I Feynman path integral and events beyond the speed of light

[DrChinese]

I still don't understand the impact of the uncertainty principle on light emission in my experiment. I am ok with light emission being random in my experiment as long as this random emission does not affect the average intensity measured at my detector. In the Feynman scenario without my constraints the grating clearly affects the probability outcome regardless of the uncertainty principle. So clearly it is my constraints that you are claiming are a problem for the experiment, but in principle I can modify my experiment so it is more and more like the Feynman scenario. I can make the path to the grating much much longer than the shortest path to the detector and thus make the uncertainty principle less and less of a factor, right?

No. It should be obvious that intensity of a contribution from spots progressively further from the "shorter" paths diminishes as distance increases. As those fringes are further away, that (positive or negative) contribution rapidly approaches 0. You will never be able to discern a statistical difference.

Just to be clear: the emission time of any photon of light cannot be resolved when its momentum is very certain. Its source position (in spacetime) cannot be certain! You must be able to understand this point. You keep acting as if the photon starts at a certain point at a certain time, traverses a specific path to the mirror, is reflected at a certain point, and then arrives at the detector at a certain place at a certain time. Don't you see all the incorrect reasoning going on? None of this happens in the quantum world we are discussing.

 

[kurt101]

No. It should be obvious that intensity of a contribution from spots progressively further from the "shorter" paths diminishes as distance increases. As those fringes are further away, that (positive or negative) contribution rapidly approaches 0. You will never be able to discern a statistical difference.

Ok, this makes some sense and Feynman did have this quote that may hint at what you are telling me:

"rule—what actually happens—is much simpler: a photon that reaches the detector has a nearly equal chance of going on any path, so all the little arrows have nearly the same length. (There are, in reality, some very slight variations in length due to the various angles and distances involved, but they are so minor that I am going to ignore them.)" - Feynman, Richard P.. QED: The Strange Theory of Light and Matter (Princeton Science Library) (p. 41). Princeton University Press. Kindle Edition.

Just to be clear: the emission time of any photon of light cannot be resolved when its momentum is very certain. Its source position (in spacetime) cannot be certain! You must be able to understand this point. You keep acting as if the photon starts at a certain point at a certain time, traverses a specific path to the mirror, is reflected at a certain point, and then arrives at the detector at a certain place at a certain time. Don't you see all the incorrect reasoning going on? None of this happens in the quantum world we are discussing.

Ok, I will take this point as it is. I will try to find some book or source that explains this more. I don't understand what its impact to the the experiment is.

I also am not getting a clear answer as I would like from you on this, but I think you are telling me that if I ran my experiment, that I would see no difference between scenarios A and B, but I would be wrong to conclude this shows that I removed the contribution of the grating of scenario B because of the Quantum Mechanics uncertainty principle. Is this correct?

 

[PeterDonis]

you can also have a single-photon source which is pretty much monochromatic

Are you referring to a Fock state? That is a very different kind of state from a coherent state, and the term "monochromatic" is usually used to refer to the latter.

 

[PeterDonis]

When I draw a space time diagram (e.g. y-axis c*t and x-axis position), as y increases I add events that will occur.

No. That's not how spacetime works. In a spacetime diagram, everything that happens at all times of relevance for the experiment is included. That includes any detections at the end of the experiment. You have to include it all, and then look at paths through spacetime and compute amplitudes. You don't look at things "event by event" as "time" goes on; you include everything that happens at all times all at once.

 

[PeterDonis]

if I am considering the experiment, up to only 840 ns

You can't; that's not how spacetime works. You can't get any valid answers from just looking at part of the experiment at a time. You have to look at the entire experiment.

 

[kurt101]

I learned diffraction gratings are used in fiberoptic cables https://en.wikipedia.org/wiki/Arrayed_waveguide_grating

Is there any fundamental difference between my experiment and this simple setup? :
I put a photon source through a beam splitter where one direction takes a much longer path than the other path and goes through one of these fiber optic cables with diffraction gratings and then eventually recombines with the shorter path

If I start listening for a signal on the combined path, I expect that I am going to get the expected signal at half the intensity from the shorter path until some time much later when the longer path recombines. Maybe when it recombines I will get some uncertain difference between the two paths. If this is fundamentally the same as my experiment, but just fewer paths, maybe it can help me understand where the uncertainty principle impacts my experiment.

And in regards to the Feynman path integral approach; if I keep adding paths like this and recombine them as I choose then am I implementing the Feynman path integral algorithm in a direct fashion? If so then maybe it can help me understand some aspects of the Feynman path integral approach that I have been struggling with.

 

[vanhees71]

Perhaps it helps a lot to simply treat the diffraction problem classically, and with the phasor method advocated in Feynman's popular-science-level QED book, you don't do anything else than that.

An intuitive idea is to simply argue with Huygens's principle. The idea is that you have some light source before the slits and you approximately assume that the corresponding light waves travel undisturbed until they reach the obstacle. Then according to Huygens from each point in the opening emerges a spherical wave, and the total em. field is the superposition of all these spherical waves.

A more formal derivation is due to Kirchhoff, who just considers the Helmholtz equation, i.e., the equation for a single-frequency wave mode and again assuming that the wave travel undisturbed from the source to the slits. Then you use Green's integral theorem with one of the functions the unknown wave field in the other the free-space Green's function for the Helmholtz equation, which in three spatial dimensions in fact is just a spherical wave, which justifies Huygens's principle for three spatial dimensions. With this you come very far, and you can apply further approximations like the usual Fresnel and Fraunhofer diffraction. In the latter approximation you end up with the simple rule that the diffraction pattern (amplitude of an em.-wave field component) behind the slits is given by the Fourier transform of the spatial pattern of the slits (indeed what you measure behind the slits is the intensity, i.e., the amplitude squared),

Last but not least this is still not the mathematically fully consistent solution, for which you'd need the true Green's function with the appropriate boundary conditions for the slits. Sommerfeld has solved the exact diffraction problem including questions of polarization for the most simple case of a single edge (i.e., one half-space filled with absorbing material). It's pretty tough problem which in full generality for more complicated geometries of the diffracting obstacles can only be solved numerically.

A very good treatment of the diffraction problem in classical E&M can be found in Sommerfeld, Lectures on Theoretical Physics, vol. 4 (optics).

 

[DrChinese]

Is there any fundamental difference between my experiment and this simple setup? :
I put a photon source through a beam splitter where one direction takes a much longer path than the other path and goes through one of these fiber optic cables with diffraction gratings and then eventually recombines with the shorter path

If I start listening for a signal on the combined path, I expect that I am going to get the expected signal at half the intensity from the shorter path until some time much later when the longer path recombines. Maybe when it recombines I will get some uncertain difference between the two paths. If this is fundamentally the same as my experiment, but just fewer paths, maybe it can help me understand where the uncertainty principle impacts my experiment.

And in regards to the Feynman path integral approach; if I keep adding paths like this and recombine them as I choose then am I implementing the Feynman path integral algorithm in a direct fashion? If so then maybe it can help me understand some aspects of the Feynman path integral approach that I have been struggling with.

There are some similarities with your ideas, at least with making the path length differences explicit and feasible. But guess what? There are no quantum effects to discuss unless the resulting photon detection results in indistinguishability (of which path). For example, for them to be indistinguishable, they should be in phase. There must be enough uncertainty in time of emission/arrival that you can't determine which path they took.

And now, no surprise here: there are again no FTL effects to discuss.

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BTW, you keep mentioning "840 nanoseconds" (also 1200 ns) for your detection window. The difference between these would be enough for light to travel about the length of a football field. As I mentioned previously, the actual time window you'd need is more on the scale of attoseconds which is 12 powers of ten shorter. You can of course use a time window as wide as you suggest, you just won't measure any difference in intensity.