Quantum Decoherence and deBroglie-Bohm theory

[stevendaryl]

Yes, the single particle case works, and people have thought about these things. Struyve and De Baere's oversimplification should be considered shorthand that people have thought about these things.

In the context of responding to someone's claim that Bohm is not equivalent to standard quantum mechanics, it seems to me to be extremely misleading to use such a shorthand. Do you know of a link to a more detailed explanation about the equivalence of Bohm and standard QM? Thanks.

 

[stevendaryl]

In the context of responding to someone's claim that Bohm is not equivalent to standard quantum mechanics, it seems to me to be extremely misleading to use such a shorthand. Do you know of a link to a more detailed explanation about the equivalence of Bohm and standard QM? Thanks.

Based on a little Googling, I can't find an online paper explaining how Bohm's theory reproduces standard quantum mechanics when measurement is taken into account, although there is a discussion that mentions Bohm's proof that his theory reproduces the measurement results of quantum mechanics here:
http://publish.uwo.ca/~wmyrvold/Bohm.pdf

The particular issue is a particle in a box. Bohm's model predicts that a particle in an energy eigenstate has velocity zero (because the velocity is related to the imaginary part of the wave function, which can be taken to be zero for an energy eigenstate). But the prediction of quantum mechanics is that a measurement of momentum will yield a nonzero value. Bohm claimed that his theory makes the same prediction, but the paper doesn't reproduce the argument.

 

[atyy]

In the context of responding to someone's claim that Bohm is not equivalent to standard quantum mechanics, it seems to me to be extremely misleading to use such a shorthand. Do you know of a link to a more detailed explanation about the equivalence of Bohm and standard QM? Thanks.

Based on a little Googling, I can't find an online paper explaining how Bohm's theory reproduces standard quantum mechanics when measurement is taken into account, although there is a discussion that mentions Bohm's proof that his theory reproduces the measurement results of quantum mechanics here:
http://publish.uwo.ca/~wmyrvold/Bohm.pdf

The particular issue is a particle in a box. Bohm's model predicts that a particle in an energy eigenstate has velocity zero (because the velocity is related to the imaginary part of the wave function, which can be taken to be zero for an energy eigenstate). But the prediction of quantum mechanics is that a measurement of momentum will yield a nonzero value. Bohm claimed that his theory makes the same prediction, but the paper doesn't reproduce the argument.

I don't know if there is an iron-clad proof that Bohmian Mechanics reproduces exactly all of standard quantum mechanics. The basic heuristic is that Bohmian Mechanics can reproduce any position measurement, provided standard decoherence arguments hold. This is good enough to reproduce measurements of observables other than position, because such observables are measured via position measurements. For example, in the single slit experiment, the transverse momentum just after the slit is measured via a position measurement at infinity, because the far field Fraunhofer limit is essentially a Fourier transform of the wave function just after the slit. This takes care of unitary evolution and measurements of observables, leaving wave function collapse. Wave function collapse is taken care of in Bohmian Mechanics by the definite experimental outcome, allowing us to ignore irrelevant parts of the wave function after measurement. I believe that in principle Bohmian Mechanics allows recoherence, whereas standard quantum mechanics does not, if a measurement has been made, but the recoherence of Bohmian Mechanics is argued to occur on time scales that are irrelevantly large, analogous to the the irrelevance of Poincare recurrences in classical kinetic theory.

Here's my reading list for Bohmian Mechanics.

1. Simple and friendly introduction
http://arxiv.org/abs/quant--ph/0611032
What you always wanted to know about Bohmian mechanics but were afraid to ask
Oliver Passon

2. Comprehensive intermediate level introduction. Section VI is an extensive discussion of measurements.
http://arxiv.org/abs/1206.1084
Overview of Bohmian Mechanics
Xavier Oriols, Jordi Mompart

3. Extensive detailed comparison of Bohmian Mechanics and the quantum formalism, including POVMs
http://arxiv.org/abs/quant-ph/0308038
Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory
Detlef Dürr, Sheldon Goldstein, Nino Zanghì

There are older checks of most of the famous quantum phenomena, including things like Bell tests, but I don't think those are available free.

Once again I'd like to stress that the point of Bohmian Mechanics is not that it is the correct theory of nature, but rather it demonstrates that solutions to the measurement problem exist.

 

[stevendaryl]

I don't know if there is an iron-clad proof that Bohmian Mechanics reproduces exactly all of standard quantum mechanics. The basic heuristic is that Bohmian Mechanics can reproduce any position measurement, provided standard decoherence arguments hold. This is good enough to reproduce measurements of observables other than position, because such observables are measured via position measurements. For example, in the single slit experiment, the transverse momentum just after the slit is measured via a position measurement at infinity, because the far field Fraunhofer limit is essentially a Fourier transform of the wave function just after the slit. This takes care of unitary evolution and measurements of observables, leaving wave function collapse. Wave function collapse is taken care of in Bohmian Mechanics by the definite experimental outcome, allowing us to ignore irrelevant parts of the wave function after measurement. I believe that in principle Bohmian Mechanics allows recoherence, whereas standard quantum mechanics does not, if a measurement has been made, but the recoherence of Bohmian Mechanics is argued to occur on time scales that are irrelevantly large, analogous to the the irrelevance of Poincare recurrences in classical kinetic theory.

Here's my reading list for Bohmian Mechanics.

1. Simple and friendly introduction
http://arxiv.org/abs/quant--ph/0611032
What you always wanted to know about Bohmian mechanics but were afraid to ask
Oliver Passon

2. Comprehensive intermediate level introduction. Section VI is an extensive discussion of measurements.
http://arxiv.org/abs/1206.1084
Overview of Bohmian Mechanics
Xavier Oriols, Jordi Mompart

Okay, the second reference explains that the equivalence between the two is very indirect. Standard quantum mechanics uses a wave function (or more generally, a density matrix) for the system of interest. In contrast, Bohmian mechanics must always have a composite wavefunction that includes the system of interest, plus measuring devices. In that sense, it's sort of similar to the MWI (Many-World Interpretation) resolution to the measurement problem by treating the observers quantum-mechanically, as well. In a certain sense, maybe, Bohmian mechanics amounts to MWI + a choice of which world is "real".

 

[atyy]

Okay, the second reference explains that the equivalence between the two is very indirect. Standard quantum mechanics uses a wave function (or more generally, a density matrix) for the system of interest. In contrast, Bohmian mechanics must always have a composite wavefunction that includes the system of interest, plus measuring devices. In that sense, it's sort of similar to the MWI (Many-World Interpretation) resolution to the measurement problem by treating the observers quantum-mechanically, as well.

Yes, that's my understanding.

In a certain sense, maybe, Bohmian mechanics amounts to MWI + a choice of which world is "real".

Others have made similar observations. (Technically, Bohmian Mechanics and Many-Worlds are similar in that both have a wave function of the universe and use decoherence.)

Wallace's review http://arxiv.org/abs/0712.0149 mentions this critique of Bohmian Mechanics by Deutsch: "A potentially more serious flaw arises from the so-called “Everett-in-denial” objection to realism (Deutsch 1996; Zeh 1999; Brown and Wallace 2005). ... Advocates of the Everett interpretation claim that, (given functionalism) the decoherence-defined quasiclassical histories in the unitarily evolving physically real wavefunction describe — are — a multiplicity of almost-identical quasiclassical worlds; if that same unitarily-evolving physically real wavefunction is present in DBB (or any other hidden-variable theory) then so is that multiplicity of physically real worlds, and all the hidden variables do is point superfluously at one of them."

Nikolic's "Solipsistic hidden variables" http://arxiv.org/abs/1112.2034 is an amusing variant of Bohmian Mechanics in which, "Finally note that our result that different observers may live in different branches of the wave function is very similar to the many-world interpretation [16, 17], briefly discussed in Sec. 2.2. Yet, there is one crucial difference. In the many-world interpretation, there is a copy of each observer in any of the branches. In our solipsistic interpretation, for each observer there is only one copy living in only one of the branches."

 

[Demystifier]

I mean why can't we find recently published papers on dBB?

We can, if we know how to search. See e.g.
http://lanl.arxiv.org/find/quant-ph/1/ti: Bohmian/0/1/0/all/0/1
http://lanl.arxiv.org/find/quant-ph/1/ti:+AND+Broglie+Bohm/0/1/0/all/0/1
http://lanl.arxiv.org/find/quant-ph/1/ti:+AND+pilot+wave/0/1/0/all/0/1

 

[bohm2]

The particular issue is a particle in a box. Bohm's model predicts that a particle in an energy eigenstate has velocity zero (because the velocity is related to the imaginary part of the wave function, which can be taken to be zero for an energy eigenstate). But the prediction of quantum mechanics is that a measurement of momentum will yield a nonzero value. Bohm claimed that his theory makes the same prediction, but the paper doesn't reproduce the argument.

But this is just a difference in ontology between BM and Orthodox QM, but both still make the same empirical predictions.

 

[stevendaryl]

But this is just a difference in ontology between BM and Orthodox QM, but both still make the same empirical predictions.

The specific example I was talking about was an empirical prediction: The prediction of standard quantum mechanics is that if you measure the momentum of a particle in a box in the ground state, you will get p = \pm \frac{\pi \hbar}{L} where L is the length of the box. Bohm claimed that his theory gives the same result (although it's a lot more complicated to show this). Even though the velocity is zero according to his model, a measurement of position at two different times will give different results, because the measurement process disturbs the particle, and so will give a nonzero measured velocity.

 

[jtbell]

The prediction of standard quantum mechanics is that if you measure the momentum of a particle in a box in the ground state, you will get p=±πℏLp = \pm \frac{\pi \hbar}{L} where LL is the length of the box.

Have you ever seen the momentum-space wave functions for the infinite square well (a.k.a. "particle in a box")?

http://physicspages.com/2012/10/04/infinite-square-well-momentum-space-wave-functions/

They do not restrict the possible values of momentum to $\pm \frac{\pi \hbar}{L}$ for the ground state.

 

[Ilja]

Anyway, the major problem in dBB is the formulation of the theory for chiral fermions interacting with non-abelian gauge fields. So you can think of all papers about chiral fermions interacting with non-abelian gauge fields as secretly dBB papers (ok, maybe I went too far there, but it is my interpretation of the literature :D).

I like this idea of secretly dBB papers, and can tell you about one which is in a quite real sense "secretly dBB": http://arxiv.org/abs/0908.0591

It contains a method to describe a pair of Dirac fermions based on a scalar field with degenerated vacuum. The latter one can be handled with straightforward Bohmian field theory.

 

[Ilja]

The point is, people mostly like dBB because it preserves(at least partly) the classical nature of physical laws... But I just don't see the reason why so many people should be so insistent that nature is as people thought before QM.

I think it is more about having a reasonable picture of the whole thing. Instead of Copenhagen with its two worlds which conceptually are in conflict with each other. So you get a classical trajectory for the quantum domain too, a wave function for the classical domain too, and have no longer any need for separate collapse dynamics.

Also dBB doesn't seem to be able to make a good marriage with SR too. I know people had some attempts but things don't seem to fit nicely.
And quantum fluctuations. How can dBB account for particles coming from nothing when it doesn't accept uncertainty principles as they are in standard QM?

If you forget about fundamental relativity, which forbids the existence of hidden preferred frames, then there is no problem with relativity at all. Straightforward field theory with \(\mathcal{L} = \partial_t\varphi^2 - \partial_i \varphi^2 + V(\varphi)\) fits nicely into the standard dBB scheme (if quadratic in the momentum variables everything is fine).