"Iterativity" of quantum potential?

[itssilva]

I understand the topic of QM interpretations is not much beloved around PF, but I find one odd enough thing regarding the pilot-wave formulation that strikes me as a fatal flaw: for suppose you have a Hamiltonian H₀ = T + V₀, a solution of which is given by Ψ₀ = R₀e^iS₀/ħ; so you use R₀ and S₀ to calculate the quantum potential Q₀, which is supposed to be some kind of "correction" to the initial potential. However, if this were the case, defining V₁ := V₀ + Q₀ the dynamics of the system would now be governed by the Hamiltonian H₁ = T + V₁, with eigensolution Ψ₁ = R₁e^iS₁/ħ , and so on. How do advocates of the formulation explain this, and perhaps more critically, what are the consequences for pilot-wave computational simulations? I've never seen this mentioned anywhere in QM textbooks, Wikipedia and so forth, and it seems kinda important.

 

[stevendaryl]

I understand the topic of QM interpretations is not much beloved around PF, but I find one odd enough thing regarding the pilot-wave formulation that strikes me as a fatal flaw: for suppose you have a Hamiltonian H₀ = T + V₀, a solution of which is given by Ψ₀ = R₀e^iS₀/ħ; so you use R₀ and S₀ to calculate the quantum potential Q₀, which is supposed to be some kind of "correction" to the initial potential. However, if this were the case, defining V₁ := V₀ + Q₀ the dynamics of the system would now be governed by the Hamiltonian H₁ = T + V₁, with eigensolution Ψ₁ = R₁e^iS₁/ħ , and so on. How do advocates of the formulation explain this, and perhaps more critically, what are the consequences for pilot-wave computational simulations? I've never seen this mentioned anywhere in QM textbooks, Wikipedia and so forth, and it seems kinda important.

There is no iteration involved. The Schrödinger equation, with no "quantum potential" is exactly equivalent to the pair of equations for $R$ and $S$, and the equation for $S$ looks like classical mechanics except for the additional quantum potential. The iteration that you seem to be thinking of is to go back to the Schrödinger equation, and use a modified hamiltonian that includes the quantum potential? There's no reason to do that.

 

[itssilva]

the equation for $S$ looks like classical mechanics except for the additional quantum potential

As I see it, this seems to be the basis of the interpretation: as the S equation looks like the classical Hamilton-Jacobi equation with an additional potential term, if one stretches this likelihood to say that ∇S is actually the classical momentum of the particles, then one must consider the "pre-quantized" dynamics of the system to be modified as well - otherwise there is no consistency between classical and quantum formalism, as, although the quantum theory is the more fundamental one, there must be enforced a one-to-one correspondence between the math of phase space and that of Hilbert space. Also, consider this analogy: if a (real, actual) field influences particle motion, the particles "kick back" and influence the field evolution as well - which is roughly what the iteration process above conveys. If you're fine with simply calculating Q₀ and calling it a "potential", fine, but if you stretch things and claim it to provoke actual classical behavior, I find the argument to be lacking, and that is the point.

 

[stevendaryl]

As I see it, this seems to be the basis of the interpretation: as the S equation looks like the classical Hamilton-Jacobi equation with an additional potential term, if one stretches this likelihood to say that ∇S is actually the classical momentum of the particles, then one must consider the "pre-quantized" dynamics of the system to be modified as well - otherwise there is no consistency between classical and quantum formalism, as, although the quantum theory is the more fundamental one, there must be enforced a one-to-one correspondence between the math of phase space and that of Hilbert space. Also, consider this analogy: if a (real, actual) field influences particle motion, the particles "kick back" and influence the field evolution as well - which is roughly what the iteration process above conveys. If you're fine with simply calculating Q₀ and calling it a "potential", fine, but if you stretch things and claim it to provoke actual classical behavior, I find the argument to be lacking, and that is the point.

I'm not exactly sure what you're saying. The two equations for $R$ and $S$ is just a different way of writing the Schrödinger equation. It's exactly equivalent to the usual Schrödinger equation. The new feature of Bohmian mechanics is the interpretation that particles move along definite trajectories determined by the equation for $S$. The fact that the equation for $S$ looks like classical mechanics plus a "quantum potential" is cute, but it's really not particularly important. The important thing is the definite trajectories, however you calculate them.

Okay, there is another important point, which is that using the equation for $S$ to calculate trajectories is necessary for the interpretation of $R^2$ as a probability density (because $S$ and $R$ together obey a continuity equation, which you need for a probability density).

 

[itssilva]

The fact that the equation for $S$ looks like classical mechanics plus a "quantum potential" is cute, but it's really not particularly important. The important thing is the definite trajectories, however you calculate them.

But that's what I'm trying to say: the important thing is the trajectories, which in this case are calculated from the assumption that the particles are under the influence of a potential V + Q₀ (and not, as I argued from consistency, V + Q₀ + Q₁ + ...), because that's what the Bohmian interpretation sets out to do or I'm misled; to say that the form of the potential in the S equation is "really not particularly important" is the same as saying a) the form of the Hamiltonian isn't important to determine the trajectories of particles, or b) the Bohmian interpretation is based on a cute coincidence which is ultimately meaningless and therefore is useless.

As for the interpretation of the R equation as a classical continuity equation, it might be viewed as a cute coincidence as well, since the quantum formula has already been established; it only serves to push further the S equation argument. (Also, note that the fact that the two equations are coupled is unimportant for the interpretation of what they mean in this case.)

 

[Demystifier]

If you're fine with simply calculating Q0 and calling it a "potential", fine

Yes, that's exactly how pilot wave theory is understood. It is just a name for an auxiliary quantity. In fact, pilot wave theory can be formulated without using the quantum potential at all.

 

[itssilva]

Yes, that's exactly how pilot wave theory is understood. It is just a name for an auxiliary quantity. In fact, pilot wave theory can be formulated without using the quantum potential at all.

OK, but this still doesn't tell me how Bohmian theory sits with the rest of physical theory; because, as I think everyone here agrees, the quantum potential is just something that one gets out of the wavefunction that is completely general, but, as soon one uses it to compute classical trajectories, the dynamics of the system become interwoven with this act, and, unless one accepts to perform iterative corrections, I don't see how pilot-wave theory can have a unambiguous mathematical formulation with well-defined classical objects defined within a Hamiltonian formalism (according to which particle trajectories are described) and quantum objects within a Hilbert space (according to which vector states are described); bear in mind also that, in analogy with perturbation theory, one might consider corrections to the classical potential up to "lowest order", if the series can be shown to converge, and maybe it would interesting to see a computational experiment calculate new trajectories with higher terms to see if one gets better results, if hasn't been done already.

But you say that one can scratch Q in pilot wave theory; so how are trajectories calculated then? Isn't the whole point of the theory to calculate classical trajectories using a quantum correction given by some Q? What's really fundamental then?

 

[stevendaryl]

OK, but this still doesn't tell me how Bohmian theory sits with the rest of physical theory; because, as I think everyone here agrees, the quantum potential is just something that one gets out of the wavefunction that is completely general, but, as soon one uses it to compute classical trajectories, the dynamics of the system become interwoven with this act, and, unless one accepts to perform iterative corrections, I don't see how pilot-wave theory can have a unambiguous mathematical formulation with well-defined classical objects defined within a Hamiltonian formalism (according to which particle trajectories are described) and quantum objects within a Hilbert space (according to which vector states are described); bear in mind also that, in analogy with perturbation theory, one might consider corrections to the classical potential up to "lowest order", if the series can be shown to converge, and maybe it would interesting to see a computational experiment calculate new trajectories with higher terms to see if one gets better results, if hasn't been done already.

The point of writing Schrödinger's equation in the Bohmian form isn't to get a new way of calculating trajectories. People already know how to solve Schrödinger's equation. To apply Bohmian mechanics, in one approach, you just solve Schrödinger's equation the way that anybody else would. This gives you a wave function $\psi$. Then you find $R$ through the equation:

$R = \sqrt{\psi^* \psi}$

Then you find $S$ through the equation:

$S = \frac{\hbar}{i} ln(\dfrac{\psi}{R})$

Then you find the velocities through the ansatz:

$\vec{v} = \frac{1}{m} \nabla S$

This doesn't actually give you a unique trajectory; it only gives a family of trajectories that differ by the initial position, $\vec{r}_0$. So you make another ansatz, which is that the initial position is chosen according to the probability distribution $\psi^* \psi$ at time $t=0$. Those two assumptions give you a probability distribution on trajectories.

I don't understand what issue you are talking about. What are you claiming is not well-defined in this approach?

The comparison with perturbation theory is not appropriate, because perturbation theory is about approximating a solution by using a power series. The Bohmian approach isn't an approximation, or at least, it isn't assumed to be. It's assumed that the relationship between $\psi$ and $S$ is exact, and that the relationship between $S$ and trajectories is exact.

 

[Demystifier]

the quantum potential is just something that one gets out of the wavefunction that is completely general, but, as soon one uses it to compute classical trajectories

One does not use quantum potential to compute classical trajectories. One uses it to compute quantum trajectories, and they are different from the classical ones.

 

[Demystifier]

But you say that one can scratch Q in pilot wave theory; so how are trajectories calculated then? Isn't the whole point of the theory to calculate classical trajectories using a quantum correction given by some Q? What's really fundamental then?

Trajectories are calculated from the velocity equal to the gradient of the phase of the wave function. The point of this is not to calculate a correction to classical trajectories. The fundamental thing are quantum trajectories, which sometimes are not even close to the classical ones.

 

[itssilva]

I don't understand what issue you are talking about. What are you claiming is not well-defined in this approach?

The comparison with perturbation theory is not appropriate, because perturbation theory is about approximating a solution by using a power series. The Bohmian approach isn't an approximation, or at least, it isn't assumed to be

The only thing I claim is I don't know what the math that backs up this funride is. The stuff you posted I've already read about - the point is not what the theory is, but why it is interpreted the way it is; you see, in my small-minded world view there are two things that describe the universe: classical trajectories (CTs) and quantum probability (densities) (QPDs); but as Demystifier puts it:

Trajectories are calculated from the velocity equal to the gradient of the phase of the wave function. The point of this is not to calculate a correction to classical trajectories. The fundamental thing are quantum trajectories, which sometimes are not even close to the classical ones.

If one uses this theory simply to define what one means by a quantum trajectory (QT), rather than trying to establish some kind of connection to the classical ones, then what's the point? Physics is either about CTs or QPDs, there's no middle ground (unless, of course, one bothers to ellaborate); defining some kind of hybrid with no parallel to those physical objects seems rather pointless - I dare say, from the depths of my ignorance, vacuous. I ask the mathematician within: where does a QT live? I know where CTs do - phase space - , as well as vector states - Hilbert space - , but what of the QTs ? In all material I read, never once this topic was discussed (of course, maybe it exists and I simply missed it), and, though the connection between QM and CM is far from established, we have many pointers of what it is about, including (semi-)classical limiting processes.

Also, I believe the analogy with perturbation theory is appropriate because of this view I've been defending here: a sensible interpretation ought to be based on (what I believe) is common ground to all physicists, CTs and QPDs, and the scheme I sketched, in the spirit of that idea, tries to define a limiting process QT → CT so that one recuperates familiar objects from a unconventional reasoning/thinking/interpretation/structure, at the least. But bear in mind, I make no claim for the correctness of it either - far from it; haven't touched upon questions of convergence, or even well-definiteness. I've only used it as a meta-counterexample, to query about the mathematical state of this interpretation. In this sense, the comparison with perturbation methods is only in the broadest sense, as the successive "corrections" aren't guaranteed to be well-behaved in any respect, or bear any strict formal similarity with Born series or the like.

But I think I demonize myself, by doing what I did not set out to do (debate phylosophy in this forum); as to not drag you into this mire any further, perhaps I should close up shop, just asking for any info you might have on alternate views of Bohmian mechanics (same eqs., different interpretations) and/or (specially) a rigorous mathematical formalism (there should exist one at least as rigorous as standard QM). Which is what I set out do. TYVMFYT.

 

[Demystifier]

If one uses this theory simply to define what one means by a quantum trajectory (QT), rather than trying to establish some kind of connection to the classical ones, then what's the point?

The point is to resolve the measurement problem of quantum mechanics. It is a happy coincidence that one possible solution (Bohmian mechanics) looks so similar to classical mechanics, but this similarity is not essential.

 

[itssilva]

The point is to resolve the measurement problem of quantum mechanics. It is a happy coincidence that one possible solution (Bohmian mechanics) looks so similar to classical mechanics, but this similarity is not essential.

I do not see how it would solve the MP, since particle evolution is still governed by deterministic equations, but feel free to quote me the lit. on that one; you actually got me curious.

BTW, sorry to revive an old post, but since there appears to be experts on Bohmian mechanics here, I figured why not seize the opportunity, specially since all my technical foundation on this comes solely from Bohm's paper Phys. Rev., 1952, 85 (2), 166-179 (should have mentioned before). OK; so, following the logic in my 1st post, BM as it was originally formulated would be a working interpretation of what QM (not measurement, etc.) is if, in all occasions, V₀≈V₁; but, by a happy coincidence, Q∝ħ, and ħ is really small, so this kinda works out by itself conceptually; in this sense and logic, BM is thus valid at least as a semiclassical approximation. If this is true, it's relatively easy to check: see if there are computations done on problems in which semiclassical techniques are known to fail, but in which nevertheless the BM approach triumphs in a statistically significant way; does anyone know of this?

 

[Demystifier]

If this is true, it's relatively easy to check: see if there are computations done on problems in which semiclassical techniques are known to fail, but in which nevertheless the BM approach triumphs in a statistically significant way; does anyone know of this?

Yes. Take any example in non-relativistic QM where standard QM is significantly different from semi-classical approximation. The BM approach makes exactly the same measurable predictions as standard QM.

 

[itssilva]

Yes. Take any example in non-relativistic QM where standard QM is significantly different from semi-classical approximation. The BM approach makes exactly the same measurable predictions as standard QM.

Well... that's an statement, alright. Meaning that, changing the numerical value of ħ never bears any difference in outcome for the same given problem, and people tested it explicitly? IDK if there exists some metastudy of this, because, even if we're talking about classical solutions used to explain classical physics, it's still a highly non-trivial behavior (there might be some like chaos or whatnot). Impressive, nonetheless.