Contradiction between Quantum Mechanics and Special Relativity？

[sunroof]

In quantum mechanics, the phase velocity is f/k=E/p because of E=hf and p=hk where k=1/wavelength.

In special relativity, E=mc^2 and p=mV (V<c). So, E/p=cc/V >c.

To photons in media, however, the phase velocity is f/k =c/n<c.

What’s wrong ?

 

[Jonathan Scott]

In quantum mechanics, the phase velocity is f/k=E/p because of E=hf and p=hk where k=1/wavelength.

In special relativity, E=m times square of c and p=mV (V<c). So, E/p=cc/V >c.

To light waves in media, however, the phase velocity is f/k =c/n<c.

What’s wrong ?

For special relativity, E/p=c²/v even for relativistic speeds.

For both types of wave, the phase velocity for a particle without rest mass in free space is c.

In QM, the case where the speed is not c relates to a particle with non-zero rest mass with a wave propagating freely.

For light waves in media, the equation relates to a particle with zero rest mass which is not in free space, which is effectively being impeded (as if it had to travel further). This effect relates specifically to electromagnetic waves.

This means that the cases are not comparable.

 

[sunroof]

For special relativity, E/p=c²/v even for relativistic speeds.

For both types of wave, the phase velocity for a particle without rest mass in free space is c.

In QM, the case where the speed is not c relates to a particle with non-zero rest mass with a wave propagating freely.

For light waves in media, the equation relates to a particle with zero rest mass which is not in free space, which is effectively being impeded (as if it had to travel further). This effect relates specifically to electromagnetic waves.

This means that the cases are not comparable.

Thanks very much for your reply. But the answer is to avoid and not to resolve issue. The validity of E/p=c^2/V, E=hf and p=hk is independent of the value of V and rest mass. Especially, to photons in vacuum where V=c and rest mass is zero, E/p=c^2/V=c of SR is consistent with the relation E/p=hf/hk=f/k=c in QM. Why the result is comparable in vacuum but incomparable in media? Then how to describe photons in media in SR and QM to get the known f/k=c/n<c in optics? Whether it implies SR or QM is not universal, if they cannot be applied to study photons in media?

 

[clamtrox]

I think you could resolve this by saying that the photons get an effective mass from the medium so E^2 = m_{\mathrm{eff}}^2 + p^2.

 

[jtbell]

To photons in media, however, the phase velocity is f/k =c/n<c.

Individual photons always travel at speed c. In a medium, a macroscopic light wave (the collective effect of many many photons which are interacting with the medium) travels at a speed which is less than c.

See the FAQ at the top of the General Physics forum for some discussion of how light propagates in a medium.

 

[bcrowell]

You don't have to do anything this complicated to find a contradiction between qm and relativity. The simplest way to see that there's a contradiction is that in qm information is always preserved, in the sense that wavefunctions evolve according to unitary operators, whereas in relativity you can lose information by dropping it into a black hole, where it's hidden behind an event horizon.

You're not going to get a contradiction between qm and relativity from your present example because it's an example that's restricted to special relativity. Special relativity is consistent with qm.

 

[jambaugh]

You don't have to do anything this complicated to find a contradiction between qm and relativity. The simplest way to see that there's a contradiction is that in qm information is always preserved, in the sense that wavefunctions evolve according to unitary operators, whereas in relativity you can lose information by dropping it into a black hole, where it's hidden behind an event horizon.

You're not going to get a contradiction between qm and relativity from your present example because it's an example that's restricted to special relativity. Special relativity is consistent with qm.

This is not a contradiction. Conservation of information does not imply conservation of availability of that information. The very fact that black holes have entropy is that the information is "still there" but just inaccessible... unless you want to cross the event horizon after it. A black hole behaves no differently w.r.t. quantum information than does any other entropy dump.

 

[George Jones]

This is not a contradiction. Conservation of information does not imply conservation of availability of that information. The very fact that black holes have entropy is that the information is "still there" but just inaccessible... unless you want to cross the event horizon after it. A black hole behaves no differently w.r.t. quantum information than does any other entropy dump.

But classical black hole spacetimes are singular. Classical black hole spacetimes have inextendible (timelike) worldlines that have bounded 4-acceleration, i.e., there are observers in rockets that have finite thrusts who "fall off the edge of spacetime" in a finite amount of proper time.

 

[jambaugh]

But classical black hole spacetimes are singular. Classical black hole spacetimes have inextendible (timelike) worldlines that have bounded 4-acceleration, i.e., there are observers in rockets that have finite thrusts who "fall off the edge of spacetime" in a finite amount of proper time.

Correct but their finite proper time is not coincident with external observer's time. One can in principle preserve the information and make it available to an observer crossing the event horizon after waiting an indeterminate time into the (external) future. Said observer could then compare the information (e.g. compare halves of an entangled pair) in the finite time before he too is squished by the singularity.

(I'm not absolutely certain about this, it may depend on the size of the BH as to how long one can wait before the spatial separation upon entering is greater than the remaining time.)

Recall that from the external observer's time perspective an infalling object (if we ignore its grav. mass) never actually reaches the horizon so there is not a definite external time when the information carried ceases to exist. I don't see it as qualitatively different than information encoded in a pair of entangled photons flying off in opposite directions along a line not intersecting any nearby objects. One cannot access the information as one cannot catch up with it. It has by no means ceased to exist in the sense of violation of conservation of information.

 

[bcrowell]

But classical black hole spacetimes are singular. Classical black hole spacetimes have inextendible (timelike) worldlines that have bounded 4-acceleration, i.e., there are observers in rockets that have finite thrusts who "fall off the edge of spacetime" in a finite amount of proper time.

On the one hand, I'd like to use this to support my original claim by the following argument. An observer who falls past the event horizon of a black hole can toss stuff ahead of him, and he'll say that that stuff is hitting the singularity before he does. Therefore, this observer does observe the destruction of information, and he says (in his final moments), "Hey, that violates quantum mechanics!"

On the other hand, I'd like to avoid the necessity of talking about this kind of observer, because this kind of observer can observe a naked singularity. Forget about quantum mechanics, any classical field theory that has singularities in it is not really a satisfactory, self-consistent theory. So you could argue that what the observer has detected is not evidence that gr is inconsistent with qm, but simply evidence that gr is inconsistent with gr.

 

[bcrowell]

Correct but their finite proper time is not coincident with external observer's time. One can in principle preserve the information and make it available to an observer crossing the event horizon after waiting an indeterminate time into the (external) future. Said observer could then compare the information (e.g. compare halves of an entangled pair) in the finite time before he too is squished by the singularity.

(I'm not absolutely certain about this, it may depend on the size of the BH as to how long one can wait before the spatial separation upon entering is greater than the remaining time.)

Recall that from the external observer's time perspective an infalling object (if we ignore its grav. mass) never actually reaches the horizon so there is not a definite external time when the information carried ceases to exist. I don't see it as qualitatively different than information encoded in a pair of entangled photons flying off in opposite directions along a line not intersecting any nearby objects. One cannot access the information as one cannot catch up with it. It has by no means ceased to exist in the sense of violation of conservation of information.

This kind of stuff is subtle. I posted a reply to this and then deleted it because I realized it wasn't right :-) How about this. Suppose observers A and B make a suicide pact. A is going to jump into one supermassive black hole, and B is going to jump into the other. They are both going to live for a finite amount of proper time after crossing the respective event horizons, but they will never be able to communicate again. Any private knowledge A took with him is permanently unavailable to B.

I think your argument also depends on the fact that it's a static black hole solution we're talking about. I think information is clearly lost during the formation of the black hole. E.g., it seems to me that there is matter that was already behind Sgr A*'s event horizon at the moment the horizon formed, and we're never going to be able to find out about it.

 

[jambaugh]

On the one hand, I'd like to use this to support my original claim by the following argument. An observer who falls past the event horizon of a black hole can toss stuff ahead of him, and he'll say that that stuff is hitting the singularity before he does. Therefore, this observer does observe the destruction of information, and he says (in his final moments), "Hey, that violates quantum mechanics!"

On the other hand, I'd like to avoid the necessity of talking about this kind of observer, because this kind of observer can observe a naked singularity. Forget about quantum mechanics, any classical field theory that has singularities in it is not really a satisfactory, self-consistent theory. So you could argue that what the observer has detected is not evidence that gr is inconsistent with qm, but simply evidence that gr is inconsistent with gr.

I don't think this is the correct interpretation of the space-time inside the event horizon of a black-hole. The singularity is a temporal boundary at which (two) of the dimensions of the space inside the EH achieve zero extent. One doesn't "hit" the singularity in the sense of a spatial collision, rather it is more of the ultimately doomsday clock... Buzzzzzzzzzz! GAME OVER! (as the tubular space collapses to a line)

Remember the r in both the interior and exterior Schwarzschild solution is a circumferential radius i.e. it is a parameter defining how large a spatial cross section is. That this parameter is time-like in the interior shows that one should view it as a time dependent contraction of the spatial geometry. The interior solution is not stationary with regard to observers inside the EH. This is why the singularity becomes extensive in those coordinate systems which avoid the coordinate singularity at the horizon.

The singularity has no duration. It has one dimensional spatial extent...which maps back to the time-like parameters outside the horizon in that where you are at the time of singularity will depend on when you entered the BH modulo additional dynamic variations.

Subsequently you cannot "throw" an object into the singularity ahead of you. You only throw an object away from you so that it will be "over there" when the time of singularity occurs.

Now with relativity you can play with simultaneity and assert that different spatial locations achieve the singularity time one before the other, at the same time, or the other before the one. But one cannot avoid the fact that the singularity is a spatially extensive event and not a time extensive singular point in space... (assuming the interior Schwarzschild solution is the correct one.)

 

[sunroof]

I think you could resolve this by saying that the photons get an effective mass from the medium so E^2 = m_{\mathrm{eff}}^2 + p^2.

Thanks very much for your advice. I considered this interpretation long ago and gave up. Substituting E=hf and p=h/wavelength into the relativistic E=sqrt((squared pc)+(squared rest mass c^2)) will lead to a negative squared rest mass, i.e.the effective rest mass is imaginary.This is unreasonable.

 

[sunroof]

Individual photons always travel at speed c. In a medium, a macroscopic light wave (the collective effect of many many photons which are interacting with the medium) travels at a speed which is less than c.

See the FAQ at the top of the General Physics forum for some discussion of how light propagates in a medium.

It is just to simplify the problem. The difference between the dielectric constant and magnetic permeability of media and vacuum reflects the interaction. The consequence is that the light speed u is changed because it depends on these two parameters. So, we can treat photons in media as free particles whose velocity is V=u=c/n<c. It is unnecessary to consider interaction now because it is included in the change of the light speed.

 

[clamtrox]

If you are getting an imaginary mass out of it, then you are formulating it incorrectly. Things with a real nonzero mass travel slower than the speed of light. Things with an imaginary mass travel faster than the speed of light. The photon in a medium travels slower than the speed of light so it has an effective mass which is real and nonzero.