since you are its author:3) My "Bohmian mechanics for instrumentalists".
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.Hrvoje Nikolic said:3.1 All perceptibles can be reduced to macroscopic positions
They are physicaly equivalent, but may differ in some fine mathematical details.Are the three derivations mathematically equivalent?
It says that it's a microscopic observable, so it's understood that it's defined on the Hilbert space of the object measured.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?
(3) is defined on a larger Hilbert space, that is on the space of the measured system + apparatus.Also, in (3) you assume a tensor product structure compatible with this assumption.
Unitarity is of course assumed, look at (9).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!
This is linearity, not unitarity. Unitarity seems to be lost in the assumption (3).Unitarity is of course assumed, look at (9).
Please justify this step from unitary dynamics.the interaction between the measured system and the apparatus induces a unitary transition of the form
$$|k\rangle|A_0\rangle → |k′\rangle|A_k\rangle.~~~~~~~~~~~~~~~~~~~ (3)$$
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).This is linearity, not unitarity. Unitarity seems to be lost in the assumption (3).
Please justify this step!
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!
"Color" is not a physical observable but a physiological one. Maybe there's a POVM to desribe the functioning of the human eye ;-)).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.
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.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!
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.The required dynamics cannot be obtained from the dynamics of the universe by coarse-graining
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?This seems appropriate only if the selected basis is invariant under the dynamics of the system alone (before the interaction begins).
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.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.
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.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.
It is possibly problematic only in the context of the derivation from Bohmian mechanics.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
In the Stern-Gerlach experiment, there is a magnetic field, breaking the rotational symmetry needed for the invariance.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?
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 don't know, whether one can derive Born's rule at all, no matter if within BM or any other interpretation of QT.
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.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.
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.As we've discussed earlier, it's hard to conceive a measurement which cannot be somehow mapped into a position measurement.
....except for Born's rule, which therefore needs to be derived.BM has the entire quantum formalism at hand
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.)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.
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:
Please point to the page with the proof for the Born rule for general operators; I didn't see it there.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
The ontology is only for a free bosonic field. Covariant interactions as needed for the interaction with an EM field are not covered.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.
Section 3.3 of Kiefer and Joos is about QED, for which no Bohmian version exists.You are missing the point. See Sec. 3.3.
is without any supporting proof, and (18) is surely not an angular momentum!Hrvoje Nikolic said:Physically, this means that the master formula (17) [...] is valid for any measurement with clearly distinguishable outcomes.