Feynman path integral and events beyond the speed of light

[kurt101]

He describes the light source as monochromatic. This is what makes it coherent. To get quasi-monochromatic light from a non-laser source, you have to filter out most frequencies, and only keep a small portion of the frequency spectrum. That is typically still wider than for light from a laser, but the coherence length can be made longer than the distance-differences in your optical instruments. And that is enough for treating the light as quasi-monochromatic.

If a laser would work better, than I am all for a better experiment. That said, it would now be interesting to me to compared the difference between using a laser vs. filtered light source.

Let me first descibe it in Feynman's photon picture. Assume that each photon would have exactly one frequency, and nicely form a coherent superposition with itself. If different photons have slightly different frequencies, then their interference patterns are slightly different. And the sum of those interference patterns no longer shows sharp interference peaks, but appears washed out. But to see an interference pattern at all, you must look at the image produced by many photons, so all you can see is the washed out pattern.

I don't think it matters to your point, but just to be clear, I am not detecting an interference pattern in my experiment, but measuring the number of photons or accumulated intensity.

If your light has the frequency f, and you switch it on for t seconds, then you end up with an frequency distribution in the range f +- 1/t. If f is big compared to 1/t, then it makes sense to talk of quasi-monochromatic light. And if c*t is big compared to the distance-differences in your optical instruments, then treating the light as monochromatic should work well.

So if I make a more practical version of my experiment and use a 200 m mirror and assume red light (700 nm, 430e15 cycles / second) and:
put the source in the middle of the mirror, 100 m from the mirror
put the grating at the far left end of the mirror
put the detector at the far right end of the mirror, 100 m from the mirror

Then as a rough calculation:
The time for light to reach the grating is about 470 ns.
The time for light to reflect off the grating and go past the source is about 840 ns.
The time for light to reflect off the grating and reach the detector is about 1200 ns.
The time for light to take the shortest path, reflect off the mirror and reach the detector is 750 ns.

In this scenario, I would turn on the light source and look at the data from detector for up to 840 ns or 1200 ns.

So f = 430 x 10^15 cycles / second is much greater than (1 / t) = 1.2 x 10 ^ 6 and by your classification would be considered quasi-monochromatic light.

And c*t would be 252 m which seems fairly large compared to what I imagine the size of the optical instruments are and so by your classification would also be considered quasi-monochromatic light.

So given your guidance, I believe it would be justified to treat the light in this more practical scenario as monochromatic.

 

[vanhees71]

Well, you can also have a single-photon source which is pretty much monochromatic. It can't be literally monochromatic, because this would refer to an energy-momentum eigenstate of a single free photon, which is like a plane wave in classical electrodynamics and this is not a proper state in Hilbert space but a distribution in a larger space. So a true single-photon Fock state has a certain width in energy and thus also in momentum. To get a coherent quasi-monochromatic state you need a state with a very small energy width (energy uncertainty). According to the energy-time uncertainty relation (which is a complicated topic of its own!) you need a very long-lived unstable state of, e.g., an atom, which can emit a pretty monochromatic photon.

 

[Nugatory]

a laser is directional (right?)

No, not for sum-of-paths purposes. That tight directed beam that we expect from a laser is itself the result of adding the contributions from paths in all directions and having the amplitudes cancel everywhere except along the path of the beam.

 

[gentzen]

Like I said to gentzen: Another reason I thought Feynman did not imply a laser in his example is because a laser is directional (right?) and his light source was not.

I wish you good luck trying to understand that stuff. I tried my best to explain it. As long as considerations when you turn the light on or off are relevant, assuming the light to be quasi-monochromatic is problematic. I won't try any further to explain it.

 

[kurt101]

You are not thinking correctly about spacetime. Spacetime includes time. A spacetime model already includes all the information about when (i.e., at what events in spacetime) the light source is turned on. Only paths originating from those events in spacetime will have a nonzero amplitude. There is no such thing as "paths that don't yet exist through spacetime"; spacetime is not something that "changes" as things happen. A spacetime model already includes all the information about everything that happens.

Maybe I don't understand, I am saying it wrong, or something. 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. If I consider time 0, position 0, on the spacetime diagram as the start of the experiment at the light source, then the detector will be say at position 100 m away and outside of the light cone of the light source. The event where a photon from the light source reflects off of the grating and reach the detector will not happen until say y is c * 1200 ns (using the example I gave to gentzen). So if I am considering the experiment, up to only 840 ns, this event will not yet have happened. If the event has not happened, do I include it in the path integral?

 

[kurt101]

I wish you good luck trying to understand that stuff. I tried my best to explain it. As long as considerations when you turn the light on or off are relevant, assuming the light to be quasi-monochromatic is problematic. I won't try any further to explain it.

I am not sure why you sounded frustrated, but based on the context of your reply, it sounded like it was on my question about lasers being directional. Something I google confused me. Anyways it was a stupid thought and I am going to just delete that I ever said anything about it.

 

[gentzen]

I am not sure why you sounded frustrated

I didn't intent to sound frustrated. I am not frustrated. Everybody here knows that this stuff is hard to understand. Don't worry. I "unwatched" this thread, and simply decided to also explicitly say "goodbye".

 

[PeroK]

Maybe I don't understand, I am saying it wrong, or something. 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. If I consider time 0, position 0, on the spacetime diagram as the start of the experiment at the light source, then the detector will be say at position 100 m away and outside of the light cone of the light source. The event where a photon from the light source reflects off of the grating and reach the detector will not happen until say y is c * 1200 ns (using the example I gave to gentzen). So if I am considering the experiment, up to only 840 ns, this event will not yet have happened. If the event has not happened, do I include it in the path integral?

Let me try to fight fire with fire here. Let's look at the Lagrangian formulation of classical mechanics for a particle in a force field (gravity or EM or whatever).

You could describe this model as "the particle tries all possible paths and picks the one that minimises the action integral". Now we ask: while the particle is trying paths in one direction, what happens if we very quickly change the potential when the particle isn't looking? Does the particle take account of that change in the potential or not?

It's obviously absurb, because you're not supposed to take the idea that the particle is trying all paths and calculating the action integral literally.

It's the same with QED: you're not supposed to be thinking that the photon is literally traveling all these paths simultaneously - so that you can fool the particle by stopping the experiment before it's had a chance to try out all the paths. Which is effectively what you are asking.

You're not supposed to be overlaying a classical model of precise spacetime events at every location for every photon path in the calculation. Not least because in QED you have the uncertainty principle undepinning everything in any case.

Your whole line of thought is inappropriate to studying QM.

 

[kurt101]

Let me try to fight fire with fire here. Let's look at the Lagrangian formulation of classical mechanics for a particle in a force field (gravity or EM or whatever).

You could describe this model as "the particle tries all possible paths and picks the one that minimises the action integral". Now we ask: while the particle is trying paths in one direction, what happens if we very quickly change the potential when the particle isn't looking? Does the particle take account of that change in the potential or not?

I would suspect that any change you made quickly is limited by the speed of light. So this would not bother me because I can rationalize the cause.

I suspect in the path integral approach it is the same and it is limited by the speed of light. I thought Dr. Chinese more or less said this, but I am waiting for his confirmation before I am comfortable with that. I would also like to get a sense of how it is limited in the path integral approach. If someone told me we simply don't count the contributions from paths that can't happen because they violate the speed of light that would make sense to me, but so far nobody has told me this is what is done.

I understand these paths are not thought of as real paths, but used for calculation, but at the same time if the paths make a real contribution to the probability calculation then they have some aspect to them that has a real effect. So it is important to understand how that contribution is added or removed when the path integral approach is applied to an actual scenario (e.g. like my experiment).

It's obviously absurb, because you're not supposed to take the idea that the particle is trying all paths and calculating the action integral literally.

I would tend to agree that a particle trying all paths and calculating the action seems like a crazy way for the universe to work. Like I said above, I have never thought of the idea that the particle takes all paths as actually what it is doing.

It's the same with QED: you're not supposed to be thinking that the photon is literally traveling all these paths simultaneously - so that you can fool the particle by stopping the experiment before it's had a chance to try out all the paths. Which is effectively what you are asking.

I am more or less trying to ask my question in two ways:
I am trying to understand in an actual experiment when and where contributions from phenomena such as a grating begin and end.
I am trying to understand in the path integral approach how faster than light contributions from phenomena such as the grating are handled (i.e. do you remove them?).
It is my understanding that both the experiment and the path integral model should give the same result.

You're not supposed to be overlaying a classical model of precise spacetime events at every location for every photon path in the calculation. Not least because in QED you have the uncertainty principle undepinning everything in any case.

Your whole line of thought is inappropriate to studying QM.

It is not clear to me whether you are saying my experiment would not tell me anything because of the uncertainty principle or that the uncertainty principle is just something that if I am not careful could skew the results of my experiment. I thought you and everyone else were saying the later, but maybe I have misunderstood.

 

[PeroK]

I suspect in the path integral approach it is the same and it is limited by the speed of light. I thought Dr. Chinese more or less said this, but I am waiting for his confirmation before I am comfortable with that. I would also like to get a sense of how it is limited in the path integral approach. If someone told me we simply don't count the contributions from paths that can't happen because they violate the speed of light that would make sense to me, but so far nobody has told me this is what is done.

You definitely must include all paths. You cannot exclude paths that "exceed the speed of light". One way to think about this is again the UP (Uncertainty Principle). There is no definite time when the photon is omitted - that is subject to quantum uncertainty. There's also no definite time when the photon was detected - for the same reason. So, you cannot say definitely that the photon traveled faster than light - but, again, that presupposes we are talking about definite photon paths, rather than contributions to a calculation.

Moreover, the photon paths must be a heuristic calculation that replaces a full QFT calculation involving a quantised EM field and photons as excitations of that field. I.e. not the point particles of the path integral formulation.

I understand these paths are not thought of as real paths, but used for calculation, but at the same time if the paths make a real contribution to the probability calculation then they have some aspect to them that has a real effect. So it is important to understand how that contribution is added or removed when the path integral approach is applied to an actual scenario (e.g. like my experiment).

Your proposed experiment changes the scenario to one that is no longer modeled by the simple path integral approach. The EM field behaves differently; the QFT calculations are different; the simple scenario presented by Feynman no longer applies; and, the simple path integral calculations no longer apply.

Your analysis depends on the paths being "real" (even though you say they are not).

I am trying to understand in the path integral approach how faster than light contributions from phenomena such as the grating are handled (i.e. do you remove them?).

No, you do not remove them. They are not real paths.

It is not clear to me whether you are saying my experiment would not tell me anything because of the uncertainty principle or that the uncertainty principle is just something that if I am not careful could skew the results of my experiment. I thought you and everyone else were saying the later, but maybe I have misunderstood.

The way you have described and analysed your experiment ignores the concept of uncertainty in QM. You say things like:

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. If I consider time 0, position 0, on the spacetime diagram as the start of the experiment at the light source, then the detector will be say at position 100 m away and outside of the light cone of the light source. The event where a photon from the light source reflects off of the grating and reach the detector will not happen until say y is c * 1200 ns (using the example I gave to gentzen). So if I am considering the experiment, up to only 840 ns, this event will not yet have happened. If the event has not happened, do I include it in the path integral?

Where you make no concessions towards uncertainty. The photon is at point $(x_0, y_0)$ at time $t = 0$; the photon is at point $(x_1, y_1)$ at time $t = 1$ etc. That is classical physics. If you mix that in with QED, you get something that is simply a wrong analysis.

There are no "events" here; there are simply contributions to a probability amplitude. You cannot talk about these virtual paths as comprising spacetime events with well-defined coordinates.

In general, you appear to have tried to pick up some QM/QED concepts, but are generally not prepared to relinquish any notions from classical physics.

 

[vanhees71]

You shouldn't take this addition over "all photon paths" literary. In relativistic QT you deal necessarily with a QFT, particularly for massless particles. The path integrals for relativistic QFT are over field configurations rather than single-particle paths, which works for nonrelativistic QM.

 

[kurt101]

You definitely must include all paths. You cannot exclude paths that "exceed the speed of light". One way to think about this is again the UP (Uncertainty Principle). There is no definite time when the photon is omitted - that is subject to quantum uncertainty. There's also no definite time when the photon was detected - for the same reason. So, you cannot say definitely that the photon traveled faster than light - but, again, that presupposes we are talking about definite photon paths, rather than contributions to a calculation.

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?

Moreover, the photon paths must be a heuristic calculation that replaces a full QFT calculation involving a quantised EM field and photons as excitations of that field. I.e. not the point particles of the path integral formulation.Your proposed experiment changes the scenario to one that is no longer modeled by the simple path integral approach. The EM field behaves differently; the QFT calculations are different; the simple scenario presented by Feynman no longer applies; and, the simple path integral calculations no longer apply.

Your analysis depends on the paths being "real" (even though you say they are not).

No, you do not remove them. They are not real paths.

Ok, I can accept that the simple Feynman path approach described in Feynman's book" The Strange Theory of Light and Matter" can not model my experiment. Also I am taking from what you are telling me that the approach that can successfully model my experiment is sufficiently different that it would not be easy for you or anyone to explain how FTL effects from say the grating in my experiment are removed from the calculation.

 

[DrChinese]

1. It is actually the opposite in the example, the diffraction grating placed at the end of the mirror adds a contribution, but I could see where one placed at the middle of the mirror counteracts the contribution, and so I get what you are saying.

2. I am not sure if I follow what you are saying. Are you saying that given the constraints of my experiment, you will see the same results with or without the diffraction grating? (i.e. were you giving a direct answer to my question?).

3, Ultimately this is what I want to understand. I have had trouble finding clear statements in any of the descriptions of the Fenman path integral (both with math and without) that clearly say contributions prohibited by the speed of light don't count.

4. That said, I still don't understand if contributions that are prohibited by the speed of light in my example are from the light sources perspective or say the mirror's perspective. Hopefully you understand what I am saying here and this would be a part 2 question for this thread and so I am reluctant to get into it too much before I am comfortable that I understand my current question.

5. If you changed the time constraint by adding additional detection time to be sure scenarios A and B (with and without the grating) should give a different result; then ran scenarios A and B many times; then took the averages; I would expect on average you would see a consistent difference between A and B, where scenario B with the grating that add contributions would on average detect more photons than A.

1. A grating prevents a reflected contribution from the section where the grating exists. It can be a positive contribution, a negative contribution, or no contribution depending on size and positioning.

2. There are many "alternative" ways to prevent contributions from a section of the mirror. If you blocked it, for example. Or in your version of the experiment, which in my opinion is impossible to implement, there wasn't a wide enough time window for the path to be traversed, that would be an alternative method.

3. I think everyone is saying the same things: there is no FTL effects. So another issue here is that reflection of light is a more complicated process than we are describing. You can't really draw a straight line from a source to a point on the mirror and then to the detector and say "it went this way". As a result, there may appear to be paths that loosely appear to be FTL. But no such path can be demonstrated as such experimentally, because the true picture is quite different.

4. I don't follow you here. There are no FTL contributions if you get strict enough. You would be violating the uncertainty principle if you tried to assert you know a particle's position at 2 precise points in time, and momentum in between.

5. There isn't a scenario where you can distinguish A and B. There is always quantum uncertainty in addition to technological constraints. At some point as the time window shrinks, there just won't be any photons emerging from the small time window.

------------------------------------

As you continue to battle about conducting your experiment, you are actually attempting - whether you know it or not - to say that you can perform an experiment that would yield complete certainty in measurements of non-commuting observables. You either already accept that as impossible per QM, or you are tilting at a windmill using the mirror as a disguise in the process. I don't think any of us can help you much further, as it has already been demonstrated no FTL events are occurring. Once you accept that, your question is fully answered.

 

[PeroK]

I still don't understand the impact of the uncertainty principle on light emission in my experiment.

When you say:

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. If I consider time 0, position 0, on the spacetime diagram as the start of the experiment at the light source, then the detector will be say at position 100 m away and outside of the light cone of the light source. The event where a photon from the light source reflects off of the grating and reach the detector will not happen until say y is c * 1200 ns (using the example I gave to gentzen). So if I am considering the experiment, up to only 840 ns, this event will not yet have happened. If the event has not happened, do I include it in the path integral?

Then that is a classical, not a QM analysis. You cannot analyse QM experiments in those terms. It cannot be said any simpler.

 

[kurt101]

2. There are many "alternative" ways to prevent contributions from a section of the mirror. If you blocked it, for example. Or in your version of the experiment, which in my opinion is impossible to implement, there wasn't a wide enough time window for the path to be traversed, that would be an alternative method.

I am fine with the alternative of blocking the grating if that helps anything in this discussion.

I am not sure what you mean by impossible. If you mean my initial example that used times of 10 seconds, that was clearly an in principle experiment not meant to be practical. If you mean your back of an envelope calculation where you used a distance to the mirror of 1000 wavelengths and came up with 0.003 femtoseconds then that was your choice of experimental setup. I came up with some more reasonable numbers in my discussion with gentzen with a difference of 90 nanoseconds. That said, I don't see why in principle you can't make the difference in paths as long as you want.

3. I think everyone is saying the same things: there is no FTL effects. So another issue here is that reflection of light is a more complicated process than we are describing. You can't really draw a straight line from a source to a point on the mirror and then to the detector and say "it went this way". As a result, there may appear to be paths that loosely appear to be FTL. But no such path can be demonstrated as such experimentally, because the true picture is quite different.

If that is the truth that there are no FTL effects, I certainly can accept that.

I don't know what you mean by loosely appear FTL.

4. I don't follow you here. There are no FTL contributions if you get strict enough. You would be violating the uncertainty principle if you tried to assert you know a particle's position at 2 precise points in time, and momentum in between.

I have not asserted I know anything about a particle position or momentum with any certainty in anything I have presented in this thread. I have asserted that the grating has an effect on the probability at the detector and I have theorized that there is something that goes from the source to the grating to the detector that can't go faster than light. By making this "something" go on a longer path, I was thinking my experiment would demonstrate that this "something" only has an effect on the probability at the detector when allowing for enough time in this experiment for this "something" to transverse its path and make it to the detector.

5. There isn't a scenario where you can distinguish A and B. There is always quantum uncertainty in addition to technological constraints. At some point as the time window shrinks, there just won't be any photons emerging from the small time window.

I am confused what you mean that there is no scenario where you can distinguish A and B. First of all that is my goal, to show that I get the same result for scenario A and B. However if you let A and B run for long enough time, I expect it to go to the scenario that Feynman presented which he clearly says there is a difference! So clearly we are not on the same page unless you are agreeing with my expectation which it does not sound like it.

To be clear we are on the same page, let's use the more practical version of my experiment that I presented to gentzen and use a 200 m mirror and assume red light (700 nm, 430e15 cycles / second) and:
put the source in the middle of the mirror, 100 m from the mirror
put the grating at the far left end of the mirror (Scenario B only)
put the detector at the far right end of the mirror, 100 m from the mirror

Then as a rough calculation:
The time for light to reach the grating is about 470 ns.
The time for light to reflect off the grating and go past the source is about 840 ns.
The time for light to reflect off the grating and reach the detector is about 1200 ns.
The time for light to take the shortest path, reflect off the mirror and reach the detector is 750 ns.

In this experiment, I would turn on the light source and look at the data from detector for up to 840 ns.

In scenario A there is no grating. We simply turn on the light source and record the accumulated intensity at the detector for 840 ns. We run this scenario many times to get an average accumulated intensity.

In scenario B there is a grating at the far left end of the mirror. We simply turn on the light source and record the accumulated intensity at the detector for 840 ns. We run this scenario many times to get an average accumulated intensity.

If I compare scenarios A and B, my expectation is that I will NOT see any difference in the average accumulated intensity (i.e. the intensity I averaged over many trials).

If I run scenarios A and B for longer than 840 ns they will become closer and closer, the longer I run them, to the scenarios Feynman described and I will see a difference in accumulated intensity.

So now that you clearly understand what I am comparing. Can you say whether you agree or disagree with my expectations and explain why?

------------------------------------

As you continue to battle about conducting your experiment, you are actually attempting - whether you know it or not - to say that you can perform an experiment that would yield complete certainty in measurements of non-commuting observables. You either already accept that as impossible per QM, or you are tilting at a windmill using the mirror as a disguise in the process.

I don't understand what you mean when you say "whether you know it or not - to say that you can perform an experiment that would yield complete certainty in measurements of non-commuting observables",
That said if you are implying I am trying to say I can perform an experiment that would yield complete certainty in my measurements. I would not say that at all. I am collecting averages over many trials. I am not expecting individual trials to measure the same intensity, but I do expect given enough trials that I will converge on some consistent average value. Even if I am measuring random noise, I will converge on some consistent average value.

So can you explain what the non-commuting observables are in my experiment?

I don't think any of us can help you much further, as it has already been demonstrated no FTL events are occurring. Once you accept that, your question is fully answered.

My expectation has always been that there would be no FTL events and no FTL effects in this scenario. So I have no problem accepting it and a little confused that you think I would.

In my mind an FTL effect would be like the EPR scenario, but in my scenario there is no local preparation (as vanhees71 has described in other threads) between the source and the grating and so that has been my rationale for expecting no FTL effect.

So if there are clearly no FTL effects then I am mostly satisfied. I would still appreciate a definitive answer on my experimental discrepancies that I list above. If you clearly understand what I am doing in the experiment, you should be able to definitively say what the result of the experiment will be. You can tell me that I can't conclude anything useful because my result but I would still like you to tell me what the result will be.

I look forward to completing this thread as soon as I can, but I would like to end on some clarity.

 

[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.