# Bohmian Mechanics: Do photons travel faster than c in double slit experiment?

• PRODOS

#### PRODOS

In the double slit experiment, Bohmian Mechanics http://plato.stanford.edu/entries/qm-bohm/#2s" the paths of real particles traveling from the two slits to the detector to look like something like this:

The above image shows particles traveling in non-straight paths.

The diagram below represents the double slit experiment.

The two slits are at points A and B.

A hypothetical real photon goes through the slit at A and is detected at point Y on the detector screen.

Sp is a straight line from A to Y.
Represent its length as “s”.

Bp is a non-straight line from A to Y.
Represent its length as “b”.

b > s

If the time it takes a photon to leave A and be detected at Y could be measured, and ...

If this was found to be:

s/c

(c is the speed of light)

Then would it be reasonable to conclude the following about the hypothetical real photon?

Either:

• The photon has in fact traveled in a straight line from A to Y - rather than in a non-straight line as depicted by Bohminan Mechanics

Or:

• The photon has at some point along its trajectory traveled faster than the speed of light

I would appreciate comments and corrections regarding this hypothetical real photon scenario.

Thanks.

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The diagram you have provided is only supposed tp apply to electrons (particles with mass), not photons (which obviously have no mass).

As far as I know Bohmian Mechanics does not describe photons, because to do that you need QFT, not just QM. This concurs with what is said in the link you posted:

"Like nonrelativistic quantum theory, of which it is a version, Bohmian mechanics is incompatible with special relativity, a central principle of physics: it is not Lorentz invariant. Nor can Bohmian mechanics easily be modified to become Lorentz invariant."

Therefore Bohmian Mechanics fails to describe the most important developments in the last 60 years of physics (those arising from Quantum Field Theory) and it surprises me that the moderators on these forums allow it to be discussed at all since it is so far outside the mainstream.

"Like nonrelativistic quantum theory, of which it is a version, Bohmian mechanics is incompatible with special relativity, a central principle of physics: it is not Lorentz invariant. Nor can Bohmian mechanics easily be modified to become Lorentz invariant."

Therefore Bohmian Mechanics fails to describe the most important developments in the last 60 years of physics (those arising from Quantum Field Theory) and it surprises me that the moderators on these forums allow it to be discussed at all since it is so far outside the mainstream.

Get over your anger, man. BM (including its QFT generalizations) is completely observationally compatible with relativity, which you would know if you didn't refuse to read about it on principle.

Here is a paper "http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVM-449V3XH-3&_user=1495569&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000053194&_version=1&_urlVersion=0&_userid=1495569&md5=946c43811645f1f658f3c8f9b764731d"" by Ghose et al. which contains the correct diagram (indeed the one above is for electrons) and a full explanation for the original poster.

Although it is difficult to make BM fundamentally Lorentz invariant, it is Lorentz invariant on the average (and therefore observationally indistinguishable from the usual stuff). One can just as logically conclude that Lorentz invariance fails at this deeper level (why not?). This very possibility is being actively considered in other contexts, in particular particle physics and quantum gravity, where some expect LI to fail at the Planck scale. There's a minor industry of "emergent relativity" with various tests going on (see e.g. http://arxiv.org/abs/0707.2673" [Broken] and refs. therein). Funny how there's much less dogma about this in some other areas of physics.

Also - tell me again why the original question is a problem? Time of flight questions are meaningless in standard QM because the particles aren't there unless you look at them and therefore don't have trajectories (and you don't have a time operator). It is only in BM where the question makes any sense. Indeed one could in principle make deductions similar to what you say, but there is no experimental evidence whatsoever that "photons" travel in straight lines or at a speed different from c. Or am I missing something?

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Good afternoon.

The following image is from from the article, Bohmian trajectories for photons, by Partha Ghose, A.S. Majumdar, S. Guhab, J. Saub, Publication: 19 November 2001, Physics Letters A 290 (2001) 205–213 found @ http://web.mit.edu/saikat/www/research/files/Bohmian-traj_PLA2001.pdf

Its accompanying caption says: "Fig. 2. Bohmian trajectories for a pair of photons passing through two identical slits. Note that there is no crossing of trajectories between the upper and lower half planes."

I note that the photon trajectories - representing real particle photons according to Bohmian Mechanics - are clearly shown as being non-straight.

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The diagram you have provided is only supposed tp apply to electrons (particles with mass), not photons (which obviously have no mass).

As far as I know Bohmian Mechanics does not describe photons, because to do that you need QFT, not just QM. This concurs with what is said in the link you posted:

"Like nonrelativistic quantum theory, of which it is a version, Bohmian mechanics is incompatible with special relativity, a central principle of physics: it is not Lorentz invariant. Nor can Bohmian mechanics easily be modified to become Lorentz invariant."

Therefore Bohmian Mechanics fails to describe the most important developments in the last 60 years of physics (those arising from Quantum Field Theory) and it surprises me that the moderators on these forums allow it to be discussed at all since it is so far outside the mainstream.

Pilot wave theory is not as far from the mainstream as you suggest. While it is not Lorentz covariant and needs a preferred frame, Lorentz covariant quantum theories can be described with pilot wave theories, some of them (in particular scalar fields and the EM field) even in a more or less straightforward way. Theories for fermions exist too.

Although it is difficult to make BM fundamentally Lorentz invariant ...
Nothing is difficult when you know how to do it.
This refers to making BM fundamentally Lorentz invariant as well:
http://xxx.lanl.gov/abs/0811.1905 [Int. J. Quantum Inf. 7 (2009) 595-602]

Pilot wave theory ... While it is not Lorentz covariant and needs a preferred frame
Not necessarily:
http://xxx.lanl.gov/abs/0811.1905 [Int. J. Quantum Inf. 7 (2009) 595-602]

Therefore Bohmian Mechanics fails to describe the most important developments in the last 60 years of physics (those arising from Quantum Field Theory)
No it doesn't:
http://xxx.lanl.gov/abs/0904.2287

I note that the photon trajectories - representing real particle photons according to Bohmian Mechanics - are clearly shown as being non-straight.

You say this like it's obvious that 'photon trajectories' should be straight in this situation. Did I misunderstand this, or do you have a particular reason for thinking so?

PRODOS, in some situations the Bohmian velocities of photons may be larger than c. Nevertheless, this is not a problem. By using the general theory of quantum measurements, it can be shown that the Bohmian velocity of photons is exactly equal to c whenever you measure this velocity.

Good afternoon zenith8,

I note that the photon trajectories - representing real particle photons according to Bohmian Mechanics - are clearly shown as being non-straight.
You say this like it's obvious that 'photon trajectories' should be straight in this situation. Did I misunderstand this, or do you have a particular reason for thinking so?

No, I don't think it's at all obvious that the photon trajectories are straight.

My statement was intended to "update" my initial post which pictured Bohmian electron particle paths instead of photon particle paths.

i.e. To indicated that ...

Although the photon trajectories look different from those of electrons, since they're still "non-straight" my question remains unaffected.

For clarity, I'd better present the question again with the correct picture ...
[... Updated version of original post to show correct Bohmian photon particle paths, including some minor edits to text ...]

In the double slit experiment, http://plato.stanford.edu/entries/qm-bohm/#2s" the paths of real particle photons traveling from the two slits to the detector to look something like this:

http://web.mit.edu/saikat/www/research/files/Bohmian-traj_PLA2001.pdf
(Thanks to https://www.physicsforums.com/member.php?u=168088" for pointing me to this paper)

The above image shows photon particles traveling in non-straight paths.

The diagram below represents the double slit experiment.

The two slits are at points A and B.

A hypothetical real photon goes through the slit at A and is detected at point Y on the detector screen.

Sp is a straight line from A to Y.
Represent its length as “s”.

Bp is a non-straight line from A to Y.
Represent its length as “b”.

b > s

If the time it takes a photon to leave A and be detected at Y could be measured, and ...

If this was found to be:

s/c

(c is the speed of light)

Then would it be reasonable to conclude the following about the hypothetical real photon?

Either:

• The photon has in fact traveled in a straight line from A to Y - rather than in a non-straight line as depicted by Bohminan Mechanics

Or:

• The photon has at some point along its trajectory traveled faster than the speed of light

I'm interested in this issue because I'm trying to understand and compare the various physics theories, models, and interpretations.

That's basically why I registered for PhysicsForums.com in the first place. I would like to also mention that, as well as the wonderful resources, and diverse perspectives available @ PF, what clinched it for me was reading the posts of https://www.physicsforums.com/member.php?u=323" His calm, fair, thoughtful approach appealed to me greatly.

I agree with your earlier note, that ...
Time of flight questions are meaningless in standard QM because the particles aren't there unless you look at them and therefore don't have trajectories (and you don't have a time operator). It is only in BM where the question makes any sense.

Thanks.

Best Wishes,

Prodos
Melbourne, Australia

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DrChinese I have noticed as well is very understanding of the layman's curiosity. I read an article on quantum entangled particles and became interested out of curiosity if it would ever lead to FTL communication via digital signal. He was very understanding of my question and helped me understand what it was I was looking into. This also opened a Pandora's box of other questions for me as well. Some of the greatest physicists of our "time" have described some of these kinds questions as being "Impossible to comprehend fully" "Beyond the reach of metaphor, visualization, or even language itself."
Take the two slit experiment for example.
You watch/observe the particle and it passes through like a bullet.
No one watching and it acts like a wave that can go crazy and pass through both slits at the same time.
Now if this kind of thing baffles these guys how am I ever to play catch up?

The number 1 question I had was ... "Why do these particles/photons act this way?" Its as if they are smiling for the camera.

Some of the greatest physicists of our "time" have described some of these kinds questions as being "Impossible to comprehend fully" "Beyond the reach of metaphor, visualization, or even language itself."
Take the two slit experiment for example.
You watch/observe the particle and it passes through like a bullet.
No one watching and it acts like a wave that can go crazy and pass through both slits at the same time.
Now if this kind of thing baffles these guys how am I ever to play catch up?

The number 1 question I had was ... "Why do these particles/photons act this way?" Its as if they are smiling for the camera.

Hi Belzy,

I think this is precisely the point of the de Broglie/Bohm interpretation. Questions like yours can be given perfectly natural, even obvious answers, formulated in terms of fully visualizable continuously existing entities and with precisely-defined mathematics. And given that it is entirely observationally equivalent to the standard viewpoint, one might as well teach it that way in particular so that beginners don't get so confused. It simply is not necessary to make QM (particularly the non-relativistic kind) into the 'weird, mysterious discipline that nobody understands' of popular legend.

In the case of the two-slit experiment:

While each particle track passes through just one of the slits, the wave passes through both; the interference profile that consequently develops in the wave generates a similar pattern in the particle trajectories guided by the wave.

It is interesting to compare Feynman's commentary ("Nobody knows any machinery" etc.) with that of John Bell's:

"Is it not clear from the smallness of the scintillation on the screen that we have to do with a particle? And is it not clear, from the diffraction and interference patterns, that the motion of the particle is directed by a wave? De Broglie showed in detail how the motion of a particle, passing through just one of two holes in the screen, could be influenced by waves propagating through both holes. And so influenced that the particle does not go where the waves cancel out, but is attracted to where they cooperate. This idea seems to me so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored."

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DrChinese I have noticed as well is very understanding of the layman's curiosity.

If you mean that I am not as understanding as our worthy Oriental friend, then you are right. I try, but he knows much more than I do.

Time of flight questions are meaningless in standard QM because the particles aren't there unless you look at them and therefore don't have trajectories (and you don't have a time operator). It is only in BM where the question makes any sense. Indeed one could in principle make deductions similar to what you say, but there is no experimental evidence whatsoever that "photons" travel in straight lines or at a speed different from c. Or am I missing something?

What I was trying to say at the end there is simply - 'Yes, Bohmian mechanics makes time-of-flight predictions (that standard QM doesn't), and yes this in principle constitutes an experimental test of it, but I am not aware of any such experiments'. What I should perhaps have made clearer is that this is because I am not at all familiar with the experimental literature on this subject. Can anyone summarize what has been done on this?

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If you mean that I am not as understanding as our worthy Oriental friend, then you are right. I try, but he knows much more than I do.

What I trying to say at the end there is simply - 'Yes, Bohmian mechanics makes time-of-flight predictions (that standard QM doesn't), and yes this in principle constitutes an experimental test of it, but I am not aware of any such experiments'. What I should perhaps have made clearer is that this is because I am not at all familiar with the experimental literature on this subject. Can anyone summarize what has been done on this?

Its not that, I was just showing appreciation to his understanding of my personal curiosity is all.

PRODOS, in some situations the Bohmian velocities of photons may be larger than c.

Do you mean “moderately” faster-than-c velocities as in the Bohmian photon trajectories of the double slit experiment? (Is it reasonable to refer to these photon speeds as "moderately" faster than c or does that misrepresent them in some fundamental way?)

Or do you mean the many-times-faster-than-light velocities that are sometimes put forward to account for nonlocality in EPR type experiment?

Or both?

If you mean “moderately” faster-than-light, a la double slit experiment, could you possibly indicate the kinds of situations where, according to Bohmian Mechanics, this occurs or might occur, please?

Nevertheless, this is not a problem. By using the general theory of quantum measurements, it can be shown that the Bohmian velocity of photons is exactly equal to c whenever you measure this velocity.

Is the application of the general theory of quantum measurement to these cases a recent development in Bohmian Mechanics or is it an established/standard aspect of Bohmian Mechanics?

Best Wishes,

Prodos
Melbourne, Australia

PRODOS, this velocity can be many times faster than c. In fact, it can even be infinite. Nevertheless, it has nothing to do with nonlocality in EPR experiments because the exchange of photons is NOT the mechanism of communication between entangled particles.
A typical situation in which such ultra-high velocities occur is a wave function which is a superposition of two different frequencies.

Concerning the application of quantum measurements to such situations, I would say that it is a standard aspect of BM, although it has been explicitly stated relatively recently.

What I was trying to say at the end there is simply - 'Yes, Bohmian mechanics makes time-of-flight predictions (that standard QM doesn't), and yes this in principle constitutes an experimental test of it, but I am not aware of any such experiments'.

Unfortunately, the trajectories of BM are unobservable, so, while time of flight makes sense as really existing, we cannot observe it. At least not in quantum equilibrium.

Any experimental literature claiming to have made experimental tests between BM and QM is based on bad theory. All what follows from the accurate interpretation is that we have yet another common confirmation of BM as well as QM.

Unfortunately, the trajectories of BM are unobservable, so, while time of flight makes sense as really existing, we cannot observe it. At least not in quantum equilibrium.
Let me further explain it. To measure time, we need a clock. But any measurement, including a measurement by a clock, is actually a measurement of the position of something (e.g., the needle of the clock). On the other hand, measurements of positions in BM allways give the same results as those in standard QM.

Good evening.

PRODOS, this velocity can be many times faster than c. In fact, it can even be infinite.

According to Bohmian Mechanics, the velocity of a particle photon – under some circumstances – can be many times greater than c. Correct?

Nevertheless, it has nothing to do with nonlocality in EPR experiments because the exchange of photons is NOT the mechanism of communication between entangled particles.

Okay. That’s an important point. Thanks for making it very clear.

A typical situation in which such ultra-high velocities occur is a wave function which is a superposition of two different frequencies.

Such a superposition of different frequencies is happening in the region between the slits and the detector in the double slit experiment. Correct?

Question: Stepping away from the double slit experiment for a moment ... Is it the case that, when we can plot a non-straight photon trajectory, we can infer a faster-than-light photon? But when we can plot a straight-line photon trajectory we infer an exactly-c-velocity? (I’m excluding photons being refracted, reflected, scattered, etc.)

Question: In the specific case of a typical photon double slit experiment, is it possible to indicate the approximate range of photon velocities that might be specified by Bohmian Mechanics? For instance: "more than c, but less than twice c"? (I understand that if we took an actual measurement at any point between slits and detector we would always measure c.)

Many thanks.

According to Bohmian Mechanics, the velocity of a particle photon – under some circumstances – can be many times greater than c. Correct?
Correct.

Such a superposition of different frequencies is happening in the region between the slits and the detector in the double slit experiment. Correct?
Not correct. In the two-slit experiment the frequency is usually well defined.

Question: Stepping away from the double slit experiment for a moment ... Is it the case that, when we can plot a non-straight photon trajectory, we can infer a faster-than-light photon? But when we can plot a straight-line photon trajectory we infer an exactly-c-velocity? (I’m excluding photons being refracted, reflected, scattered, etc.)
Not correct. Photon can change the direction of its motion, but the absolute value of the velocity can still be c. Or it can move along a straight line but still change its velocity.

Question: In the specific case of a typical photon double slit experiment, is it possible to indicate the approximate range of photon velocities that might be specified by Bohmian Mechanics? For instance: "more than c, but less than twice c"? (I understand that if we took an actual measurement at any point between slits and detector we would always measure c.)
It is certainly possible, but not without explicit calculation. Which I haven't done.

I think this is precisely the point of the de Broglie/Bohm interpretation. Questions like yours can be given perfectly natural, even obvious answers, formulated in terms of fully visualizable continuously existing entities and with precisely-defined mathematics. And given that it is entirely observationally equivalent to the standard viewpoint, one might as well teach it that way in particular so that beginners don't get so confused. It simply is not necessary to make QM (particularly the non-relativistic kind) into the 'weird, mysterious discipline that nobody understands' of popular legend.[/I]"

I don't hide the mystery of quantum mechanics from my students, rather I START my QM course with it! My students read “Bringing home the atomic world: Quantum mysteries for anybody,” N.D. Mermin, Am. J. Phys. 49, Oct 1981, 940-943 and Quantum Mechanics and Experience, David Z. Albert, Harvard Univ Press, 1992, before we even start in Shankar's text. We also reproduce all the calculations and data analysis in “Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory,” D. Dehlinger and M.W. Mitchell, Am. J. Phys. 70, Sep 2002, 903-910.

If you don't acknowledge the ontological challenges implied by QM, you've reduced the course to another exercise in mathematical physics a la classical dynamics. Don't get me wrong, I like teaching classical dynamics, but it does not suggest a paradigm shift. Many in the foundations community believe the "weirdness of QM" suggests we are on the verge of a paradigm shift akin to that from Aristotle's teleological approach to Newton's mechanical approach. Why hide this from the students?

I don't hide the mystery of quantum mechanics from my students, rather I START my QM course with it! ... Why hide this from the students?

Of course - you're quite right. But the point is that normally the thing that is hidden from the students is that one can understand QM in terms of causally connected events.

The only thing that students normally learn about the two-slit experiment - for example - is how it is completely impossible to understand. Why not teach them that - if one merely states that particles exist - then the results of the experiment are obvious. Then, if you like, you can add - well, if one wishes to believe that the wave function is all there is, then it is not clear why the results are the way they are. Then the students can choose which viewpoint they prefer.

Tell me - do you mention the de Broglie-Bohm theory in your course. If not, tell me - why do you not present both points of view?

See http://www.nature.com/nature/journal/v406/n6792/full/406123a0.html" for an interesting perspective on this.

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Of course - you're quite right. But the point is that normally the thing that is hidden from the students is that one can understand QM in terms of causally connected events.

The only thing that students normally learn about the two-slit experiment - for example - is how it is completely impossible to understand. Why not teach them that - if one merely states that particles exist - then the results of the experiment are obvious. Then, if you like, you can add - well, if one wishes to believe that the wave function is all there is, then it is not clear why the results are the way they are. Then the students can choose which viewpoint they prefer.

Tell me - do you mention the de Broglie-Bohm theory in your course. If not, tell me - why do you not present both points of view?

See http://www.nature.com/nature/journal/v406/n6792/full/406123a0.html" for an interesting perspective on this.

I introduce them conceptually to Bohmian mechanics, Everett’s Many Worlds, Price’s backwards causation, Cramer’s advanced action, Aharonov’s two-vector formalism, and the Copenhagen interpretation per Mermin, i.e., “shut up and calculate” (N. D. Mermin, Physics Today 57, #5, 10-11 (2004)). I’ve been involved with the foundations of physics for 15 years and I don’t have the impression that any of these approaches has won a majority of support from the foundations community (the Copenhagen interpretation is the overwhelming attitude among physicists in general). I would say Many Worlds has the largest following at this point, but still less than 10%.

When pushed to explain the more problematic experiments, e.g., interaction-free measurement (A. Elitzur & L. Vaidman, Found. Phys. 23, 1993, 987-997), the quantum liar paradox (Lucien Hardy, Phys. Lett. A 167, 1992, 11-16) and delayed choice, “Time and the Quantum: Erasing the Past and Impacting the Future,” Y. Aharonov & M.S. Zubairy, Science 307, 11 Feb 2005, 875-879; Anton Zeilinger, “Why the quantum? ‘It’ from ‘bit’? A participatory universe? Three far-reaching challenges from John Archibald Wheeler and their relation to experiment,” in Science and Ultimate Reality: Quantum Theory, Cosmology and Complexity, John D. Barrow, Paul C.W. Davies and Charles L. Harper, Jr. (eds.), (Cambridge Univ Press, Cambridge, 2004), pp 201-220, the Bohmians resort to a preferred frame and the foundations community is just not interested in giving up relativity. Frankly, I see Bohmian mechanics as the modern equivalent of “save the appearances” where today it’s a mechanistic view that physicists are trying to save. The following quote captures my position:

“In the past, fundamental new discoveries have led to changes – including theoretical, technological, and conceptual changes – that could not even be imagined when the discoveries were first made. The discovery that we live in a universe that, deep down, allows for Bell-like influences strikes me as just such a fundamental, important new discovery. … If I am right about this, then we are living in a period that is in many ways like that of the early 1600s. At that time, new discoveries, such as those involving Galileo and the telescope, eventually led to an entirely new way of thinking about the sort of universe we live in. Today, at the very least, the discovery of Bell-like influences forces us to give up the Newtonian view that the universe is entirely a mechanistic universe. And I suspect this is only the tip of the iceberg, and that this discovery, like those in the 1600s, will lead to a quite different view of the sort of universe in which we live.” Worldviews: An Introduction to the History and Philosophy of Science, Richard DeWitt, Blackwell Publishing, 2004, p 304.

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the Bohmians resort to a preferred frame and the foundations community is just not interested in giving up relativity.
Recently, it has been found out how to formulate Bohmian mechanics without a preferred frame:
http://xxx.lanl.gov/abs/0811.1905 [Int. J. Quantum Inf. 7 (2009) 595-602]

the quantum liar paradox (Lucien Hardy, Phys. Lett. A 167, 1992, 11-16)
This paper is about empty waves. Can you explain what it has to do with the liar paradox?

This paper is about empty waves. Can you explain what it has to do with the liar paradox?

Hardy's experiment is the basis for creating the entangled state for quantum liar, Elitzur and Dolev call the atoms entangled in the MZI per Hardy's technique, "Hardy atoms." I should have given an Elitzur reference to the full quantum liar description:

Sorry.

Thanks for the paper!

One comment: All such paradoxes emerge from the assumption that photon is ONE object. In the Bohmian interpretation there are actually two objects (wave function AND particle) and all such paradoxes trivially disappear.

But of course, most physicists find simpler to deal with a number of paradoxes than to accept a terribly complicated hypothesis that the photon consists of two objects.

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Whether or not these photon paths predict FTL velocities is, as I see it, irrelevant, considering that the upper bound of velocity is c, we can affirm with some certain that light will never travel faster than itself. I suspect that this apparent increase in velocity is merely a mathematical anomaly.

I reproduce my diagram from the beginning of this thread ...

The line Sp is a straight line from slit A to Y.

Denote its length as “s”.
Note: In this diagram, Sp is not intended to represent a particle trajectory. It is merely a straight ruler.

A photon travels a non-straight-line Bohmian trajectory from A to Y.

I’ll denote its trajectory as Bp and the length of the trajectory as “b”.
However: The Bp shown in this diagram is not in any way meant to be an accurate depiction of what is predicted by Bohmian Mechanics. I have merely drawn an arbitrary non-straight line. (I hope this isn’t off-putting or inadvertently introduces any confusion into the issues I’m going to ask about.)

Question 1:

Is the time it takes for a photon particle to complete the non-straight Bohmian path, Bp, from A to Y equivalent to:

s/c ... ?

Question 2:

Does every distinct location Y on the detector have a unique photon trajectory associated with it?
i.e. Any photon particle that goes from A and ends up at Y, will have traveled the same route?
i.e. There is only one distinct trajectory-shape – only one Bp - allowed for every location Y on the detector screen?

Question 3:

The speed of a photon traveling the Bohmian path Bp (inaccurately depicted in the above diagram) is faster than c. Is this a constant speed along the whole of Bp? Or does it (or might it) vary at different points along the path, Bp?

Question 4:

Is the speed (or range of speeds) of any photon along every point on the trajectory the same?
Or, put another way, can different photons traveling the same path, Bp, have different speeds (or ranges of speeds) along the way?

Question 5:

Is the speed of the faster-than-c photon:

bc/s ... ?

b is the length of the Bohmian trajectory.
c is the speed of light.
s is the length of the straight ruler.

Alternatively, is bc/s the average speed of the photon particle?

Once again, thanks very much for assistance in better understanding these issues.

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In the double slit experiment, Bohmian Mechanics http://plato.stanford.edu/entries/qm-bohm/#2s" the paths of real particles traveling from the two slits to the detector to look like something like this:

The above image shows particles traveling in non-straight paths.

It may surprise Demystifier but you can actually define a Bohmian QM
without anything moving faster than c. (for mass and mass less particles)

Because particles can't move faster as light but voids can.
To illustrate this, below the void "0" moves 8 positions to the right.

1101111111111 ==> 1111111111011

But this is just the same a eight ones "11111111" moving only a
single position to the left, that is at 1/8 of the speed. The void
can in principle move at any speed but the guiding wave is which
controls the "motion" of the void is limited by c.

Regards, Hans

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PRODOS, from drawing trajectories in space, nothing can be concluded about their time dependence. Consequently, nothing can be concluded about their velocities (except their space direction).

Concerning Question 2, the answer is yes if the wave function has a definite frequency. But in general it does not need to be so. Trajectories attaining A (or Y) at different times may have different shapes in space as well.

PRODOS, from drawing trajectories in space, nothing can be concluded about their time dependence. Consequently, nothing can be concluded about their velocities (except their space direction).

Concerning Question 2, the answer is yes if the wave function has a definite frequency. But in general it does not need to be so. Trajectories attaining A (or Y) at different times may have different shapes in space as well.

Thanks Demystifier.

As usual, you've given me plenty to think about and to study further.

Thanks for the paper!

One comment: All such paradoxes emerge from the assumption that photon is ONE object. In the Bohmian interpretation there are actually two objects (wave function AND particle) and all such paradoxes trivially disappear.

But of course, most physicists find simpler to deal with a number of paradoxes than to accept a terribly complicated hypothesis that the photon consists of two objects.

From talking to members of the foundations community, I have the impression that their complaint isn't with the BM wave per se, but with its superluminal influences. Most of the physicists I've spoken with at foundations conferences are just not willing to invoke "spooky action at a distance" to resolve QM paradoxes.

From talking to members of the foundations community, I have the impression that their complaint isn't with the BM wave per se, but with its superluminal influences. Most of the physicists I've spoken with at foundations conferences are just not willing to invoke "spooky action at a distance" to resolve QM paradoxes.
But Bell has shown that superluminal influences are not an exclusive property of BM per se, but of ANY theory (compatible with QM) claiming that reality (i.e., objective physical properties existing even without measurements) exists. My impression is that most physicists who complain about superluminal BM influences are not aware of this general Bell's result.