Path integral implies superluminal motion?

[substitute materials]

In the path integral formalism, where we treat a photon as if it takes every possible path, aren't the possible paths limited by the speed of light?

If we were to perform the double slit experiment, and shield the detector after a specified time frame to limit the time for a photon to make the journey, would this affect the results by eliminating the more circuitous paths? If the detector shield was activated exactly after the straight line light travel distance had elapsed, then wouldn't only classical straight line photon paths be possible? Would this affect the results, measurably or otherwise?

This recent experiment indicates that with a third slit, non-classical paths should have a measurable influence if they are present. I take it we have not yet measured the contribution by non-classical paths then?

http://physicsworld.com/cws/article/news/2014/sep/25/photons-weave-their-way-through-a-triple-slit

 

[mfb]

Photons don't have a well-defined position, and this is critical to understand the double-slit experiment (if you don't want to consider the whole thing as wave phenomenon, where it is clear that waves don't have a single position). Messing around with slits within some interesting time frame will influence the results, with the details depending on what exactly you do.

Paths with superluminal motion don't contribute to the overall result.

 

[substitute materials]

Alright then, thanks!

Any thoughts on the linked experiment? My take, please correct me if I'm wrong, is that up to now the resolution of the experiment has been below the threshold that would show the influence of non-classical paths, which occur with very small probability.

 

[mfb]

It is a non-classical path for particles only, but there the regular double-slit effect is non-classical as well. For waves the newly observed path is also present in a classical theory.

It is nice to have it observed, but it is not something completely new. With ultra-fast shutters you could remove this path, but it would change all other paths as well.

 

[ftr]

Alright then, thanks!

Any thoughts on the linked experiment? My take, please correct me if I'm wrong, is that up to now the resolution of the experiment has been below the threshold that would show the influence of non-classical paths, which occur with very small probability.

The standard PI approach assumes that paths do not loop,i.e. do not go back and then forward. The idea of this article is what if they do.

 

[Strilanc]

I always thought the longer paths were just contributions from cases where the photon was emitted earlier.

If you did a triple-slit experiment, where the photon can take a zig-zag path through slit 1, back through slit 2, and then back through slit 3 before hitting the screen, I would have expected doing the experiment in sufficiently quick bursts (blocking the slits and turning off the light source in between) vs continuously keeping them open; to affect the interference pattern. Because it would in effect allow you to separate the longer paths from the shorter paths, preventing them from interfering. Is that correct?

 

[mfb]

A shorter coherence length (light bulb instead of a laser) will reduce the contribution of those higher-order effects, yes.

 

[Demystifier]

Paths with superluminal motion don't contribute to the overall result.

Are you sure? Can you support it by a reference?

 

[mfb]

Superluminal paths are part of the integral (and contribute within the integral), but there is no observation you can make that would have any superluminal measurement result: quantum mechanics is local*.

*some interpretations are non-local, but that is a problem of those interpretations, and not the discussion topic here.

Not an academic reference, but here is a blog article discussing this (section "The relativistic limits on speed are taken care of automatically").

 

[Demystifier]

Superluminal paths are part of the integral (and contribute within the integral), but there is no observation you can make that would have any superluminal measurement result: quantum mechanics is local*.

*some interpretations are non-local, but that is a problem of those interpretations, and not the discussion topic here.

Not an academic reference, but here is a blog article discussing this (section "The relativistic limits on speed are taken care of automatically").

I agree with that. But let me quote from the blog:
"No "regulation" of the violent behavior of the path integral is needed. Quite on the contrary. Any "intervention" into the path integral that would drop some histories that fail to obey certain inequalities - that you incorrectly assume must be imposed on the individual basis - will result in a violation of the consistency rules such as the conservation of probabilities."

So, if one simply removes the superluminal paths from the integral, one gets a wrong result. In other words, superluminal paths also contribute to the final physical result, despite the fact that the physical result does not involve physical superluminal motions of the wave. Do you agree?

 

[mfb]

I guess it gets a matter of semantics. You can always get rid of specific superluminal paths by looking at intermediate results.

 

[substitute materials]

A shorter coherence length (light bulb instead of a laser) will reduce the contribution of those higher-order effects, yes.

This suggests that physically limiting the situation to exclude some trajectories changes the result, right?

 

[substitute materials]

I agree with that. But let me quote from the blog:
"No "regulation" of the violent behavior of the path integral is needed. Quite on the contrary. Any "intervention" into the path integral that would drop some histories that fail to obey certain inequalities - that you incorrectly assume must be imposed on the individual basis - will result in a violation of the consistency rules such as the conservation of probabilities."

So, if one simply removes the superluminal paths from the integral, one gets a wrong result. In other words, superluminal paths also contribute to the final physical result, despite the fact that the physical result does not involve physical superluminal motions of the wave. Do you agree?

But we're not supposed to cherry pick trajectories mathematically, as the blog states it will lead to incorrect results in the path integral calculation.

This leads me back to confusion.

 

[mfb]

This suggests that physically limiting the situation to exclude some trajectories changes the result, right?

Of course.

 

[substitute materials]

Right sorry, I meant limiting some non-classical trajectories

 

[vanhees71]

I think, it's important to stress that the path integrals for relativistic QT should be taken as pathintegrals over field configurations, not paths in space as in non-relativistic QT. The reason is the same as in the operator formalism: A single-particle treatment a la "1st quantization" is not adequate for relativistic QT. The reason is that in the interacting case, particle numbers are not conserved (only certain charges of the Standard Model like electric charge) and you need QFT to take the possible annihilation and creation processes properly into account. Particularly photons are easily annihilated and created. There's no restriction concerning any charge-like conserved quantity for them, and in addition they are massless, i.e., even energy conservation cannot restrict the possible photon production/annihilation processes considerably.

 

[Demystifier]

I think, it's important to stress that the path integrals for relativistic QT should be taken as pathintegrals over field configurations, not paths in space as in non-relativistic QT. The reason is the same as in the operator formalism: A single-particle treatment a la "1st quantization" is not adequate for relativistic QT. The reason is that in the interacting case, particle numbers are not conserved (only certain charges of the Standard Model like electric charge) and you need QFT to take the possible annihilation and creation processes properly into account. Particularly photons are easily annihilated and created. There's no restriction concerning any charge-like conserved quantity for them, and in addition they are massless, i.e., even energy conservation cannot restrict the possible photon production/annihilation processes considerably.

That's all true, but it doesn't help to answer the question.

First, even QFT can be formulated in terms of path integrals, and for bosonic fields the "integral" really means integral (i.e. continuous sum)*. So you sum over various field configurations, and some of those configurations propagate faster than light. Yet, one must not omit them.

Second, whatever the physical interpretation of "first-quantized" Klein-Gordon equation might or might not be (for instance, one can think of it as a classical field theory), one can always solve this equation mathematically by using the method of path integrals. It is perfectly legitimate to ask whether superluminal paths contribute or not.

*For fermionic fields the "integral" does not really mean the integral or any kind of sum. A long time ago I have attempted to represent fermionic integrals as true sums
https://arxiv.org/abs/hep-th/0210307
but a smart PRL referee found an error which I was not able to fix.