes in phase, and [d] for "longer" paths compared to the classical one, a small change of trajectory makes a big change in the action compared to Planck's constant, so [e] for sets of small changes in "longer" paths the phases change rapidly and thus are highly likely to cancel out. So most of those infinities are effectively irrelevant and we can focus on the smaller infinities that make a marginal-to-relevant change to the total probability amplitude and figure out a way to calculate it from them (e.g. by sampling).

raattgift 4y ago

> a huge surplus of nonsense in the theory It gets even worse if one starts thinking about the flashlight's filament or semiconductor, the wall, stratified Jupiter with its optical depth, the flashlight-wielder's visual system, and everything in between all being (sets of) quantum objects. Maybe the thing to do in this example is to assume correspondence and think classically, and insist on using a fewer-quantum-number example if one wants to think quantumly. However I tend to admire detailed wrestling with everything-is-quantum in an example like this. Maybe start by remembering that the spot-on-wall-to-retina is relevant, not just the flashlight-to-wall.