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

[Max™]

Yeah, and even then it isn't so much superluminal influence as it is non-local effects.

It isn't sending signals between point A and point B at v > c, it is influence between point A and point B in t = (more or less) 0, as if the two points were not separated in a spacelike manner.

There are other possibilities as well, for example if influence were transmitted from point A to point B at v < c, but t < 0, farewell causality you say, but the effects would be as far as I know indistinguishable from A to B in t = 0.

Bohm's interpretation is useful to help consider what is actually happening, just as the Copenhagen is, but it is unlikely that they are remotely accurate phenomenological descriptions of reality.

 

[RUTA]

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.

I'm sure you're right in general, I've met physicists who have never heard of EPR-Bell phenomena. But, those in the foundations community are more knowledgeable and do appreciate that Bell's theorem entails giving up separability or locality -- unless, for example, there is information flow from the future (which is the basis for many so-called "backwards causation" interpretations, see the focus issue in Studies in History and Philosophy of Modern Physics 39, Nov 08). There is no interpretation that generates consensus in the community at this point. I suspect the only way a new interpretation will find widespread acceptance is that it yields new physics, because the conceptual cost must be outweighed by the scientific value and, as Bell has shown, there will be conceptual cost.

 

[metacristi]

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.

Well one can still argue that we deal with 'non-classical correlations' without any 'action-at-distance' involved. How can Nature conspire, in non-classical ways, to produce the appearance of action-at-distance is another thing but I wouldn't take lightly the 'conspirational hypothesis'; superdeterminism for example is fully compatible with all experiments conducted so far (it may lack 'le bon sens', to quote Duhem, but counterfactual definiteness is by no means untouchable).

And a few questions regarding Bohmian Mechanics (as I see you know it very well):

1. How it is explained within BM the fact that electrons do not fall into the nucleus? (as far as I know Vigier proposed the hypothesis that a radiation of very long wavelength is indeed emitted by the electrons but we aren't yet capable to measure it).

2. If the guiding wave affect the particle shouldn't the particle affect the wave as well? If the answer is yes how is this happening?

3. Why is Lorentz invariance so important? As far as I know there are indeed reformulations of Special Relativity - involving some changes at the level of the basic postulates but indistinguishable at the experimental level from the classical solution (based on the usual Lorentz transformations) - where a preferred frame of reference is accepted (thus fully compatible with Bohmian Mechanics). But my point is that although (local) Lorentz invariance is part of the accepted methodologies in Physics (I accept that it has a provisional edge over the alternatives) it is by no means required with necessity by the observed facts (anyway the Brans-Dicke theory etc are still with us). I wouldn't be surprised if a final (successful, the 'winner') TOE is not Lorentz invariant...

4. Why the density of probability P tends towards |PSI|^2, isn't this an ad-hoc hypothesis? (some accuse the ad-hoc nature of Bohm's explanation, namely that the value of P is pushed towards |PSI|^2 by aleatory interactions and the quantum dynamics, more or less like in the classical statistical mechanics).

5. Do you believe that a non-trivial change at the level of the basic postulates of QM could make a difference between the existing interpretations of QM, by pointing towards a clear 'winner' (possible an unconceived yet alternative)?

6. Are there other promising (non-local) hidden variables programs?

7. The spin?...but of course there is no real problem to consider it a phenomenological property

 

[Demystifier]

1. How it is explained within BM the fact that electrons do not fall into the nucleus?

The quantum force (i.e., the force resulting from the quantum potential) is such that it prevents them to fall into the nucleus.

2. If the guiding wave affect the particle shouldn't the particle affect the wave as well? If the answer is yes how is this happening?

The answer is - no. (Otherwise, it would be in contradiction with standard QM.)

3. Why is Lorentz invariance so important? As far as I know there are indeed reformulations of Special Relativity - involving some changes at the level of the basic postulates but indistinguishable at the experimental level from the classical solution (based on the usual Lorentz transformations) - where a preferred frame of reference is accepted (thus fully compatible with Bohmian Mechanics). But my point is that although (local) Lorentz invariance is part of the accepted methodologies in Physics (I accept that it has a provisional edge over the alternatives) it is by no means required with necessity by the observed facts (anyway the Brans-Dicke theory etc are still with us). I wouldn't be surprised if a final (successful, the 'winner') TOE is not Lorentz invariant...

Fundamental Lorentz invariance is not necessary. However, since we need Lorentz invariance at least at some phenomenological level, a fundamental theory can explain phenomenological Lorentz invariance more easily if the fundamental theory itself is also Lorentz invariant.

4. Why the density of probability P tends towards |PSI|^2, isn't this an ad-hoc hypothesis? (some accuse the ad-hoc nature of Bohm's explanation, namely that the value of P is pushed towards |PSI|^2 by aleatory interactions and the quantum dynamics, more or less like in the classical statistical mechanics).

Valentini has found a quantum version of the Boltzmann H-theorem, showing that the distribution will approach the equilibrium distribution |PSI|^2 even if it starts from an arbitrary distribution.

5. Do you believe that a non-trivial change at the level of the basic postulates of QM could make a difference between the existing interpretations of QM, by pointing towards a clear 'winner' (possible an unconceived yet alternative)?

Yes I do. For a recent attempt in that direction see
http://xxx.lanl.gov/abs/0811.1905 [Int. J. Quantum Inf. 7 (2009) 595-602]

6. Are there other promising (non-local) hidden variables programs?

There certainly are other nonlocal hidden variables programs. Personally, I don't find them very promissing.

7. The spin?...but of course there is no real problem to consider it a phenomenological property

According to the Bohmian interpretation, particles do not really have spin. Instead, the appearance (or illusion) of spin is caused by the wave function, i.e., by quantum forces that change particle motions in such a way that they make illusion that particles have spin.

 

[ytuab]

The quantum force (i.e., the force resulting from the quantum potential) is such that it prevents them to fall into the nucleus.

According to the Bohmian interpretation, particles do not really have spin. Instead, the appearance (or illusion) of spin is caused by the wave function, i.e., by quantum forces that change particle motions in such a way that they make illusion that particles have spin.

I don't know about Bohmian Mechanics(BM) very well. Please tell me some things.

Bohmian Mechanics is based on the quantum mechanics, isn't it?
The orbital angular momentum of the electron in the S orbital is zero.
So the electron in the S orbital is not moving in BM? (If the quantum forse prevents the electron to fall into the nucleus in S orbital.)

A nucleus with a large charge will cause an electron to have a high velocity. A higher electron velocity means an increased electron relativistic mass, as a result the electrons will be near the nucleus more of the time and thereby contract the radius for small principal quantum numbers. http://en.wikipedia.org/wiki/Relativ...ntum_chemistry

The relativistic effect shows that the electron is actually moving fast.
How do you explain these relativistic phenomina and electronic motion in BM?

If the size of the electron is too small(about classical radius), by equating the angular momentum of the spinning sphere to 1/2 hbar, the sphere speed leads to 100 times the speed of light. How Big is one electron in BM?

The spin g factor is about 2. This means that the mass and charge of one electron is separated . How do you explain this g factor?

 

[Demystifier]

Bohmian Mechanics is based on the quantum mechanics, isn't it?
The orbital angular momentum of the electron in the S orbital is zero.
So the electron in the S orbital is not moving in BM? (If the quantum forse prevents the electron to fall into the nucleus in S orbital.)

I said that electrons do not move according to the classical laws. I did not say that they do not move at all.

A nucleus with a large charge will cause an electron to have a high velocity. A higher electron velocity means an increased electron relativistic mass, as a result the electrons will be near the nucleus more of the time and thereby contract the radius for small principal quantum numbers. http://en.wikipedia.org/wiki/Relativ...ntum_chemistry

The relativistic effect shows that the electron is actually moving fast.
How do you explain these relativistic phenomina and electronic motion in BM?

The relativistic effects that you refer to are properties of the wave function. In BM, the wave function is exactly the same as that in standard QM. However, in BM the wave function of the electron is not identified with the electron itself. Yet, the wave function determines the motion of the electron, in such a way that observable properties of the electron are exactly the same as those predicted by standard QM.

If the size of the electron is too small(about classical radius), by equating the angular momentum of the spinning sphere to 1/2 hbar, the sphere speed leads to 100 times the speed of light. How Big is one electron in BM?

In BM, electron is pointlike. It does not spin.

The spin g factor is about 2. This means that the mass and charge of one electron is separated . How do you explain this g factor?

BM explains the g factor in the same way as standard QM does - as a property of the wave function.

In all your questions, you seem to be missing the crucial point of BM: the electron and the wave function of the electron are not the same. Yet, the latter determines the motion of the former.

 

[ytuab]

I said that electrons do not move according to the classical laws. I did not say that they do not move at all.

In BM, electron is pointlike. It does not spin.

BM explains the g factor in the same way as standard QM does - as a property of the wave function.

In all your questions, you seem to be missing the crucial point of BM: the electron and the wave function of the electron are not the same. Yet, the latter determines the motion of the former.

I think that you say Bohmian Mechanics is the same thing as the quantum mechanics.
What's the meaning of the BM?

You say the electron and the wave function are not the same.
So, does the electron have the charge and mass? Or does the wave function have the charge and mass? What is the electron? (mass ? charge? )

Spin has a magnetic moment. The magnetic moment is caused by the movement of the charge. Do you say the wave function has the charge?

 

[Hans de Vries]

If the size of the electron is too small(about classical radius), by equating the angular momentum of the spinning sphere to 1/2 hbar, the sphere speed leads to 100 times the speed of light. How Big is one electron in BM?

The spin g factor is about 2. This means that the mass and charge of one electron is separated . How do you explain this g factor?

This is a familiar argument, but it makes the unnecessary assumption that the
net remaining charge that we measure is entirely responsible for the magnetic
moment (which is indeed impossible).

However. in a vacuum which admits both positive and negative charges there
is more charge available to generate magnetic fields as the net charge density,
which is merely the difference between the positive and negative charges.

As far as Quantum Electro Dynamics goes: The remaining net charge of an
electron is an exceedingly small part of the "real charge" of the electron which
is shielded by opposite charges due to vacuum polarization.


Regards, Hans

 

[ytuab]

This is a familiar argument, but it makes the unnecessary assumption that the
net remaining charge that we measure is entirely responsible for the magnetic
moment (which is indeed impossible).

However. in a vacuum which admits both positive and negative charges there
is more charge available to generate magnetic fields as the net charge density,
which is merely the difference between the positive and negative charges.

As far as Quantum Electro Dynamics goes: The remaining net charge of an
electron is an exceedingly small part of the "real charge" of the electron which
is shielded by opposite charges due to vacuum polarization.

Regards, Hans

It seems that the Bohmian mechanics is the same thing as quantum mechanics and QED.
So Why does the Bohmian mechanics(BM) exist?

And Demystifier said that the electron and the wavefunction are not the same thing in BM.
So what is the electron(charge and mass)?

If both the electron and the wavefunction have the charge and mass, what is the difference between them?

And do you say that the charge and mass of the electron is meaningless?
The effective charge and mass which causes spin magnetic moment can change?

 

[Hans de Vries]

It seems that the Bohmian mechanics is the same thing as quantum mechanics and QED.
So Why does the Bohmian mechanics(BM) exist?

And Demystifier said that the electron and the wavefunction are not the same thing in BM.
So what is the electron(charge and mass)?

If both the electron and the wavefunction have the charge and mass, what is the difference between them?

And do you say that the charge and mass of the electron is meaningless?
The effective charge and mass which causes spin magnetic moment can change?

I'll let Demystifier speak for Bohemian Mechanics since this is his area.

In QED (or in molecular modeling) you have to assign to each point of a wave
function quantities like:

- charge density
- current density
- axial current density
- magnetic moment density (magnetization)
- polarization (electric moment density due to a moving magnetic moment density)
- mass density
- energy and momentum density
- spin density
- Hamiltonian and Lagrangian density

These quantities are expressions using Dirac's formulation of the
electron wave-function. The field quantities like charge/current
density and magnetic moment density result in electromagnetic
fields which have to be taken into account when calculating
interactions.


Regards, Hans

 

[Demystifier]

I think that you say Bohmian Mechanics is the same thing as the quantum mechanics.

No, I do not say that.

You say the electron and the wave function are not the same.
So, does the electron have the charge and mass? Or does the wave function have the charge and mass? What is the electron? (mass ? charge? )

In a version of the theory in which the effects of quantum field theory are ignored, the electron has both charge and mass.

Spin has a magnetic moment. The magnetic moment is caused by the movement of the charge. Do you say the wave function has the charge?

The magnetic moment is caused by the movement of the charge in classical mechanics. However, it is not (necessarily) so in quantum mechanics.

 

[ytuab]

I said that electrons do not move according to the classical laws. I did not say that they do not move at all.

The relativistic effects that you refer to are properties of the wave function. In BM, the wave function is exactly the same as that in standard QM.

In BM, electron is pointlike. It does not spin.

In a version of the theory in which the effects of quantum field theory are ignored, the electron has both charge and mass.

You said the relativistic effects are properties of the wave function. So the relativistic effect is not directly related to the velocity of the electron? Is that really so?

In the relativistic theory, the velocity of the particles is equal to the velocity of us (in the opposite direction).
If the relativistic effect is not related to the velocity of the electron, when we move faster, doesn't the relativistic effect of the electron change ? (from our viewpoint)

If you say the electron is actually moving in BM, how is the elecron in the S orbital moving?
Why doesn't it radiate energy ? (the electron has charge as you said.)

And you said the electron is pointlike. So by equating the angular momentum of the spinning sphere to 1/2 hbar, the sphere speed leads to more than 100 times the speed of light.
You said the electron has mass. Is the mass of electron not related to the 1/2 hbar?
What causes 1/2 hbar?

And the spinning electron will go back to itself when it is rotated by an angle of 4pai (not 2 pai). If the electron is pointlike, is there a differense between 2pai and 4pai?

 

[Demystifier]

You said the relativistic effects are properties of the wave function. So the relativistic effect is not directly related to the velocity of the electron? Is that really so?

Yes, that is so.

In the relativistic theory, the velocity of the particles is equal to the velocity of us (in the opposite direction).
If the relativistic effect is not related to the velocity of the electron, when we move faster, doesn't the relativistic effect of the electron change ? (from our viewpoint)

Yes it does. But it can also be described by the relativistic change of the wave function.

If you say the electron is actually moving in BM, how is the elecron in the S orbital moving?
Why doesn't it radiate energy ? (the electron has charge as you said.)

For a proper treatment, radiation must also be described quantum mechanically, not in terms of continuous radiation, but in terms of photons. Thus, you need a Bohmian interpretation of quantum field theory, which is a more difficult subject. If you are interested, I can give you appropriate references.

And you said the electron is pointlike. So by equating the angular momentum of the spinning sphere to 1/2 hbar, the sphere speed leads to more than 100 times the speed of light.
You said the electron has mass. Is the mass of electron not related to the 1/2 hbar?
What causes 1/2 hbar?

Mass of electron is not related to the 1/2 hbar. 1/2 hbar is caused by a 4-component (or 2-component in a non-relativistic limit) spinor wave function described by the Dirac equation.

And the spinning electron will go back to itself when it is rotated by an angle of 4pai (not 2 pai). If the electron is pointlike, is there a differense between 2pai and 4pai?

No, it is the spinor wave function that goes back to itself when rotated by an angle of 4pi (not 2pi).

 

[ytuab]

"In the relativistic theory, the velocity of the particles is equal to the velocity of us (in the opposite direction).
If the relativistic effect is not related to the velocity of the electron, when we move faster, doesn't the relativistic effect of the electron change ? (from our viewpoint) "

Yes it does. But it can also be described by the relativistic change of the wave function.

"And the spinning electron will go back to itself when it is rotated by an angle of 4pai (not 2 pai). If the electron is pointlike, is there a differense between 2pai and 4pai?"

No, it is the spinor wave function that goes back to itself when rotated by an angle of 4pi (not 2pi).

The spin is an actual thing (not only spinor wave).

Some experiments (which were independently performed by different teams) showed that the spinning neutrons went back to original forms when they are "actulally" rotated by an angle of 4pai (not 2pai).
(H. Rauch et al. Phys.Lett. 54A (1975) 425)

Do you say the spin of the neutron is different from the spin of the electron?

And how is the electron actulally moving in the S orbital in BM?

 

[Demystifier]

The spin is an actual thing (not only spinor wave).

Not really. Spin is measured by the Stern-Gerlach apparatus which actually determines the POSITION of the particle. We only INTERPRET the change of position as being due to the intrinsic magnetic moment.

Some experiments (which were independently performed by different teams) showed that the spinning neutrons went back to original forms when they are "actulally" rotated by an angle of 4pai (not 2pai).
(H. Rauch et al. Phys.Lett. 54A (1975) 425)

I will take a look and comment later.
Edit: I've just did it. They don't really measure any rotation of the neutrons. They just measure interference, which is theoretically explained by rotations of wave functions.

Do you say the spin of the neutron is different from the spin of the electron?

Not really. (Although there are some differences because neutron is a composite particle.)

And how is the electron actulally moving in the S orbital in BM?

Here is a deal: Write me the wave function for the S orbital and then I will tell you how the electron moves.

 

[Demystifier]

I'll let Demystifier speak for Bohemian Mechanics since this is his area.

Actually, my area is Bohmian (not Bohemian) mechanics.
But you may be surprised that I made a contribution to the "Bohemian" as well:
http://xxx.lanl.gov/pdf/physics/0702069 [Am.J.Phys.76:143-146,2008]
(See the lyrics after the Abstract.)

 

[Hans de Vries]

Actually, my area is Bohmian (not Bohemian) mechanics.

Oops

But you may be surprised that I made a contribution to the "Bohemian" as well:
http://xxx.lanl.gov/pdf/physics/0702069 [Am.J.Phys.76:143-146,2008]
(See the lyrics after the Abstract.)

 

[ytuab]

I will take a look and comment later.

Edit: I've just did it. They don't really measure any rotation of the neutrons. They just measure interference, which is theoretically explained by rotations of wave functions.

The experiment (Phys Lett. 54A 1975 425) measured spatial rotation of the neutrons.

Did you really read this paper or something about this experiment?

They actually "spatially" rotated the neutrons by an angle of 4pai using "rotation by precession" in this real world . (The result is just consistent with 4 pai. It is not accidential.)
Spatiall rotation is consistent with the rotaions of spinor wavefunction ( you say spatial rotation is different from the rotation of the spinor wavefunction.)

Of course, we can not look at the spinning neutrons directly.
So they measured interference. (you say only about this last step. you didn't say the method of rotaion (this is most important.))

In my personal view, I DON'T believe "the spin of the electron".
But you believe in the quantum mechanics. So you misunderstand this experiment.

 

[Hans de Vries]

The experiment (Phys Lett. 54A 1975 425) measured spatial rotation of the neutrons.

They actually "spatially" rotated the neutrons by an angle of 4pai using "rotation by precession" in this real world . (The result is just consistent with 4 pai. It is not accidential.)

The 4pi rule is valid in all rotational directions. If the rotation is around
the spin pointer then you can measure only the interference. (The spin
pointer doesn't change direction) However, a rotation around any other
axis will rotate the spin pointer itself and change the direction of the
magnetization which is measurable.

Regards, Hans

 

[ytuab]

The 4pi rule is valid in all rotational directions. If the rotation is around
the spin pointer then you can measure only the interference. (The spin
pointer doesn't change direction) However, a rotation around any other
axis will rotate the spin pointer itself and change the direction of the
magnetization which is measurable.

Regards, Hans

I see.
What you say is indeed interesting.

But I didn't know about the other direction.

 

[Demystifier]

In my personal view, I DON'T believe "the spin of the electron".

What exactly do you mean by that?

 

[ytuab]

What exactly do you mean by that?

I don't believe "the spin of the electron".

Because as I have stated before, "spin" has some serious problems.

For example, the electron is too small. So by equating the spin angular momentum to 1/2 hbar, spinning sphere speed leads to 100 times the speed of light. (If the electron size is classical radius.)

The classical radius size is as big as nuclear size of the hydrogen .

Spinning electoron has magnetic moment and spin angular momentum.
And one electron has the charge and mass.
Magnetic moment is basically caused by the rotaion of the charge, isn't it?
Angular momentum is basically caused by the rotaion of the mass, isn't it?

Probably Nobody believes "spin" is a actual spinning of the electron.
In spite of that, why do we call it "spin"?
It is inconsistent.

 

[Demystifier]

I don't believe "the spin of the electron".

Because as I have stated before, "spin" has some serious problems.

For example, the electron is too small. So by equating the spin angular momentum to 1/2 hbar, spinning sphere speed leads to 100 times the speed of light. (If the electron size is classical radius.)

The classical radius size is as big as nuclear size of the hydrogen .

Spinning electoron has magnetic moment and spin angular momentum.
And one electron has the charge and mass.
Magnetic moment is basically caused by the rotaion of the charge, isn't it?
Angular momentum is basically caused by the rotaion of the mass, isn't it?

Probably Nobody believes "spin" is a actual spinning of the electron.
In spite of that, why do we call it "spin"?
It is inconsistent.

I completely agree that spin as a kind of rotational motion does not exist. Still, the effect that we call "spin" exists. We should probably use a different word for this effect, but it is just a matter of terminology, not of physics.

Nevertheless, there are reasons for calling it "spin": The sum of this and the ordinary angular momentum is conserved.

 

[ArjenDijksman]

2. If the guiding wave affect the particle shouldn't the particle affect the wave as well? If the answer is yes how is this happening?

The answer is - no. (Otherwise, it would be in contradiction with standard QM.)

I wasn't aware of this detail of Bohmian quantum mechanics and I don't understand why a particle couldn't affect its pilot wave. Couder and Fort have shown experimentally that the particle and its steering wave must be considered as a whole. If the particle ceases to exist, the pilot wave disappears. And reversely, if the pilot wave is disturbed, the particle is unable to survive: http://www.physorg.com/news78650511.html.

 

[Demystifier]

I don't understand why a particle couldn't affect its pilot wave.

If it did, then pilot wave would not satisfy the Schrodinger equation. On the other hand, all experiments confirm that the wave satisfies the Schrodinger equation.

 

[ArjenDijksman]

Hello Demystifier. Thanks for your answer.

If it did, then pilot wave would not satisfy the Schrodinger equation. On the other hand, all experiments confirm that the wave satisfies the Schrodinger equation.

Could you be more precise? The pilot wave and the particle satisfy the same Schrödinger equation, the wave as a whole and the particle as a quantum entity. So if the particle affects the wave, the wave still satisfies the Schrödinger equation. For example, if the particle is a little bit disturbed before the two slits, the pilot wave will readjust its phase to the new state of the particle, satisfying of course the perturbed Schrödinger equation.

Regards,
Arjen

 

[zenith8]

I don't understand why a particle couldn't affect its pilot wave.

In classical physics there is an interplay between particle and field - each generates the dynamics of the other. In de Broglie-Bohm pilot wave theory $$\Psi$$ acts on the positions of particles but, evolving as it does autonomously via Schroedinger's equation, it is not acted upon by the particles.

One may think this is unaesthetic, but while it may be reasonable to require reciprocity of actions in classical theory, this cannot be regarded as a logical requirement of all theories that employ the particle and field concepts, especially one involving a nonclassical field.

However, as you imply, there is in fact a kind of back-action. This arises from the standard notion that the shape of the quantum field of a particle is determined by the shape of the environment (which consists of many particles, and is part of the boundary conditions put into the Schroedinger equation before solving it, even in conventional QM).

Normally in QM this 'back-action' is not taken into account. The wave guides the particles but back-action of the particle onto the wave is not systematically calculated. Of course, the back-action is physically real since the particle movement determines the initial conditions for the next round of Schroedinger calculation, but there is no systematic way to characterize such feedback. The reason this works in practice is that the back-action may not exert any systematic effect.

There is a fair of amount of interesting speculation lurking in dark corners of the internet that there is actually a systematic effect in systems which are self-organizing. That is - 'life' is what happens when a physical system uses its own nonlocality in its organization (Note to moderators: don't ban me - I'm just repeating what I heard). In this case a feedback loop is created, as follows: system configures itself so as to set up its own pilot wave, which in turn directly affects its physical configuration, which then affects its non-local pilot wave, which affects the configuration etc..

This sort of thing has never been systematically developed in the pilot-wave literature, largely because the people who talk about it on the internet are in the main well-known wackos (I won't name names, because it's probably against the rules). However, there is something interesting in this idea of 'back-action'. Bohm and Hiley even mention it in their Undivided Universe testbook - though in a very different context. (p. 345-346 if you're interested).

There is also quite a lot of speculation that the two-way traffic between pilot-wave and particle configurations provides a possible mechanism for consciousness. See Paavo Pylkkanen's Mind, Matter, and the Implicate Order book (2007) or the Cambridge Pilot-wave http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" (lectures 7 and 8).

Peter Holland has explored some deeper ideas related to this question in his work on a possible Hamiltonian formulation of pilot-wave theory. They're a bit technical, but see the following papers:

Hamiltonian theory of wave and particle in quantum mechanics I: Liouville's theorem and the interpretation of the de Broglie-Bohm theory P. Holland (2001)

Hamiltonian theory of wave and particle in quantum mechanics II: Hamilton-Jacobi theory and particle back-reaction P. Holland (2001)

 

[Demystifier]

Could you be more precise? The pilot wave and the particle satisfy the same Schrödinger equation, the wave as a whole and the particle as a quantum entity. So if the particle affects the wave, the wave still satisfies the Schrödinger equation. For example, if the particle is a little bit disturbed before the two slits, the pilot wave will readjust its phase to the new state of the particle, satisfying of course the perturbed Schrödinger equation.

Can you write down the equations that support your claims above? More precisely, can you write down the equation that describes how particle affects the wave? Without the equations, any further discussion of that would be pointless.

 

[ArjenDijksman]

However, there is something interesting in this idea of 'back-action'. Bohm and Hiley even mention it in their Undivided Universe testbook - though in a very different context. (p. 345-346 if you're interested).

Thanks for your complete answer. Because mutual back action between wave and particle is common good in classical physics, it's surprising that neither De Broglie nor Bohm developed the "two way relationship between wave and particle" (p.346) apart from the short section you mentioned in The Undivided Universe. All the more because that would have given the wealth of "experimental clues" they were longing for, through parametric adjustment of macroscopic pilot-wave experiments like those of Couder and Fort mentioned earlier.

Can you write down the equations that support your claims above? More precisely, can you write down the equation that describes how particle affects the wave? Without the equations, any further discussion of that would be pointless.

You're right, it's better to be precise. The process I visualized was the following:

• State vector |$$\Psi$$> of (any) quantum particle satisfies i.hbar.d|$$\Psi$$>/dt = H.|$$\Psi$$>, where H is the hamiltonian matrix.
• At the same time wave-function $$\Psi$$(x,t) of the pilot wave satisfies i.hbar.d$$\Psi$$(x,t)/dt = H.$$\Psi$$(x,t).

Suppose the particle is affected by the interaction with another particle (Compton scattering, collision...):

• New state vector |$$\Psi'$$> of particle satisfies i.hbar.d|$$\Psi'$$>/dt = (H+H').|$$\Psi$$'>, where H' is the part of the hamiltonian caused by interaction with other particle.
• The old pilot wave being completely out of phase with the particle, the new pilot-wave will have as wave-function $$\Psi'$$(x,t) satisfying i.hbar.d$$\Psi'$$(x,t)/dt = (H+H').$$\Psi$$'(x,t).

 

[Demystifier]

You're right, it's better to be precise. The process I visualized was the following:

• State vector |$$\Psi$$> of (any) quantum particle satisfies i.hbar.d|$$\Psi$$>/dt = H.|$$\Psi$$>, where H is the hamiltonian matrix.
• At the same time wave-function $$\Psi$$(x,t) of the pilot wave satisfies i.hbar.d$$\Psi$$(x,t)/dt = H.$$\Psi$$(x,t).

Suppose the particle is affected by the interaction with another particle (Compton scattering, collision...):

• New state vector |$$\Psi'$$> of particle satisfies i.hbar.d|$$\Psi'$$>/dt = (H+H').|$$\Psi$$'>, where H' is the part of the hamiltonian caused by interaction with other particle.
• The old pilot wave being completely out of phase with the particle, the new pilot-wave will have as wave-function $$\Psi'$$(x,t) satisfying i.hbar.d$$\Psi'$$(x,t)/dt = (H+H').$$\Psi$$'(x,t).

OK, now I see what you mean.
My answer is the following. There is no "old" and "new" wave function. There is only one wave function that corresponds to your Psi'. At early times the influence of H' on Psi' may be negligible so that Psi' can be approximated by Psi at early times. Still, the wave function is only one.