Questions on the Bohmian mechanics interpretation for instrumentalists

[kurt101]

TL;DR Summary

I have questions regarding the paper titled "Bohmian mechanics for instrumentalists" by Hrvoje Nikoli´c

Regarding the paper Bohmian mechanics for instrumentalists

I like the ideas put forward in the paper because it provides a basis for an intuitive and logical explanation of quantum mechanics. I have many questions about it, but I will start with 3.

Question 1:
In this interpretation, what is the difference between the fundamental Bohmian point like particle, the Ether, and the guiding wave?

There are various reasons this is unclear to me, but at one point the paper says:
"What we propose here is that the Earth (and everything else) is made of ether."

Based on this statement, I would conclude the ether includes both the Bohmian point like particle and the guiding wave. Is this correct?

Question 2:

The paper says:
"But we stress that the law (22) with (26) is non-local when the wave function is in the entangled state. The velocity of any particle at time t depends on the positions of all particles at the same time t, no matter how far the other particles are. We also stress that this non-local interaction is the only interaction between Bohmian point-like particles. When the wave function is not entangled, then Bohmian point-like particles do not interact with each other at all. In this sense, Bohmian particles have only non-local interactions."

What is it meant by non-local interactions? Does it mean non-local in the way described by Bell in his paper BERTLMANN’S SOCKS AND THE NATURE OF REALITY when he says "casual influences - do go faster than light"
Here is more context from this Bell paper:
"Thirdly, it may be that we have to admit that causal influences - do go faster than light. The role of Lorentz invariance in the completed theory would then be very problematic. An "ether" would be the cheapest solution".

Question 3:

My understanding is that entanglement requires local interaction in order to prepare it. Are the Bohmian point like particles in this interpretation entangled and if so how are they prepared?

 

[Elias1960]

Regarding the paper Bohmian mechanics for instrumentalists

Having looked into the (imho very good) paper, I see only one minor objection: it exaggerates the problems of BM with QFT in the introduction. The standard reference

Bohm.D., Hiley, B.J., Kaloyerou, P.N. (1987). An ontological basis for the quantum theory, Phys. Reports 144(6), 321-375

to BFT is missed.

 

[Elias1960]

Question 1:
In this interpretation, what is the difference between the fundamental Bohmian point like particle, the Ether, and the guiding wave?

A part of the problem appears in classical mechanics. Namely, one has to define what is the configuration space Q of the classical theory. If it is defined, the classical solution is a trajectory $q(t)\in Q$. But the Q may be something very different. In variant 1 of the paper it would be phonon particles, in variant two atoms, in variant 3 elementary particles, in variant 4 something more fundamental, an analog of the atoms in comparison with phonons. In all these variants, we have some particles, but even this is not obligatory. In principle, the configuration space maybe something different, in a field theory the configuration is a function u(x) defined everywhere in space, in a corresponding lattice discretization it may be a function in every lattice node u(n).

Instead, the wave function is something completely different. It is not a trajectory in configuration space $q(t)\in Q$, but a function on the configuration space $\psi: Q\to \mathbb{C}$ defined for every imaginable configuration $q\in Q$ a complex number $\psi(q)\in \mathbb{C}$.

(Note that the square of this function, $\rho(q) = |\psi(q)|^2$ defines the probability of the configuration having this particular value, and part of the Schroedinger equation is a continuity equation for this probability density
$$\partial_t \rho(q,t) + \nabla (\rho(q)\vec{v}(q)) = 0$$
which, as already suggested by its name, allows us to assume that a continuous trajectory $q(t)\in Q$ exists. So, functions of this type appear in classical probability theory too.)

So, whatever the configuration space Q, the wave function will be a function of this type. This is completely independent of our theories about what is the configuration space Q.

So, if the configuration of the world is defined by the configuration of some ether, we may describe this with condensed matter variables $\rho(x), v^i(x)$ and so on, this would be a field theory, or it could be a theory of an atomic ether with positions of all ether atoms as the configuration.

The wave function would be in both cases different, in the field-theoretic variant, it would be a functional (a function on a space of functions), in the atomic ether variant simply a many-particle wave function.

I would conclude the ether includes both the Bohmian point like particle and the guiding wave. Is this correct?

I would say the ether itself is defined by the particular configuration of the ether taken alone. The wave function exists in BM too, is part of the complete description of physics.

 

[Demystifier]

Question 1:
In this interpretation, what is the difference between the fundamental Bohmian point like particle, the Ether, and the guiding wave?

The ether is a collection of many fundamental Bohmian particles. Those particles are guided by the wave function.

Question 2:
What is it meant by non-local interactions? Does it mean non-local in the way described by Bell in his paper BERTLMANN’S SOCKS AND THE NATURE OF REALITY when he says "casual influences - do go faster than light"

Essentially, yes.

Question 3:
My understanding is that entanglement requires local interaction in order to prepare it. Are the Bohmian point like particles in this interpretation entangled and if so how are they prepared?

The entangled wave function is prepared by a local Hamiltonian. But the entangled wave function is not a local object, you cannot tell what is the value of the wave function at a given position in the 3-dimensional space. Since the Bohmian particles are guided by this nonlocal wave function, the interactions between particles are not local. Bohmian particles are not entangled, they are guided by entangled wave function. Preparation is an athropomorphic concept, it's something that humans do in a controlled way. Humans can control effective Hamiltonians that prepare wave functions, but humans cannot control initial positions of Bohmian particles, so Bohmian particles are not prepared.

 

[kurt101]

The ether is a collection of many fundamental Bohmian particles. Those particles are guided by the wave function.

Is this interpretation saying the guiding wave for the fundamental Bohmian particle is the same as the wave function used in non-relativistic quantum mechanics? If so, then won't this interpretation fail to make the same predictions as QFT?

The entangled wave function is prepared by a local Hamiltonian. But the entangled wave function is not a local object, you cannot tell what is the value of the wave function at a given position in the 3-dimensional space. Since the Bohmian particles are guided by this nonlocal wave function, the interactions between particles are not local. Bohmian particles are not entangled, they are guided by entangled wave function. Preparation is an athropomorphic concept, it's something that humans do in a controlled way. Humans can control effective Hamiltonians that prepare wave functions, but humans cannot control initial positions of Bohmian particles, so Bohmian particles are not prepared.

A guiding wave with non-local properties that operates on a fundamental Bohmian particle does not make sense to me for a number of reasons. In this interpretation, why is entanglement a property of the guiding wave and not a property of the fundamental Bohmian particle?

My understanding is a photon does not have a wave function in QM. What is a photon guided by in this interpretation?

 

[Elias1960]

Is this interpretation saying the guiding wave for the fundamental Bohmian particle is the same as the wave function used in non-relativistic quantum mechanics? If so, then won't this interpretation fail to make the same predictions as QFT?

It depends on the particular classical theory of these Bohmian particles. In the simplest case, this theory defines some lattice of such Bohmian particles that oscillate around their fixed average position. This would give three independent types of sound waves. The large distance continuous limit of these sound wave equations are simple wave equations, thus, they have the same form $(\partial_t^2 - \Delta)u^i(x,t) = 0$ as usual for wave equations. This classical equation itself has relativistic symmetry: If you apply the Lorentz transformation (with c as the speed of sound) to one solution you get another, Doppler-shifted solution.

So, this Bohmian theory gives a QFT with three scalar fields.

Three scalar fields is, of course, not what we observe in reality. The job would be now to find such an ether model that works similarly but gives the SM plus gravity fields. Such a proposal has been made and published but has been simply ignored. For gravity, in
I. Schmelzer, A Generalization of the Lorentz Ether to Gravity with General-Relativistic Limit, Advances in Applied Clifford Algebras 22, 1 (2012), p. 203-242, resp. arxiv:gr-qc/0205035.
For the SM, in
I. Schmelzer, A Condensed Matter Interpretation of SM Fermions and Gauge Fields, Foundations of Physics, vol. 39, nr. 1, p. 73 (2009), resp. arxiv:0908.0591.
BM is not mentioned there, but nothing prevents to apply BM to these proposals in the same way.

My understanding is a photon does not have a wave function in QM. What is a photon guided by in this interpretation?

The answer is simply that nobody cares, for the same reason that nobody cares about a wave function for phonons in quantum condensed matter theory. To define a BM interpretation of some QM, you have to fix the configuration space. Only for the configuration space a trajectory $q(t)\in Q$ exists. BM does not define any trajectories for other variables, like $p(t)$ or $E(t)$. All the other variables are contextual, that means, they have no hidden trajectory, and a measurement result depends also on the Bohmian configuration of the "measurement device".

A guiding wave with non-local properties that operates on a fundamental Bohmian particle does not make sense to me for a number of reasons. In this interpretation, why is entanglement a property of the guiding wave and not a property of the fundamental Bohmian particle?

This is already a question of type "why is BM as it is", which does not have an answer beyond "this is what BM postulates". You may not like this, but this is not a valid objection, and does not show that BM makes no sense.

 

[Demystifier]

Is this interpretation saying the guiding wave for the fundamental Bohmian particle is the same as the wave function used in non-relativistic quantum mechanics? If so, then won't this interpretation fail to make the same predictions as QFT?

My understanding is a photon does not have a wave function in QM. What is a photon guided by in this interpretation?

Did you read Sec. 5 of the paper? All those questions are answered there.

A guiding wave with non-local properties that operates on a fundamental Bohmian particle does not make sense to me for a number of reasons. In this interpretation, why is entanglement a property of the guiding wave and not a property of the fundamental Bohmian particle?

Because the entanglement is by definition a state in the Hilbert space that cannot be written as a product of states in its subspaces. The wave is nothing but a suitably represented state in the Hilbert space. The Bohmian particle is mathematically something else.

 

[kurt101]

Did you read Sec. 5 of the paper? All those questions are answered there.

Yes I have read Sec. 5 and the entire paper many times.

The paper says:
"The BM trajectories are only needed for truly elementary particles. We do not know yet what the theory of those truly elementary particles is, but in principle it is possible that the truly elementary particles are not described by relativistic QFT."

Saying "We do not know yet what the theory of those truly elementary particles is", is one of several reasons that made me think this paper was not actually proposing an interpretation where the fundamental guiding wave (in position space) for the "truly elementary particle" is the wave function. That is why I am asking questions to clarify what is fundamentally guiding the fundamental particles in position space for this interpretation. Because saying it is fundamentally the wave function is why I was not receptive to BM in the first place. BM is just using the crutch of QM and failing to explain anything much deeper than QM.

While this paper does explain in several ways why this interpretation of BM is more fundamental than QFT. It does not explain how it can make the same accurate predictions of experiments like the double slit experiment with photons that QFT can. If this BM interpretation relies on the wave function and the photon does not have a wave function, how can it predict anything about the photon?

I. Schmelzer, A Generalization of the Lorentz Ether to Gravity with General-Relativistic Limit, Advances in Applied Clifford Algebras 22, 1 (2012), p. 203-242, resp. arxiv:gr-qc/0205035.

Thanks, very interesting paper!

The answer is simply that nobody cares, for the same reason that nobody cares about a wave function for phonons in quantum condensed matter theory. To define a BM interpretation of some QM, you have to fix the configuration space. Only for the configuration space a trajectory $q(t)\in Q$ exists. BM does not define any trajectories for other variables, like $p(t)$ or $E(t)$. All the other variables are contextual, that means, they have no hidden trajectory, and a measurement result depends also on the Bohmian configuration of the "measurement device".

Really? Nobody cares that BM can't explain how a photon is guided? Maybe you can explain this answer a little more. I am not familiar with the reason that nobody cares about the wave function for phonons in quantum condensed matter theory and how not caring applies to Bohmians not caring about how the photon is guided.

 

[Elias1960]

Really? Nobody cares that BM can't explain how a photon is guided? Maybe you can explain this answer a little more. I am not familiar with the reason that nobody cares about the wave function for phonons in quantum condensed matter theory and how not caring applies to Bohmians not caring about how the photon is guided.

Ok, it is not completely correct. Those Bohmians who think bosons should be described by particle positions of photons have to care. Those who prefer a Bohmian field theory at least for bosons don't have to, because for them the "position" is replaced by the configuration, and the configuration is the EM field $A_\mu(x)$. And the trajectory of the Bohmian particles is, then, a trajectory in the field configuration $A_\mu(x,t)$. The photons are nothing but energy levels of fields with a particular momentum, simply quantum effects of no fundamental importance. For the phonons of condensed matter theory the situation is similar. Here BM also cares only about how the fundamental particles move, not how the phonons move. First of all, because the phonons do not even have well-defined positions. Phonons are oscillations of a lattice, and you cannot localize such an oscillation with more accuracy than the lattice distance. What, then, approximately could identify a phonon position is a quite complicate function of the p and q of the atoms of the lattice, thus, they simply don't have Bohmian trajectories.

The only Bohmian trajectories are trajectories in the configuration space $q(t) \in Q$. So, if you describe the same theory with different configuration spaces, you get different Bohmian interpretations with different Bohmian trajectories.

 

[Demystifier]

If this BM interpretation relies on the wave function and the photon does not have a wave function, how can it predict anything about the photon?

Bohmian mechanics does not make measurable predictions on microscopic objects such as photons and electrons. It makes measurable predictions on macroscopic objects, called perceptibles in the paper. For instance, it makes measurable predictions on photon detectors. But according to Bohmian mechanics, a macroscopic object is made of many microscopic particles. The paper proposes that even a single photon is made of many fundamental microscopic particles, in the same sense in which a single phonon is made of many atoms. It is those fundamental microscopic particles that have a fundamental wave function.

That being said, note that it is not entirely true that a photon does not have a wave function. It does, in the same sense in which a phonon has a wave function as in Eq. (34). But that wave function is not fundamental and it does not have the usual (either probabilistic or Bohmian) physical interpretation.

 

[kurt101]

The entangled wave function is prepared by a local Hamiltonian. But the entangled wave function is not a local object, you cannot tell what is the value of the wave function at a given position in the 3-dimensional space. Since the Bohmian particles are guided by this nonlocal wave function, the interactions between particles are not local. Bohmian particles are not entangled, they are guided by entangled wave function. Preparation is an athropomorphic concept, it's something that humans do in a controlled way. Humans can control effective Hamiltonians that prepare wave functions, but humans cannot control initial positions of Bohmian particles, so Bohmian particles are not prepared.

For this interpretation, can the Hamiltonian be considered the entire universe? And is it correct to say that every fundamental BM particle in the universe is guided by non-local interaction with every other BM particle in the universe? Based on your answer to a previous question, I am taking non-local interaction to be "casual influences - do go faster than light".

The paper says "We also stress that this non-local interaction is the only interaction between Bohmian point-like particles. " For this interpretation is it correct to say there are no interactions between BM particles that are not instant (i.e. faster than light)?

The paper says "When the wave function is not entangled, then Bohmian point-like particles do not interact with each other at all." When is it the case that the wave function is not entangled for a BM particle?

 

[Demystifier]

For this interpretation, can the Hamiltonian be considered the entire universe?

For the preparation interpretation, no.

And is it correct to say that every fundamental BM particle in the universe is guided by non-local interaction with every other BM particle in the universe?

No, it's not the usual way of talking about it.

The paper says "We also stress that this non-local interaction is the only interaction between Bohmian point-like particles. " For this interpretation is it correct to say there are no interactions between BM particles that are not instant (i.e. faster than light)?

Yes.

The paper says "When the wave function is not entangled, then Bohmian point-like particles do not interact with each other at all." When is it the case that the wave function is not entangled for a BM particle?

Mathematically, e.g. when $\Psi(x_1,x_2)=\psi_1(x_1)\psi_2(x_2)$. Physically, it can be the case when the effective Hamiltonian does not contain interaction between $x_1$ and $x_2$. Do you understand the difference between effective Hamiltonian and fundamental Hamiltonian?

 

[kurt101]

Mathematically, e.g. when $\Psi(x_1,x_2)=\psi_1(x_1)\psi_2(x_2)$. Physically, it can be the case when the effective Hamiltonian does not contain interaction between $x_1$ and $x_2$.

Sorry, I should have gotten that from your previous answer "Because the entanglement is by definition a state in the Hilbert space that cannot be written as a product of states in its subspaces.". This makes sense now.

Do you understand the difference between effective Hamiltonian and fundamental Hamiltonian?

Not really and I only read about it after you asked. My crude understanding is the effective Hamiltonian approximates the fundamental Hamiltonian by estimating or neglecting terms that don't contribute very much to the fundamental Hamiltonian.

 

[Demystifier]

Not really and I only read about it after you asked. My crude understanding is the effective Hamiltonian approximates the fundamental Hamiltonian by estimating or neglecting terms that don't contribute very much to the fundamental Hamiltonian.

True, but from that it follows that:
1. Effective Hamiltonian may be expressed in terms of different degrees of freedom than the fundamental one. For instance, one may be expressed in terms of particles, and the other in terms of fields.
2. Effective Hamiltonians can sometimes be manipulated (changed by humans in the laboratory), while the fundamental ones cannot.