Undergrad How is realism understood in QM?

[A. Neumaier]

"Motion" is defined as a continuous process, not 'motion' done in quantum jumps. How can motion in DeBB for an electron be defined, if it goes from A to B without traveling the distance from A to B?

The solution of any differential equation results in continuous trajectories.

 

[Demystifier]

Thanks. I hadn"t noticed this.

But the treatment is nonrelativistic, and would lead to the well-known infinities in a relativistic treatment to one loop order. Did anyone develop a renormalized version of this, or is Bohm"s treatment still the state of the art?

I think nobody renormalized it in a way you would like to. People usually regularize it by some lattice-type regularization and assume that a sufficiently fine lattice gives sufficiently fine results. I know you disagree, but we already discussed it several times so let us not go into it once again.

 

[EPR]

The solution of any differential equation results in continuous trajectories.

What about tunneling? Which DeBB equation of continuous motion describes it?

 

[A. Neumaier]

What about tunneling? Which DeBB equation of continuous motion describes it?

The equations are always the same, except that the potential changes.

The electron moves continuously either through the barrier or along a reflected path, depending on the initial conditions. I don"t know whether anyone has looked at this problem numerically so that one could see how the motion proceeds.

 

[A. Neumaier]

People usually regularize it by some lattice-type regularization

Bohm's version??

I think they abandon Bohm's version and introduce other beable degrees of freedom, and do not keep Bohm's beables.

 

[EPR]

The equations are always the same, except that the potential changes.

The electron moves continuously either through the barrier or along a reflected path, depending on the initial conditions. I don"t know whether anyone has looked at this problem numerically so that one could see how the motion proceeds.

I am struggling to come to grips with "electron goes through barrier without like charges repelling and pushing the charges apart".

 

[PeterDonis]

I am struggling to come to grips with "electron goes through barrier without like charges repelling and pushing the charges apart".

Remember that the quantum potential is also present in the equations of motion, so you cannot use your classical intuitions about what will happen.

 

[EPR]

Remember that the quantum potential is also present in the equations of motion, so you cannot use your classical intuitions about what will happen.

Fine, but then how is it continuous motion? How is it continuous Bohmian motion if a particle goes from A to B without traversing the distance between A and B?

 

[PeterDonis]

how is it continuous motion?

Because, as has already been noted, the motion is the solution of a differential equation.

How is it continuous Bohmian motion if a particle goes from A to B without traversing the distance between A and B?

Nobody said the particle did not traverse the distance between A and B. In the Bohmian interpretation, it does. It just doesn't obey the classical equations of motion when doing so, because the Bohmian equations of motion include the quantum potential. The quantum potential is not something magical that converts continuous motion to discrete jumps. It's just a potential that is present in the quantum differential equation that is not present in the classical differential equation.

 

[EPR]

Nobody said the particle did not traverse the distance between A and B. In the Bohmian interpretation, it does. It just doesn't obey the classical equations of motion when doing so, because the Bohmian equations of motion include the quantum potential.

Assuming that the above is true and the electron travels from A to B continuously, how is it able to do quantum tunneling?
You are saying it goes from A to B continuously but also does so in discreet steps due to the quantum potential(e.g. during tunneling). Both can't be true.

 

[PeterDonis]

Assuming that the above is true and the electron travels from A to B continuously, how is it able to do quantum tunneling?

Um, by following a Bohmian trajectory from A to B?

You are saying it goes from A to B continuously

Yes.

but also does so in discreet steps due to the quantum potential(e.g. during tunneling).

No. I have said no such thing. Go back and read what I said again. "Quantum potential" does not mean "discrete steps". It is just a term in a differential equation with continuous solutions.

 

[EPR]

Maybe i didn't express myself clearly so far and so had to google a Bohmian trajectory.

These non-classical trajectories appear to be mapped out by a nonlocal guiding wave. Assuming nonlocality, there is no issue mapping out a continuous trajectory(average paths) that also corresponds to the orthodox quantum rules.

 

[PeterDonis]

These non-classical trajectories appear to be mapped out by a nonlocal guiding wave.

Yes, that is the effect of the quantum potential, which is nonlocal.

 

[Demystifier]

Bohm's version??

I think they abandon Bohm's version and introduce other beable degrees of freedom, and do not keep Bohm's beables.

Can you give an example?

 

[A. Neumaier]

Can you give an example?

Replacing Bohm's continuum model by a lattice model does not preserve Bohm's beables.

 

[Demystifier]

Replacing Bohm's continuum model by a lattice model does not preserve Bohm's beables.

I think changing a continuum field with a field on a lattice is a minor change of beables.

 

[A. Neumaier]

I think changing a continuum field with a field on a lattice is a minor change of beables.

The change robs momentum modes from being beables, hence is drastic.

 

[Demystifier]

The change robs momentum modes from being beables, hence is drastic.

I think those two field ontologies are equivalent. If one knows the actual value of $\tilde{\phi}(k)$ and if one defines
$$\phi(x)\equiv \int dk\, e^{ikx} \tilde{\phi}(k)$$
then one also knows the actual value of $\phi(x)$.

It may be illuminating to see what is it analogous to in particle ontology. It is not analogous to proposing that the beable is the particle momentum $(p_x,p_y,p_z)$ instead of the particle position $(x,y,z)$. Instead, it is analogous to proposing that the beables are the spherical position coordinates $(r,\theta,\varphi)$ instead of Cartesian position coordinates $(x,y,z)$.

Bohm in his paper is not explicit about that, but I think it's implicit in his Eq. (A1). Moreover, in the book by Bohm and Hiley (The Undivided Universe) it is quite clear that those two ontologies are considered equivalent.

 

[A. Neumaier]

I think those two field ontologies are equivalent. If one knows the actual value of $\tilde{\phi}(k)$ and if one defines
$$\phi(x)\equiv \int dk\, e^{ikx} \tilde{\phi}(k)$$
then one also knows the actual value of $\phi(x)$.

This works only in the continuum.

From $x$ limited to a finite lattice (as in lattice models) one cannot go back to Bohm"s momenta $p$. Thus lattice ontologies are essentially different from Bohm"s ontology.

 

[Demystifier]

This works only in the continuum.

From $x$ limited to a finite lattice (as in lattice models) one cannot go back to Bohm"s momenta $p$. Thus lattice ontologies are essentially different from Bohm"s ontology.

I think you are nitpicking. It's well known how to write the corresponding equation on the lattice. For a scalar field that's indeed very simple.

 

[A. Neumaier]

I think you are nitpicking. It's well known how to write the corresponding equation on the lattice. For a scalar field that's indeed very simple.

How do you get beables with arbitrary wave vector from the beables of a scalar field theory on a finite lattice? It is well-known to be impossible by Nyquist's theorem.

 

[Demystifier]

How do you get beables with arbitrary wave vector from the beables of a scalar field theory on a finite lattice? It is well-known to be impossible by Nyquist's theorem.

The following is taken from the book "Quantum Fields on a Lattice" by Montvay and Munster. I think Eqs. (2.4) and (2.6) answer your question. Or do I miss something?

 

[A. Neumaier]

The following is taken from the book "Quantum Fields on a Lattice" by Montvay and Munster. I think Eqs. (2.4) and (2.6) answer your question. Or do I miss something?

What you miss is that $p$ is restricted to a finite number of values, unlike Bohm's beables, which are defined for all $p$. Thus a lattice theory has too few beables compared to Bohm's.

 

[A. Neumaier]

The following is taken from the book "Quantum Fields on a Lattice" by Montvay and Munster. I think Eqs. (2.4) and (2.6) answer your question. Or do I miss something?

In addition, space is not periodic, and the argument you quoted only works in the periodic case.

 

[Demystifier]

What you miss is that $p$ is restricted to a finite number of values, unlike Bohm's beables, which are defined for all $p$. Thus a lattice theory has too few beables compared to Bohm's.

Bohm's beables on a continuum are defined for all $p$, but Bohm's beables on a lattice are defined for some $p$ only. Bohm himself haven't study the lattice explicitly, but it's straightforward to modify the theory by replacing the continuum with the lattice, so it's justified to still refer to the resulting theory as "Bohm's".

Perhaps the source of our disagreement is philosophical. While for you the continuum and the lattice are conceptually totally different things, for me the continuum is conceptually just a lattice in the limit $a\to 0$.

So from my point of view, your objection is just irrelevant nitpicking.

 

[Elias1960]

Replacing Bohm's continuum model by a lattice model does not preserve Bohm's beables.

In a field theory, the field beables are $\phi(x)$. In the corresponding lattice model, the corresponding beables are $\phi(n)$ for $n\in\mathbb{Z}^3_N\subset \mathbb{R}^3$. So it is simply a subset. There is also no problem with the momentum variables, which are in both cases simply $\pi(x) = \dot{\phi}(x)$ (and have nothing to do with the momentum of the particles, which are in a field ontology irrelevant pseudo-particles similar to phonons).

(Not sure what you name "Bohm's beables", but in

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

field beables are used.)

 

[A. Neumaier]

Not sure what you name "Bohm's beables"

You'd read my posts within their context. See Post #59. Bohm's beables are the transverse momentum modes of the vector potential. These cannot be defined on a finite, nonperiodic lattice.

 

[Elias1960]

You'd read my posts within their context. See Post #59. Bohm's beables are the transverse momentum modes of the vector potential. These cannot be defined on a finite, nonperiodic lattice.

Ok. That means, another proposal made by Bohm et al later is better, it does not have your problem:

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

Of course, for vector gauge fields the Wilsonian lattice variant is even better than the straightforward one.

 

[A. Neumaier]

Ok. That means, another proposal made by Bohm et al later is better, it does not have your problem:

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

We were discussing an electron in an electromagnetic field, where @Demystifier referred in post #58 to this paper of Bohm. Instead, you propose references for scalar field theories that don't apply.

Of course, for vector gauge fields the Wilsonian lattice variant is even better than the straightforward one.

Where does this variant treat an electron in an electromagnetic field? Please refer to a paper that compared to #58 gives an improved answer.

 

[Elias1960]

We were discussing an electron in an electromagnetic field, where @Demystifier referred in post #58 to this paper of Bohm. Instead, you propose references for scalar field theories that don't apply.

The formulas have been given, for simplicity, for scalar field theory, but it is said that the field ontology works for other bosonic fields too. But, ok, here is a paper which describes explicitly the EM field:

Kaloyerou, P.N. (1994). The causal interpretation of the electromagnetic field. Physics Reports 244, 287-358