How to derive Born's rule for arbitrary observables from Bohmian mechanics?

[A. Neumaier]

TL;DR Summary

Whether Born's rule for arbitrary observables follows from BM (with quantum equilibrium assumption) is unclear to me.

Where can I find a reference to a derivation of Born's rule for arbitrary observables from Bohmian mechanics?

 

[Demystifier]

You can find it in many places, including:
1) P. Holland, The Quantum Theory of Motion, Chapter 8.
2) https://www.physicsforums.com/threads/spin-in-bohmian-mechanics.861549/#post-5406709
3) My "Bohmian mechanics for instrumentalists".

EDIT: A month after this post, after an extensive discussion with @A. Neumaier, I add that this refers to any observable that can be measured in practice. It does not refer to any self-adjoint operator, because some self-adjoint operators are not measurable in practice.

 

[A. Neumaier]

Are the three derivations mathematically equivalent?

I looked at https://arxiv.org/pdf/1811.11643.pdf =

3) My "Bohmian mechanics for instrumentalists".

since you are its author:

3.1 All perceptibles can be reduced to macroscopic positions

This is not strictly on topic here but how do you reduce to position the color of an object (surely a perceptible) sitting at the macroscopic position x? It is not created in the eyes, but later in the brain, by a process nobody really understands. The explanation given in the third paragraph explains nothing.

The derivation of Born's rule for an observable $K$ starts at the bottom of p.5. You forget to say
on which Hilbert space $K$ is defined - that of the object measured or that of the universe, whose wave function provides the position dynamics? I take it to be the former since in the latter,
$K$ would be heavily degenerate. Also, in (3) you assume a tensor product structure compatible with this assumption.

But (3) is not what unitary dynamics says. The latter maps the product state into a superposition of product states! This can be seen by writing down a formula for the Hamiltonian responsible for the interaction and considering a small time step in the Born approximation. Thus your derivation is based on assuming already a nonunitary dynamics!

 

[Demystifier]

Are the three derivations mathematically equivalent?

They are physicaly equivalent, but may differ in some fine mathematical details.

You forget to say on which Hilbert space $K$ is defined - that of the object measured or that of the universe, whose wave function provides the position dynamics?

It says that it's a microscopic observable, so it's understood that it's defined on the Hilbert space of the object measured.

Also, in (3) you assume a tensor product structure compatible with this assumption.

(3) is defined on a larger Hilbert space, that is on the space of the measured system + apparatus.

But (3) is not what unitary dynamics says. The latter maps the product state into a superposition of product states! This can be seen by writing down a formula for the Hamiltonian responsible for the interaction and considering a small time step in the Born approximation. Thus your derivation is based on assuming already a nonunitary dynamics!

Unitarity is of course assumed, look at (9).

 

[A. Neumaier]

Unitarity is of course assumed, look at (9).

This is linearity, not unitarity. Unitarity seems to be lost in the assumption (3).

the interaction between the measured system and the apparatus induces a unitary transition of the form
$$|k\rangle|A_0\rangle → |k′\rangle|A_k\rangle.~~~~~~~~~~~~~~~~~~~ (3)$$

Please justify this step from unitary dynamics.

 

[Demystifier]

This is linearity, not unitarity. Unitarity seems to be lost in the assumption (3).

Please justify this step!

I didn't explain it in detail because it is pretty much standard in the quantum theory of measurement. See e.g. https://arxiv.org/abs/quant-ph/9803052 Eq. (2).

 

[A. Neumaier]

I didn't explain it in detail because it is pretty much standard in the quantum theory of measurement. See e.g. https://arxiv.org/abs/quant-ph/9803052 Eq. (2).

But the argument (2) given there is for an idealized model case, where (in the interaction picture), the interaction has no off-diagonal terms in the selected basis. This seems appropriate only if the selected basis is invariant under the dynamics of the system alone (before the interaction begins). Thus if the system is a particle and angular momentum is to be measured, this assumption does not work!

The required dynamics cannot be obtained from the dynamics of the universe by coarse-graining, as required for Bohmian mechanics.

Or at least I'd like to see an argument how this special situation can come about in the case of an angular momentum measurement!

 

[vanhees71]

Are the three derivations mathematically equivalent?

I looked at https://arxiv.org/pdf/1811.11643.pdf =

since you are its author:

This is not strictly on topic here but how do you reduce to position the color of an object (surely a perceptible) sitting at the macroscopic position x? It is not created in the eyes, but later in the brain, by a process nobody really understands. The explanation given in the third paragraph explains nothing.

"Color" is not a physical observable but a physiological one. Maybe there's a POVM to desribe the functioning of the human eye ;-)).

 

[Demystifier]

Or at least I'd like to see an argument how this special situation can come about in the case of an angular momentum measurement!

I don't know how exactly the (orbital) angular momentum is measured in practice, i.e. what kind of interaction is used for that. But this question is not specific to Bohmian mechanics, this question is independent on the interpretation. Indeed, there is absolutely nothing specifically Bohmian about Eq. (3) in my paper or Eq. (2) in the other paper I mentioned. If you tell me in more detail how the angular momentum is measured in practice, I will be able to tell you in more detail how Bohmian mechanics explains this. And whatever the answer (to the question how exactly the angular momentum is measured) is, I am pretty much confident that it fits to the general measurement scheme explained in Sec. 3.3 of my paper.

The required dynamics cannot be obtained from the dynamics of the universe by coarse-graining

Are you saying that standard QM cannot explain the measurement of angular momentum? Note that the whole Sec. 3 is not about Bohmian mechanics, but about standard QM.

 

[Demystifier]

Maybe there's a POVM to desribe the functioning of the human eye

The POVM scheme is so general that it would be a miracle if such a POVM did not exist.

 

[A. Neumaier]

"Color" is not a physical observable but a physiological one.

I only claimed that it is a perceptible in the sense of the paper. The term is used only there, nowhere else.

 

[Demystifier]

This seems appropriate only if the selected basis is invariant under the dynamics of the system alone (before the interaction begins).

Actually, I don't see what exactly is your problem. The angular momentum is conserved, i.e. the basis consisting of angular-momentum eigenstates in invariant under dynamics. Do I miss something?

 

[vanhees71]

I also don't see what's problematic with angular-momentum measurements, and as usual, to be able to analyze it in detail one must look at the specific experimental setup with which you measure the angular momentum. One example for measuring total angular momenta of atoms is the Stern-Gerlach experiment of course or is it again the presumed problem that in reality you measure the magnetic moment and need to know the gyrofactors to "recalibrate" it to the angular-momentum "scale" (in multiples of $\hbar/2$)?

 

[A. Neumaier]

Are you saying that standard QM cannot explain the measurement of angular momentum? Note that the whole Sec. 3 is not about Bohmian mechanics, but about standard QM.

No, I only claimed that the measurement of angular momentum cannot be described in the interaction picture by a Hamiltonian of the form (1) considered in the decoherence paper by Kiefer and Joos that you had cited.

Thus I conclude that your argument for derive Born's rule in Bohmian mechanics does not apply for angular momentum measurements.

Note that the Kiefer and Joos paper didn't claim to give a general measurement theory but only an idealized model for measuring the position of a dust grain in which one gets the decoherence property (6) without having to do any significant analysis. True decoherence is significantly more complicated.

If you tell me in more detail how the angular momentum is measured in practice, I will be able to tell you in more detail how Bohmian mechanics explains this.

My query in post #1 was about deriving the Born rule for arbitrary observables. Thus angular momentum should be a special case. If there is no derivation for arbitrary observables then the question is unresolved in general, and the derivation for each particular observable is its own research project.

I also don't see what's problematic with angular-momentum measurements, and as usual, to be able to analyze it in detail one must look at the specific experimental setup with which you measure the angular momentum. One example for measuring total angular momenta of atoms is the Stern-Gerlach experiment

It is possibly problematic only in the context of the derivation from Bohmian mechanics.

Actually, I don't see what exactly is your problem. The angular momentum is conserved, i.e. the basis consisting of angular-momentum eigenstates in invariant under dynamics. Do I miss something?

In the Stern-Gerlach experiment, there is a magnetic field, breaking the rotational symmetry needed for the invariance.

 

[vanhees71]

I don't know, whether one can derive Born's rule at all, no matter if within BM or any other interpretation of QT. I consider it one of the necessary independent postulates.

What do you and @Demystifier mean by "invariance of the angular-momentum eigenstates". Of course they are invariant. There's a unique set of orthonormal eigenvectors of $\hat{\vec{J}}^2$ and $\hat{J}_z$. Whether or not there's a symmetry is determined by the Hamiltonian. Of course having a magnetic moment in a magnetic field, you have an interaction term of the type $\hat{H}=-g q \mu_{\text{B}} \vec{B}(\hat{\vec{x}}) \cdot \hat{\vec{s}}$ (nonrelativistic QM). Of course now rotational symmetry is broken, and it better be if you want to measure angular momentum with help of this interaction. In the SG this "symmetry breaking" is precisely what you want, leading to an entanglement between the spin component in direction of the $\vec{B}$-field and position.

 

[A. Neumaier]

I don't know, whether one can derive Born's rule at all, no matter if within BM or any other interpretation of QT.

BM assumes only the unitary dynamics of the wave function guiding the particles, and claims to reproduce from this standard quantum mechanics, which includes the Born rule for arbitrary observables. Thus BM must derive the Born rule for arbitrary observables.

I had queried in post #1 where this claim is substantiated, and got as answer an argument specifically made for the position measurement of a dust grain but claimed (sofar without any justification) to be valid arbitrarily.

 

[Demystifier]

Thus I conclude that your argument for derive Born's rule in Bohmian mechanics does not apply for angular momentum measurements.

You are missing the point. See Sec. 3.3.

 

[vanhees71]

Good point, but BM has the entire quantum formlism at hand (it's, despite the preachers of the gospel of BM claim otherwise, simply standard QM with the Bohmian trajectories as an addition on top). So maybe one can derive Born's rule for observables somehow by mapping measurement procedures for these observables to position measurements. As we've discussed earlier, it's hard to conceive a measurement which cannot be somehow mapped into a position measurement.

Indeed as usual the SGE is a paradigmatic example: The measured observable (spin-$z$ component) is entangled with a position observable ($z$ component of the position), and the "perceptible" is indeed given by the photographs of silver atoms on a glass plate made visible by Stern and Gerlach using a photo-developer treatment of the glass plates (nowadays in the standard lab experiment at universities one uses some electronic detector, a Langmuir-Taylor detector, but that doesn't matter for this general discussion, it's just registering the presence of an atoms at the place where the detector is located). In this way indeed the measurement of the spin component is translated into a 1-to-1 equivalent position measurement. The 1-to-1 nature of this "translation" is due to entanglement.

Of course, this is only the most simple paradigmatic case, and I'm not sure whether this can be made general for all observables, but it's at least quite plausible.

 

[Demystifier]

and got as answer an argument specifically made for the position measurement of a dust grain but claimed (sofar without any justification) to be valid arbitrarily.

If it's not valid arbitrarily, then it's not merely a problem for Bohmian mechanics. It is a problem for the quantum theory of measurement in general, as physics currently understands it. It would make much more sense if you would rephrase your question accordingly.

 

[Demystifier]

As we've discussed earlier, it's hard to conceive a measurement which cannot be somehow mapped into a position measurement.

I think one of the @A. Neumaier 's problems is precisely the opposite, to conceive how most measurements can be mapped into position measurements. It seems that he thinks that SG is an exception, rather than a rule. That's probably the reason why, in his thermal interpretation, he introduces a separate ontological quantity for each observable.

 

[charters]

At least some (most in my experience) Bohmian interpreters take the view that the only physically possible measurements are position measurements. See https://arxiv.org/abs/1805.07120

 

[Elias1960]

The classical reference to the original question of proof for the Born rule for general operators is

Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of "hidden" variables II, Phys.Rev. 85(2), 180-193

For the question about colors counting as positions, I would refer to

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

where for bosonic fields a field ontology is proposed. So, we do not have to look for photon positions, they become as irrelevant as phonon positions, but, instead, what really exists in Bohmian field theory are the EM fields themselves. So, the EM fields E and H are defined by the configuration.

 

[A. Neumaier]

BM has the entire quantum formalism at hand

...except for Born's rule, which therefore needs to be derived.

If it's not valid arbitrarily, then it's not merely a problem for Bohmian mechanics. It is a problem for the quantum theory of measurement in general, as physics currently understands it.

No, because in the traditional interpretations, Born's rule is assumed to hold for all measurements. Thus there is no problem. (The other, well-known general problem of unique outcomes if one insists on unitarity alone is not a problem in the Copenhagen or statistical interpretation.)

At least some (most in my experience) Bohmian interpreters take the view that the only physically possible measurements are position measurements. See https://arxiv.org/abs/1805.07120

This would be consistent with the fact that what seems to be a spin measurement in the BM account of the Stern-Gerlach experiment has nothing to do with the particle spin but is in fact only a measurement of position:

In the analysis of

• T. Norsen, The pilot-wave perspective on spin. American Journal of Physics, 82 (2014), 337-348.

Figure 2 suggests that rather than measuring spin it measures starting in the upper part of the SG arrangement, independent of spin!

 

[A. Neumaier]

The classical reference to the original question of proof for the Born rule for general operators is

Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of "hidden" variables II, Phys.Rev. 85(2), 180-193

Please point to the page with the proof for the Born rule for general operators; I didn't see it there.

For the question about colors counting as positions, I would refer to

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

where for bosonic fields a field ontology is proposed. So, we do not have to look for photon positions, they become as irrelevant as phonon positions, but, instead, what really exists in Bohmian field theory are the EM fields themselves. So, the EM fields E and H are defined by the configuration.

The ontology is only for a free bosonic field. Covariant interactions as needed for the interaction with an EM field are not covered.

 

[A. Neumaier]

You are missing the point. See Sec. 3.3.

Section 3.3 of Kiefer and Joos is about QED, for which no Bohmian version exists.

Section 3.3 of your paper is based on (17), generalizing (11) on no clear grounds. Since (11) cannot be trusted in general (being based on the assumption (3) for position measurements of a dust grain in Kiefer and Joos), why should I trust (17)? Your claim

Physically, this means that the master formula (17) [...] is valid for any measurement with clearly distinguishable outcomes.

is without any supporting proof, and (18) is surely not an angular momentum!

 

[Elias1960]

Please point to the page with the proof for the Born rule for general operators; I didn't see it there.

Part II sec. 2 Quantum theory of measurements p.180

Let us now consider an observation designed to measure an arbitrary (hermitian) "observable" Q, associated with an electron. ...

Of course, this is a physical paper, thus, with physical requirements for proofs instead of mathematical.

The ontology is only for a free bosonic field. Covariant interactions as needed for the interaction with an EM field are not covered.

Do you think they are somehow problematic? Interaction terms are products of the local field values. That's clearly a problem leading to infinities if you want to have them for arbitrary distances, but not if you accept that you have anyway only an effective field theory. From a Bohmian field-theoretic point of view, no momentum terms are involved, so they are all part of V(q), which is not restricted. So, the straightforward field theory terms will be fine. BFT also does not need any fundamental gauge or relativistic covariance, all that one can hope for is a theory with a preferred frame, where the preferred frame remains unobservable because of relativistic symmetry of the observables.

 

[Demystifier]

No, because in the traditional interpretations, Born's rule is assumed to hold for all measurements. Thus there is no problem.

You should distinguish standard QM from standard theory of quantum measurements. The standard QM does not contain the theory of quantum measurements, standard QM treats the measuring apparatus as classical and does not attempt to write its wave function. The standard theory of quantum measurement, on the other hand, treats the measuring apparatus as quantum and "my" Eq. (3) is a standard formula in that theory. Bohmian mechanics uses this standard theory of quantum measurements, but this theory exists even without Bohmian mechanics. It is used also in decoherence theory, many-worlds interpretation, Schrodinger-cat and Wigner-friend thought experiments, etc.

 

[vanhees71]

I think one of the @A. Neumaier 's problems is precisely the opposite, to conceive how most measurements can be mapped into position measurements. It seems that he thinks that SG is an exception, rather than a rule. That's probably the reason why, in his thermal interpretation, he introduces a separate ontological quantity for each observable.

An observable is defined by a measurement procedure, and as I said, it's hard to find an example, where finally a measurement procedure doesn't somehow at the end lead to a position measurement.

Take a particle detector like a cloud chamber. There a particle ionizes the vapor molecules around it, which leads to condensation of droplets, leaving a track, which can be photographed or filmed, i.e., to localize the particle it's due to some interaction with the vapor molecules and subsequent amplification of the track by the condensation of the droplets, whose position (sic!) then can be meausured by classical means. Applying a magnetic field through the measurement of the cloud-chamber track's curvature (position measurement!) leads to measurement of the particle's momentum.

I'd rather like to see an example, where a measurement is done without reference to a position measurement at the very end ;-)).

 

[Demystifier]

Section 3.3 of your paper is based on (17), generalizing (11) on no clear grounds. Since (11) cannot be trusted in general (being based on the assumption (3) for position measurements of a dust grain in Kiefer and Joos), why should I trust (17)?

Suppose that (17) is not true. Then what else the right-hand side of (17) could be?

and (18) is surely not an angular momentum!

It is the macroscopic observable that describes perceptible outcomes associated with a measurement of angular momentum.

 

[Demystifier]

Thus if the system is a particle and angular momentum is to be measured, this assumption does not work!

Suppose that the measurement of angular momentum is a projective measurement. If Eq. (3) in my paper can be wrong, then what else the right-hand side of Eq. (3) could be?

 

[Demystifier]

@A. Neumaier by now it's quite clear that your true problem is not to understand Bohmian mechanics per se, but to understand quantum theory of measurements. Hence I would suggest you to redirect your curiosity in this more general direction. For instance, you can start with https://arxiv.org/abs/1406.5535 with emphasis on Sec. 6.1. The author is a world renowned quantum experimentalist who also has a good understanding of the theory.

 

[vanhees71]

Great reference! I'd change the title to

"Quantum Measurements: a modern view for quantum optics mathematicians"!

 

[DarMM]

@A. Neumaier by now it's quite clear that your true problem is not to understand Bohmian mechanics per se, but to understand quantum theory of measurements. Hence I would suggest you to redirect your curiosity in this more general direction. For instance, you can start with https://arxiv.org/abs/1406.5535 with emphasis on Sec. 6.1. The author is a world renowned quantum experimentalist who also has a good understanding of the theory.

That is a great paper. I like the example where collapse occurs when nothing happens. Very useful for demonstrating how one can see it as a (generalized) Bayesian updating.

 

[kith]

I like the paper, too. His statement that tracing over the environment corresponds to a measurement on the environment, forgetting the results and using the average over all possible outcomes (= the reduced density matrix) as the new system state makes the difference between proper and improper mixtures and the relationship between the partial trace and the Born rule much clearer than most texts on decoherence which I've read so far.

 

[A. Neumaier]

Interaction terms are products of the local field values. That's clearly a problem leading to infinities if you want to have them for arbitrary distances, but not if you accept that you have anyway only an effective field theory. From a Bohmian field-theoretic point of view, no momentum terms are involved, so they are all part of V(q), which is not restricted. So, the straightforward field theory terms will be fine. BFT also does not need any fundamental gauge or relativistic covariance, all that one can hope for is a theory with a preferred frame, where the preferred frame remains unobservable because of relativistic symmetry of the observables.

In an effective theory with cutoff there is no relativistic covariance, hence no relativistic symmetry, hence preferred frames are in principle observable.