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

#### A. Neumaier

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?

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#### Demystifier

2018 Award
You can find it in many places, including:
1) P. Holland, The Quantum Theory of Motion, Chapter 8.
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.

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#### 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:
Hrvoje Nikolic said:
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

2018 Award
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

2018 Award
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

Gold Member
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

2018 Award
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.

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#### Demystifier

2018 Award
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

2018 Award
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

Gold Member
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.

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#### vanhees71

Gold Member
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.

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#### Demystifier

2018 Award
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

Gold Member
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

2018 Award
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

2018 Award
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
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
Hrvoje Nikolic said:
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!

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