For understanding my question I invite everybody to read the example.
In the Bohmian mechanics (BM) the velocity formula gives an infinite value to the velocity of the Bohmian particle at points where the wave-function vanishes, but the gradient doesn't vanish.
Do we have any proof that this is wrong, i.e. that attributing superluminal velocity to a particle is wrong? Could it be that this feature of the BM is a flaw, and implies that BM is wrong?
EXAMPLE:
In a Hanbury-Brown and Twiss (HB&T) type experiment with a pair of identical photons, one passing through a slit A and the other passing through a slit B - see attach - the wave-function of the pair looks as follows:
Ψ(r1, r2) = 2-½ {exp[ik|r1 - rA| exp(ik|r2 - rB|) + exp(ik|r1 - rB|) exp(ik|r2 - rA|) }
= 2-½ exp[iθ(r1, r2)] cos[ϕ(r1, r2)]
where r1 , r2, denore the positions of the photons, and rA, rB, the positions of the slits.
ϕ(r1, r2) = (κ/2) { (|r1 - rA| + |r2 - rB|) - (|r1 - rB|) + |r2 - rA|) }.
If ϕ is an odd integer multiple of π/2, the wave-function vanishes. Assume that so happens for the pair of points P'1 and P'2. If one places detectors at these two points and, say, the detector at P'1 makes a detection, the detector at P'2 remains silent. By the BM, the particle moving towards P'2 jumps over this point with an infinite velocity, and this is why it cannot be detected.
The problem is that the presence of the detector at P'2 reduces the probability Prob(P'1, Q2) of joint detection at P'1 and Q2, where Q2 is any point below P'2. This probability shows no more interference effect, it is given only by the cross waves from A to Q2 and from B to P'1.
Obviously, the detector at P'1 although cannot detect the particle passing through it, but yes stops something. The probability of joint detection in P'1 and Q2 decreases due to the detector at P'2 not only by the disappearence iof the nterference, but also below the sum of the isolated probabilities of detection from the crossed rays and detection from the direct rays.
Dear Sophia!
Short answer: A real particle (waht means an object with rest mass) can never move with luminal or superluminal velocity.
This is only possible in theory or in mathematics. But the reality is not a mathematical derivation and also not a creation of human or godly ideas.
Best wishes! Hans
Dear Sophia,
In principle there are nothing which forbids to have a superluminical particle. But only if such particle never was in one state of motion equal or lower that the one associated to the velocity of light. These are the tachyonic particles with must survive in the non causal region of the spacetime. For the moment this is a pure speculation that everybody avoids to find as solution.
Check out Cherenkov radiation, for motion faster than light in a material medium.
Dear Juan,
The Cherenkov radiation never gets a velocity of light higher than in the vacuum, in fact this radiation appears because the charged particle which have to radiate when it moves at a velocity higher than the velocity of light in such medium, but which is always lower than the velocity of light in vacuum. Special relativity is followed in all its terms.
@Juan Weisz
The issue here is not the light velocity in media, neither is here a question about tunneling.
The issue is whether a theory admitting that the light velocity in vacuum, is wrong. I added right now an example of strange behavior of the particle said to have a superluminal velocity.
Dear Hans, Daniel, Juan, an the other users interested in my question!
Please pay attention to the example I added to the question. It shows that the particle assumed by the BM to acquire a superluminal velocity at points where the wave-function vanishes, has a quite contradictory behavior.
Since a Bohmian particle cannot be observed to have superluminal speed, there is no contradiction with relativity.
I guess this is akin to the situation in tunneling, where particles seem to pass a potential wall in zero time (or, according to newer results, in a very small time interval corresponding to many times the speed of light). This does not lead to a contradiction with relativity either, because no observable particle velocity exceeds the speed of light.
Daniel, my answer already implied your comment. Sorry if I rambled away from
the original issue. I do not know anything about truely superluminal stuff.
Tunneling time is a bit mysterious,there is not a single accepted criteria for that.
Dear Klaus,
I would be interested that you read the example in my question. In that example it is shown that if a photon is detected at P'1, the other photon is not detected at P'2 because the wave-function vanishes for this pair of points. But, if the detector is moved from P'2 to a lower point, Q2, s.t. Prob(P'1, Q2) ≠ 0, the photon is detected at Q2. Bohm's mechanics explains this fact by claiming that the particle jumps over P'2 at superluminal speed, and that's why it can't be detected at P'2.
But, the things are not so simple. If a detector is left at P'2, the probability of joint detection Prob(P'1, Q2) decreases from the value it has if there is no detector at P'2. The formulas show that only the cross rays contribute to Prob(P'1, Q2) in this case. That means, in less cases a particle arrives at Q2 in combination with a particle arriving at P'1. The conclusion is that the detector at P'2 , stops the the impinging superluminal particle, but doesn't report it. How so?
Dear Juan,
You wrote it so shortly than I thought that you considered Cherenkov radiation related with relativistic velocities higher than the light. Relativity considers as a limit the velocity of the light in vacuum.
Dear Sofia,
By the BM, the particle moving towards P'2 jumps over this point with an infinite velocity, and this is why it cannot be detected.
the problem here is that there is no particle at all which travels at indefinite speed. It is as if the particle made a jump at superluminal speed, but this did not occur. What can occur instead is that the wave function associated , for which a speed cannot be defined, collapses in a differnet portion of space than expected.
This example has similarities to the experiment in which the wavefunction of particulary treated photons collapses at two distance places according to behaviour of detectors. There is nothing in that case which actually "travels" but there is an action (detection) on an entity (wavefunction) which determines the place where the collapse occurs.
Dear Stefano,
It's not I who says that a particle may have superluminal velocity. It's the Bohmian mechanics (BM), which says that there where the wave-function is null, the paticle passes through those points at superluminal velocity. Essentially, they explain that the particle cannot be detected at those points because it passes through them so quickly (at superluminal velocity) that it can't be caught.
The Bohmian particle is not an ordinary particle as in the non-quantum physics.
You say "the problem here is that there is no particle at all which travels at indefinite speed." What I am trying to do, is to prove that the hypothesis of superluminal velocity is impossible. The basic argument in my proof is that the QM calculus shows that a detector placed at a point where the wave-function is null, seems to stop the particle. This seems a contradiction, that the detector stops the particle but can't detect it.
Please try to follow the rationale in my example. I placed it here for getting suggestions, criticism, etc. Maybe somebody would find some weak point in it, or would suggest clarifications, improvements.
Dear Sofia,
I don't know the Bohm's quantum mechanics enough to understand the superluminical argument. The main achievement of his model was the Aharonov-Bohm effect where the potentials can take a set of singularities for taking into account the places where the fields are zero. Thus if you use Lorentz equation for transport of one electric chage (which only takes into account the fields) you could say that nothing could affect the electric charge while if you employe quantum mechanics you would observe a change of phase (just a translation) on the spectra. This obviously doesn't takes any assumption of superluminical behaviour.
Let me give you one counterexample of a region where the particle cannot stay and which is not related with superluminical transport. In a semiconductor you have a gap where the particles cannot survive and an distinguish the conducting and valence bands.
Dear Daniel,
Thanks for trying to help. Please see, my question is related to an article. The example there is of Hanbury-Brown and Twiss type, as I wrote in the question. So, semiconductors, or Bohm-Aharonove, are not connected with it. But, I wrote you a message with details, please look at it.
Dear Sophia,
here is the link of a Phiscist dealing with the interpretation of QM... I don't share his point of view but it is interesting to read.
Article Making Sense of Bell's Theorem and Quantum Nonlocality
When you look at the particle energies, imaginary answers occur at faster than light speed. You have the choice of adding more dimensions and trying again to get a real energy spectrum, or accepting that the physical event is not real.
My work shows that FTL does not occur, because the increase of kinetic stress energy causes the local light speed to increase as measured by the particle frame of reference, a concept allowed by Albert Einstein and endorsed by Max Born. Doppler shifting is helpful to show that FTL does not occur, because the forward looking temperature goes to infinity at light speed. FTL is temperature greater than infinity, which has no meaning.
Before these conditions can be reached GR must modify to include QM, which means the standard model doesn't describe everything correctly in that energy range.
Scale relativity, TGD, and squeezed quantum states lead to possibilities of 4D tunneling through 6D folded space, apparent FTL from a distance, but not locally measured FTL. Example is scale 4 which occurs at high energies and velocities close to light speed, giving predictions of Galilean flat space inside a warp field elongated to worm hole shape by frame dragging of local stress energy fields.
Scale 5 is the infinite temperature and total destruction which prevents local FTL.
Otherwise the FTL math is fun to play with.
In non-relativistic mechanics, whether classical or quantum, the velocity of light doesn't play any particular role. So it's meaningless to attempt to assign it any significance in such a context. The quantum states that don't have a classical limit aren't special in that regard, either. So it doesn't make sense, either, to claim that non-relativistic quantum mechanics has conceptual issues, just because it isn't consistent with special relativity. It's the fact that it is a non-relativistic description, not that it's a quantum description, that makes it inconsistent with special relativity-which is to be expected. That doesn't mean that it is inconsistent as such, though.
So arbitrarily large velocities aren't a distinguishing feature of Bohmian mechanics. They're irrelevant to discussing its consistency.
The fact that the wavefunction vanishes, by itself doesn't mean anything, because that statement can be shown not to have any effect on any observable quantity-since the wavefunction itself only makes sense for constructing the probability density.
From the two-particle wavefunction it's completely straightforward to compute the joint probability density-which is what matters for the physics-and, from that, any correlation desired. These correlations describe all the physics of the problem.
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