Undergrad Can Photons travel faster than c?

[Electric to be]

I've looked up this question on the web, and I've gotten seeming conflicting answers.

According to Feynman's path integral - to find the probability of a photon being at A at time 1 and B at time 2 can be determined by taking an integral of the photon traveling over all possible paths. I understand that these paths more than anything are mathematical constructs and should not be taken super literally (since classical trajectories don't really make sense for quantum particles)

Regardless, I have seen on the internet that for points outside of the light cone, the integral results in very low probability of photons being detected, but still nonzero. This effect apparently becomes more significant for very small distances, and goes to zero at larger distances. This would mean there is an exponentially decaying probability of the photon having a speed greater than c. However, in other places I've seen people say pretty strictly that no it is not possible for the photon to have even slightly larger values of c. Even still, I've seen people say that while the photon may travel faster than c, information cannot. This I don't understand. If a photon is known to have been emitted at time 1, and absorbed at time 2, and this is faster than the speed of light - how did information not travel at the same speed? Though I know this has some implications on Causality..

So what is the truth?

Thanks.

 

[NFuller]

No particle can travel faster than the speed of light in vacuum. I suspect that you are seeing a non vanishing probability density outside the light cone because you are not using the relativistic formulation of quantum mechanics. Are you solving the Schrödinger Eq., the Dirac Eq., or the Klein Gordon Eq. ?

 

[Electric to be]

No particle can travel faster than the speed of light in vacuum. I suspect that you are seeing a non vanishing probability density outside the light cone because you are not using the relativistic formulation of quantum mechanics. Are you solving the Schrödinger Eq., the Dirac Eq., or the Klein Gordon Eq. ?

Well I'm not solving anything - I was referring to what others have said about the matter. I was referring to the path integral in quantum field theory ( relativistic) for a photon.

 

[NFuller]

If it's quantum field theory they are probably solving the Klein-Gordon equation. This equation is Lorentz invariant and should not allow faster than light solutions.

One thing to consider is that these paths are not really true particle paths. If they were then this would be classical mechanics and not quantum mechanics. It turns out however, that when you use the path integral formalism for a quantum particle you get the same probability density as you would if you use a propagator (the traditional approach). Some of these paths are classically forbidden, such as the faster than light paths you mention, but they are all required to sum to the correct probability amplitude.

Now all this assumes that your point "b" is in the light cone centered at point "a". If your end point "b" moves outside of this light cone then all of these paths should cancel each other out to give a total probability amplitude of zero.

 

[Electric to be]

If it's quantum field theory they are probably solving the Klein-Gordon equation. This equation is Lorentz invariant and should not allow faster than light solutions.

One thing to consider is that these paths are not really true particle paths. If they were then this would be classical mechanics and not quantum mechanics. It turns out however, that when you use the path integral formalism for a quantum particle you get the same probability density as you would if you use a propagator (the traditional approach). Some of these paths are classically forbidden, such as the faster than light paths you mention, but they are all required to sum to the correct probability amplitude.

Now all this assumes that your point "b" is in the light cone centered at point "a". If your end point "b" moves outside of this light cone then all of these paths should cancel each other out to give a total probability amplitude of zero.

I'm no expert in QM, but I always thought QFT was a separate animal, and left behind Schrodinger, Dirac, and Klein-Gordon equations for the "path integral formalism". But that is slightly digressing.

While I'm not sure how reliable, here is an external link :

https://www.quora.com/When-analysin...ove-faster-than-light-to-reach-B-at-the#!n=12

I've seen in other locations as well, that the summation due to classically forbidden terms are small, but the overall summation is still nonzero.

 

[NFuller]

the summation due to classically forbidden terms are small, but the overall summation is still nonzero.

Right, this is what I meant by saying "they are all required to sum to the correct probability amplitude." When the particle moves from point a to b, it does not do so by following some path between the two points but instead it propagates as a field. The path integrals aren't necessarily the actual paths being followed by the particle so there is no reason to believe anything is traveling faster than light.

The important part is that if point b is outside of point a's light cone then the probability density at point b is zero even if the probabilities for individual paths are non-zero. You might think "hey if some paths have probabilities that are non-zero then do some paths need negative probabilities to get a total of zero at point b?" The answer is of course YES. The Klein-Gordon equation allows for negative probability densities, this is related to antiparticles.

 

[Electric to be]

Right, this is what I meant by saying "they are all required to sum to the correct probability amplitude." When the particle moves from point a to b, it does not do so by following some path between the two points but instead it propagates as a field. The path integrals aren't necessarily the actual paths being followed by the particle so there is no reason to believe anything is traveling faster than light.

The important part is that if point b is outside of point a's light cone then the probability density at point b is zero even if the probabilities for individual paths are non-zero. You might think "hey if some paths have probabilities that are non-zero then do some paths need negative probabilities to get a total of zero at point b?" The answer is of course YES. The Klein-Gordon equation allows for negative probability densities, this is related to antiparticles.

But the point he was trying to make was that the overall summation (sum of all the paths) was non-zero, so the probability density would be non-zero. He goes on to explain that even so this doesn't violate Causality.

 

[NFuller]

I think the confusion may be coming from the fact that the path integral formulation used often relies on a WKB approximation. This means that the propagator may give non vanishing solutions outside the light cone where the approximation is not as good. The exact solution however will always be zero outside the light cone. Here is a link to a paper going into some detail on this. https://arxiv.org/pdf/gr-qc/9210019.pdf

 

[PeterDonis]

I have seen on the internet

Where? Please give a reference. And if it isn't a textbook or peer-reviewed paper, be prepared to be told that it isn't a valid reference and you should look at textbooks or peer-reviewed papers.

 

[Electric to be]

I think the confusion may be coming from the fact that the path integral formulation used often relies on a WKB approximation. This means that the propagator may give non vanishing solutions outside the light cone where the approximation is not as good. The exact solution however will always be zero outside the light cone. Here is a link to a paper going into some detail on this. https://arxiv.org/pdf/gr-qc/9210019.pdf

And this applies for photons in particular?

Where? Please give a reference. And if it isn't a textbook or peer-reviewed paper, be prepared to be told that it isn't a valid reference and you should look at textbooks or peer-reviewed papers.

Well unfortunately I don't have any at the moment. The idea of what I was talking about is posted above in the Quora link, though. However, apparently Feynman said this exact same thing. That there is a probability of this occurring, however small, especially at small distances. He says: "The amplitudes for these possibilities are very small compared to the contribution from speed c; in fact, they cancel out when light travels over long distances."

 

[NFuller]

And this applies for photons in particular?

It would apply to any particle.

 

[Electric to be]

It would apply to any particle.

Got it. However, as I posted above Feynman said (extended version):

"It may surprise you that there is an amplitude for a photon to go at speeds faster or slower than the conventional speed, c. The amplitudes for these possibilities are very small compared to the contribution from speed c; in fact, they cancel out when light travels over long distances. However when distances are short... these other possibilities become vitally important and must be considered." Pg 89 QED

Implying at shorter distances, they wouldn't necessarily cancel out. Is this in contrast with what you said?

 

[PeterDonis]

unfortunately I don't have any at the moment

This is an "I" level thread. That means you are assumed to have undergraduate level knowledge of the subject matter. Such knowledge is normally obtained by studying at least some textbooks or peer-reviewed papers. Have you studied any?

 

[PeterDonis]

The idea of what I was talking about is posted above in the Quora link, though.

Note carefully this statement from the Quora link:

"no measurement can affect any other measurement outside of its lightcone."

What that means is that, if we have two measurements that are spacelike separated, their results can't affect each other. So if one "measurement" is the emission of a photon, and the other "measurement" is the absorption of a photon, neither one can affect the other. That is why it is said that no information can be transmitted in this manner--information transmission requires the "source" to be able to affect the "receiver" in some controllable way.

Also, you have to be careful not to assume that the photon emitted at one event and the photon detected at the other event are "the same" photon. Photon's don't have little identity labels on them. You can say that a photon was emitted at event A and a photon was detected at event B, but you can't say the two are "the same" photon.

Pg 89 QED

As good as this book is, it's still a pop science book, not a textbook or peer-reviewed paper. As such, it leaves out things. For example, it says that "these other possibilities are vitally important and must be considered", but it doesn't (IIRC) say exactly what they must be considered for.

 

[Electric to be]

Note carefully this statement from the Quora link:

"no measurement can affect any other measurement outside of its lightcone."

What that means is that, if we have two measurements that are spacelike separated, their results can't affect each other. So if one "measurement" is the emission of a photon, and the other "measurement" is the absorption of a photon, neither one can affect the other. That is why it is said that no information can be transmitted in this manner--information transmission requires the "source" to be able to affect the "receiver" in some controllable way.

Also, you have to be careful not to assume that the photon emitted at one event and the photon detected at the other event are "the same" photon. Photon's don't have little identity labels on them. You can say that a photon was emitted at event A and a photon was detected at event B, but you can't say the two are "the same" photon.As good as this book is, it's still a pop science book, not a textbook or peer-reviewed paper. As such, it leaves out things. For example, it says that "these other possibilities are vitally important and must be considered", but it doesn't (IIRC) say exactly what they must be considered for.

Well alright. I have seen similar statements on forums such as PhysicsStack and while I agree it may not be a peer reviewed source it's just confusing hearing this seemingly conflicting statements - especially from the likes of somebody like Feynman. What is your take on the matter? Is a non-zero probability of detecting outside of the light cone nonsense? Or is something like what was said on Quora close to the truth - photons may travel faster than c, but will not violate causality? I just seem to be getting somewhat conflicting answers.

 

[PeterDonis]

Is a non-zero probability of detecting outside of the light cone nonsense?

No.

photons may travel faster than c, but will not violate causality?

No. You have assumed that the photon emitted and the photon detected are "the same photon", which "travels" between the two events. You can't assume that. Go back and read my previous post again.

I just seem to be getting somewhat conflicting answers.

No, you're failing to recognize that you are making implicit assumptions which are false.

 

[Electric to be]

No.
No. You have assumed that the photon emitted and the photon detected are "the same photon", which "travels" between the two events. You can't assume that. Go back and read my previous post again.
No, you're failing to recognize that you are making implicit assumptions which are false.

Well then how can we ever talk about the concept of photons moving at a particular speed, if it is not the time between the movement of nonzero probability? It doesn't matter to me if it's one photon or another, if the probability of photon being detected at point B after being emitted at point A at time X seconds, then that is good enough for the speed of a traveling photon?

 

[PeterDonis]

how can we ever talk about the concept of photons moving at a particular speed

In certain situations, this is a good enough approximation and it greatly simplifies the analysis. But it is only an approximation, and in many situations, such as the one under discussion here, it breaks down.

if the probability of photon being detected at point B after being emitted at point A becomes nonzero after x seconds, then that is good enough for the speed of a traveling photon?

No. See above.

 

[Electric to be]

In certain situations, this is a good enough approximation and it greatly simplifies the analysis. But it is only an approximation, and in many situations, such as the one under discussion here, it breaks down.
No. See above.

Alright. I guess I'll just accept that I won't really understand the matter fully since I don't have much formal training in QFT. Thanks anyway.

 

[georg gill]

The speed of photons in vacuum are derived in electromagnetism. You would first need the maxwell equations to get to that. Look for example here where they arrive at this result in (453):

http://farside.ph.utexas.edu/teaching/em/lectures/node48.html

The speed of photons is slower if it is not vacuum. This is also derivable from the fundamental electromagentic equations.

schrødinger is based on a differential equation that is based on maxwell equations. They also use the fact that the speed of photons are c in the derivation of schrødinger.

 

[Electric to be]

The speed of photons in vacuum are derived in electromagnetism. You would first need the maxwell equations to get to that. Look for example here where they arrive at this result in (453):

http://farside.ph.utexas.edu/teaching/em/lectures/node48.html

The speed of photons is slower if it is not vacuum. This is also derivable from the fundamental electromagentic equations.

schrødinger is based on a differential equation that is based on maxwell equations. They also use the fact that the speed of photons are c in the derivation of schrødinger.

I am fully aware of the wave equation in classical electromagnetism, and that light moves at c in all reference frames according to these equations. That wasn't the concern here. Classical electromagnetism (Maxwell's equations) also are not a description of photons. That requires Quantum Field Theory, and is the issue that I was concerned with here.

I'm also not sure what you mean about using Maxwell's equations in the derivation of Schrodinger's equation. Schrodinger's equation is non-relativistic. That's diverging from the topic anyways.

 

[Nugatory]

The speed of photons in vacuum are derived in electromagnetism. You would first need the maxwell equations to get to that. Look for example here where they arrive at this result in (453)
http://farside.ph.utexas.edu/teaching/em/lectures/node48.html

The word "photon" doesn't even appear in that piece, and it is quite clear that they are talking about the speed of electromagnetic waves , not photons.

schrødinger is based on a differential equation that is based on maxwell equations. They also use the fact that the speed of photons are c in the derivation of schrødinger.

There have been efforts to motivate Schrodinger's equation by analogy with E&M, but it is a huge stretch to call any of these attempts "derivations". A proper derivation can be found in advanced textbooks such as Sakurai, and there's an outline of such a derivation in the Wikipedia article for "Schrodinger equation"; neither the invariance of $c$ nor Maxwell's equations are involved.

 

[Electric to be]

In certain situations, this is a good enough approximation and it greatly simplifies the analysis. But it is only an approximation, and in many situations, such as the one under discussion here, it breaks down.
No. See above.

If I may ask though, in what situations is it valid? Why do people so often say that photons "only go at the speed of light, c" when situations like this break down this sort of notion? I know that for longer distances, the probability of detection is also essentially zero outside the lightcone. Is it valid in situations where the speed of information matches the speed of photon detection, so essentially for longer distances?

Clearly on a macroscopic classical sense, EM waves are only observed to go at C, but of course we aren't talking about photons here.

 

[Nugatory]

Why do people so often say that photons "only go at the speed of light, c"?

Because they're speaking carelessly, and they should have said "light signal" or "flash of light" or "wavefront" instead of "photons".

 

[vanhees71]

One hint: Before thinking about QFT start with classical electrodynamics, if it comes to photons. There in the very early days of special relativity this question has troubled the famous experimentalist Willy Wien: He knew very well that electromagnetic waves can have phase and group velocities larger than $c$ in media, around a resonance. The phenomenon is known as "anomalous dispersion". Principally this apparent contradiction to SRT was as quickly solved by the famous theorist Arnold Sommerfeld in an article taking about half a page to show that there is nothing traveling faster than $c$ that is not "allowed" to do so by SRT, which is no surprise since classical electrodynamics is the paradigmatic example of a relativistically covariant local classical field theory. The problem was worked out in much more detail by Sommerfeld himself and Brillouin, where they explicitly calculated the wave form of an em. wave train entering a medium in the region of frequencies of anomalous dispersion. It comes out that within their model the wave front in fact moves with $c$ (i.e., with the speed of light in vacuo), because the medium hasn't had time to respond to the field yet and thus is doing nothing to the wave front. Then you have some transient state, where very rapidly oscillating "precursors" occur (there can be two types, the Sommerfeld and the Brilloin precursor depending on the details of the parameters of the dielectric function of the medium) which then go over into the steady state, where the field inside the medium is a wave with the same frequency as the incoming wave, which is what you usually have in mind when describing the situation with the Fresnel equations: This implicitly assumes that the wave field is present for a sufficiently long time within the medium such that the transient states all are damped away, and the steady state is reached. This cannot happen at speeds faster than the speed of light in vacuum, since the wave front travels at this speed, i.e., nothing happens earlier than the time of the head of the wave needs to travel in the vacuum to reach a certain position in the medium, i.e., before that time there's neither a response of the medium nor a field different from 0.

 

[ISamson]

No particle can travel faster than the speed of light in vacuum.

Is there any experimental proof for this?
I know no particle can, but I am interested in the proof.

 

[NFuller]

Is there any experimental proof for this?

A particle accelerator.

 

[phinds]

Is there any experimental proof for this?
I know no particle can, but I am interested in the proof.

There is no such thing as a proof in physics. That's for math. You can only ask is there any observational evidence for ... and is there any observational evidence against ...

Evidence FOR something does not prove it

A particle accelerator.

Which is just observational evidence and most emphatically not a proof

 

[PeterDonis]

Is there any experimental proof for this?

No. Experiments can't "prove" anything. But they can give evidence for it. The evidence is extremely strong that no particle can travel faster than the speed of light in a vacuum. That's the best you're going to get.

 

[Dale]

Several posts about the day to day activities of physicists have been removed. Such posts would belong in a new thread in the Career Guidance section.