Speed of light with quantum path integrals

[Sophrosyne]

TL;DR Summary

Has the speed of light from one point to another been measured to be different for the different paths light can travel between them?

Richard Feynman formulated quantum path integrals to show that a single photon can theoretically travel infinitely many different paths from one point to another. The shortest path, minimizing the Lagrangian, is the one most often traveled. But certainly other paths can be taken. Using single photon emissions, this has indeed been shown to be the case.

But these other paths should take longer to get to the photometer. Has anyone shown that single photons sometimes take longer than the path minimizing the Lagrangian? Shining a bunch of photons from a single source at a single instant, has anyone been able to show that the photons arrive at the photometer not in a single instant, but in a distribution of times reflecting the probabilities of having traveled the different paths?

 

[PeterDonis]

Richard Feynman formulated quantum path integrals to show that a single photon can theoretically travel infinitely many different paths from one point to another.

No, that's not what he showed. What he showed is that you can predict probabilities of detecting photons at certain spacetime points by using path integrals. He did not show that the photon travels any of those paths in between measurements. You can't show that, because you can't make any assertions about what path the photon travels in between measurements. You can only make assertions about what gets measured.

Shining a bunch of photons from a single source at a single instant, has anyone been able to show that the photons arrive at the photometer not in a single instant, but in a distribution of times reflecting the probabilities of having traveled the different paths?

To do this, you would have to measure not just when the photons arrive at the detector, but when they are emitted from the source. Most photon sources don't allow you to measure this. AFAIK every time measurements have been made with photon sources that do allow you to measure the time of emission, the photon travel time is exactly what you would predict using the extremal path, i.e., the one on which the photon travels at the speed of light.

Note, btw, that I said "extremal", not "shortest". The spacetime squared length of the "speed of light" path is zero. Other paths that appear in the path integral could have squared lengths either greater than zero (spacelike) or less than zero (timelike). The extremum of the null path is a saddle point, not a minimum (or maximum).

 

[Sophrosyne]

No, that's not what he showed. What he showed is that you can predict probabilities of detecting photons at certain spacetime points by using path integrals. He did not show that the photon travels any of those paths in between measurements. You can't show that, because you can't make any assertions about what path the photon travels in between measurements. You can only make assertions about what gets measured.
To do this, you would have to measure not just when the photons arrive at the detector, but when they are emitted from the source. Most photon sources don't allow you to measure this. AFAIK every time measurements have been made with photon sources that do allow you to measure the time of emission, the photon travel time is exactly what you would predict using the extremal path, i.e., the one on which the photon travels at the speed of light.

Note, btw, that I said "extremal", not "shortest". The spacetime squared length of the "speed of light" path is zero. Other paths that appear in the path integral could have squared lengths either greater than zero (spacelike) or less than zero (timelike). The extremum of the null path is a saddle point, not a minimum (or maximum).

In his book Quantum Electrodynamics, Feynman gives an example of a single photon emitter, bouncing light down to a mirror 45 degrees below, which then gets picked up by a detector situated 45 degrees above the mirror on the other side, at the same level as the emitter. He shows how that 45 degree trajectory is the shortest path and therefore the one most likely crossed by the photon.

But, he then removes just that central part of the mirror which allows the photon to travel the shortest distance, and then repeats the experiment. He shows that the single photons emitted still gets detected by the detector. This goes to show that that some of these photons do travel other pathways as well.

Doesn't it? Or is it that the photons "choose" those other pathways only if that central shortest pathway is closed off to them?

 

[PeterDonis]

He shows how that 45 degree trajectory is the shortest path and therefore the one most likely crossed by the photon.

"Most likely" is not correct. A given photon does not "choose" just one of the paths. Every possible path for a given photon contributes to the total amplitude for detection of that photon.

What Feynman shows is that the shortest path is the one that makes the largest contribution to the total amplitude for photon detection when the entire mirror is present. But other paths also make contributions in this case; they're just not easily measurable in this case because their contributions are so much smaller than the contribution of the shortest path (and the paths very close to that path).

e then removes just that central part of the mirror which travels the shortest distance, and then repeats the experiment. He shows that the single photons emitted still get detected by the detector. This goes to show that that some of these photons do travel other pathways as well.

No, it shows that the other paths do in fact contribute to the total amplitude, since there is still a nonzero amplitude for detection even when you remove the "shortest" path that bounces off the center of the mirror.

Or is it that the photons "choose" those other pathways only if that central shortest pathway is closed off to them?

Again, no individual photon chooses one particular path. Every possible path for a given photon contributes to the total amplitude for detection of that photon.