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

[DarMM]

If you think so, your choice. The formulation "demonstrated the existence of" sounds dubious, not like "has constructed an example of". Whatever, if he gets the prize for solving the Millenium problem I will no longer use this claim.

"Demonstrate the existence of" is completely normal language. Do you just disagree with everything?
He has constructed the continuum limit. What hasn't been shown is that the infinite volume limit exists with a mass gap, which is required for the Millenium problem.

You were saying there is serious doubt over the existence of the continuum limit. There isn't, due to completely well defined 2D and 3D theories, as well as existence results for the 4D continuum limit. People working in constructive QFT don't have doubts over continuum QFT existing.

In fact, even if one can somehow define them, it will be not worth much, given that non-renormalizable gravity is anyway only an effective theory.

This is again a non-sequiter. We were talking about whether QED and other QFTs have continuum limits. You said there were serious doubts, there aren't. Now what, there's a problem with this because nobody has formulated Quantum Gravity or something?

 

[Elias1960]

He has constructed the continuum limit. What hasn't been shown is that the infinite volume limit exists with a mass gap, which is required for the Millenium problem.

I have tried to find the relevant paper and found this:
Balaban, T. (1987). Renormalization Group Approach to Lattice Gauge Field Theories I. Commun. Math. Phys. 109, 249-301
Balaban, T. (1988). Renormalization Group Approach to Lattice Gauge Field Theories II. Commun. Math. Phys. 116, 1-22
Are these the relevant papers?
(That would be funny if the best results about the very existence of continuous theories have been reached by the same lattice methods which Neumaier thinks cannot be applied to QED.)

You were saying there is serious doubt over the existence of the continuum limit. There isn't, due to completely well defined 2D and 3D theories, as well as existence results for the 4D continuum limit. People working in constructive QFT don't have doubts over continuum QFT existing.

Ok, I will take this into account and formulate my position in the future differently.

This is again a non-sequiter. We were talking about whether QED and other QFTs have continuum limits. You said there were serious doubts, there aren't. Now what, there's a problem with this because nobody has formulated Quantum Gravity or something?

There is no problem with this. I have based my statement about the serious problems on what I have heard about this question, from people less optimistic about this than you. I have not checked that myself given that for me it was an irrelevant side issue. If the situation is better, fine, I will remember this. But I don't have to change anything else, and the point of my remark about gravity was to explain why it is, in my opinion, only a quite irrelevant side issue.

 

[atyy]

I wouldn't say this. We have several examples of continuum theories in 2D and 3D which are well-defined. Balaban also demonstrated the existence of a continuum limit for Yang-Mills in 4D, so I don't think any serious doubt remains. It's the infinite volume limit that is more difficult.

Is it the case that the 4D limit has been established, but not the 3D limit? In describing Balaban's 3D work, http://www.claymath.org/sites/default/files/yangmills.pdf says that the contiuum limit has not been established: "This is an important step toward establishing the existence of the continuum limit on a compactified space-time. These results need to be extended to the study of expectations of gauge-invariant functions of the fields."

That article also seems to indicate that the 4D finite volume problem is open, and it is not just the infinite volume problem that remains: "These steps toward understanding quantum Yang–Mills theory lead to a vision of extending the present methods to establish a complete construction of theYang–Mills quantum field theory on a compact, four-dimensional space-time. One presumably needs to revisit known results at a deep level, simplify the methods,and extend them."

 

[DarMM]

Balaban has established that there is a continuum limit of the action, i.e. there is a well defined theory in the continuum. He never established that expectation values of gauge invariant operators are unique, nor did he prove certain analyticity properties for them.

These are usually necessary to solve what is called the finite volume case in constructive field theory, but they're not really the issues the average physicist means when they say the continuum limit. They usually mean there being something well defined and nontrivial in the continuum limit.

Thus Balaban has shown the continuum limit exists, but not demonstrated it has certain uniqueness and analytic properties for Wilson loops.

 

[atyy]

Thus Balaban has shown the continuum limit exists, but not demonstrated it has certain uniqueness and analytic properties for Wilson loops.

Have you heard the story about why he gave up working on Yang Mills? He moved house, and the movers lost the box with his notes on Yang Mills.

 

[A. Neumaier]

Have you heard the story about why he gave up working on Yang Mills? He moved house, and the movers lost the box with his notes on Yang Mills.

See here and
https://www.physicsforums.com/posts/5443402/
https://www.physicsforums.com/posts/5443430/

 

[DarMM]

Have you heard the story about why he gave up working on Yang Mills? He moved house, and the movers lost the box with his notes on Yang Mills.

Yes from yourself years ago! Makes one want to cry!

 

[A. Neumaier]

$$\rho(\vec{x},\vec{y}) =|\Psi(\vec{x},\vec{y})|^2
\simeq \sum_k|c_k|^2 |\Psi_k(\vec{x},\vec{y})|^2 ~~~~~(1)$$
In the second equality we have assumed that $A_{kq}(\vec{x})$ are macro distinct for different $k$, which we must assume if we want to have a system that can be interpreted as a measurement of $K$.

In (1) [equation label added by me] you assume without justification that the $\Psi_k(\vec{x},\vec{y})$ with different $k$ have approximately disjoint support. This is unwarranted without a convincing analysis.

 

[Demystifier]

In (1) [equation label added by me] you assume without justification that the $\Psi_k(\vec{x},\vec{y})$ with different $k$ have approximately disjoint support. This is unwarranted without a convincing analysis.

So you want to see an explicit calculation based on the theory of decoherence, right? If this is what would satisfy you, I will try to find one in the literature.

 

[A. Neumaier]

So you want to see an explicit calculation based on the theory of decoherence, right? If this is what would satisfy you, I will try to find one in the literature.

Whatever you need to justify it without assuming nondemolition. Wigner's analysis indicates to me that this is impossible.

 

[Demystifier]

Wigner's analysis indicates to me that this is impossible.

I don't understand that claim, can you explain how Wigner indicates that it is impossible?

 

[A. Neumaier]

I don't understand that claim, can you explain how Wigner indicates that it is impossible?

I had discussed this in post #42.

 

[Demystifier]

I had discussed this in post #42.

It's basically the objection that non-demolition is not a reasonable assumption. But I don't see how is that related to the assumption that detector wave functions are approximately separated in the position space.

 

[A. Neumaier]

It's basically the objection that non-demolition is not a reasonable assumption. But I don't see how is that related to the assumption that detector wave functions are approximately separated in the position space.

No. Wigner's statement (quoted at the end of my post #42) essentially says that nondemolition is a necessary condition to get the wanted decomposition. Of course you assume only that the decomposition is approximate, so the argument by Wigner is not watertight in your case.

But your argument is completely absent - you just write some formulas and then jump without further justification to the desired conclusion [namely to (1) in post #98]!

 

[Demystifier]

But your argument is completely absent - you just write some formulas and then jump without further justification to the desired conclusion [namely to (1) in post #98]!

I totally disagree, but if you think so I don't know what argument to offer without repeating myself.

 

[A. Neumaier]

In (1) [equation label added by me] you assume without justification that the $\Psi_k(\vec{x},\vec{y})$ with different $k$ have approximately disjoint support. This is unwarranted without a convincing analysis.

I totally disagree, but if you think so I don't know what argument to offer without repeating myself.

You could explain in more detail why (1) follows from your argument for the second equality.
The only way this can be concluded seems to me by assume that the $\Psi_k(\vec{x},\vec{y})$ with different $k$ have approximately disjoint support. But you didn't give an argument for the latter, you just said that we must assume it:

Instead of my Eq. (3), more generally we have
$$|k\rangle|A_0\rangle \rightarrow \sum_q a_q |q\rangle |A_{kq}\rangle~~~~ (3')$$
[...]

In the multi-position representation we have
$$\Psi(\vec{x},\vec{y})=\sum_k c_k\Psi_k(\vec{x},\vec{y})$$
where
$$\Psi_k(\vec{x},\vec{y})=\sum_q a_q \psi_q(\vec{y}) A_{kq}(\vec{x})$$
Using the Born rule in the multi-position space we have
$$\rho(\vec{x},\vec{y}) =|\Psi(\vec{x},\vec{y})|^2
\simeq \sum_k|c_k|^2 |\Psi_k(\vec{x},\vec{y})|^2$$
In the second equality we have assumed that $A_{kq}(\vec{x})$ are macro distinct for different $k$, which we must assume if we want to have a system that can be interpreted as a measurement of $k$.

You actually assume that $A_{kq}(\vec{x})$ and $A_{k'q'}(\vec{x})$ have essentially disjoint support whenever $k\ne k'$. How does this follow from the definition of the $A_{kq}$ in (3') above?

 

[Demystifier]

But you didn't give an argument for the latter, you just said that we must assume it:

I said that we must assume that if we want to have a system that can be interpreted as a measurement of $k$. Sure, there are many interactions for which this condition is not satisfied. But such interactions, whatever they may be useful for, are not useful for measurement of $k$. Hence I don't consider such interactions that are not useful for measurement of $k$. I only consider those that are useful.

You actually assume that $A_{kq}(\vec{x})$ and $A_{k'q'}(\vec{x})$ have essentially disjoint support whenever $k\ne k'$. How does this follow from the definition of the $A_{kq}$ in (3') above?

It doesn't follow from that. Instead, it follows from the assumption that an interaction that can be used for measurement of $k$ exists. It is an existence assumption. And it seems to me (correct me if I'm wrong) that you think that such an interaction doesn't even exist.

 

[A. Neumaier]

I said that we must assume that if we want to have a system that can be interpreted as a measurement of $K$. Sure, there are many interactions for which this condition is not satisfied. But such interactions, whatever they may be useful for, are not useful for measurement of $K$. Hence I don't consider such interactions that are not useful for measurement of $K$. I only consider those that are useful.It doesn't follow from that. Instead, it follows from the assumption that an interaction that can be used for measurement of $K$ exists. It is an existence assumption. And it seems to me (correct me if I'm wrong) that you think that such an interaction doesn't even exist.

This amounts to saying that, by definition, a process deserving the label measurement of a selfadjoint operator $K$ with eigenstates $|k\rangle$ is one where
$$|k\rangle|A_0\rangle \rightarrow \Psi_k $$
and the $\Psi_k(\vec{x},\vec{y})$ have essentially disjoint support. With this definition, of course everything is trivial.

This is what I mean that you essentially assume what there is to prove. With this definition, the nontrivial statement would be to show that there is a class of physically meaningful not [missing not added later] nondemolition processes where this property actually holds. What I think Wigner's argument amounts to is that this is impossible.

 

[Demystifier]

With this definition, the nontrivial statement would be to show that there is a class of physically meaningful nondemolition processes where this property actually holds. What I think Wigner's argument amounts to is that this is impossible.

I still don't understand why do you talk about non-demolition. My equation that you called (3') in #106 is not non-demolition. Consequently, my last equation in #106 is also not non-demolition. It would be non-demolition if $a_q$ whre nonzero only for $q=k$, but I do not assume that.

 

[vanhees71]

I think the trouble again is that you don't discuss some specific (idealized toy model of a) measurement of momentum (I guess that's what you mean by $k$). One way to measure the momentum of a charged particle is to use a cloud chamber and a magnetic field and then measuring the curvature of the "trajectory" (i.e., the track of the particle as indicated by the droplets formed). Then all you need is a slightly modified calculation as in the famous Mott paper adding the magnetic field to the Hamiltonian.

 

[A. Neumaier]

I still don't understand why do you talk about non-demolition. My equation that you called (3') in #106 is not non-demolition. Consequently, my last equation in #106 is also not non-demolition. It would be non-demolition if $a_q$ whre nonzero only for $q=k$, but I do not assume that.

Yes, but your demonstration of (1) consists in saying that in order to qualify for a measurement the right hand side of (3') must satisfy the postulate I stated in post #108 (which I abstracted from your treatise by dropping irrelevant calculations). Since this now amounts to a definition of what a measurement process is, the question is which real experiments fall under this definition.

Nondemolition measurements are well-known to satisfy the condition of the definition in post #108. But nondemolition experiments are only a tiny part of the collection of measurements for which the Born rule is claimed. Hence to have a credible derivation of Born's rule you need to show that at least one other physically interesting class of (therefore not nondemolition) measurements is also eligible in this definition. Wigner gives arguments that suggest to me that the latter seems impossible to do. If that were the case your definition would rule out all usual (not nondemolition) experiments from being measurements. Thus there is something nontrivial left to be shown.

 

[A. Neumaier]

I think the trouble again is that you don't discuss some specific (idealized toy model of a) measurement of momentum (I guess that's what you mean by $k$). One way to measure the momentum of a charged particle is to use a cloud chamber and a magnetic field and then measuring the curvature of the "trajectory" (i.e., the track of the particle as indicated by the droplets formed). Then all you need is a slightly modified calculation as in the famous Mott paper adding the magnetic field to the Hamiltonian.

In the paper, $k$ is a label for an eigenstate of the arbitrary Hermitian operator $K$.

A cloud chamber is a nondemolition experiment since the particle continues (after having ionized some atoms leading to the droplets) with essentially the same momentum. For the class of nondemolition measurements the derivation given is ok. But the discussion is about whether Born's rule can also be derived in Bohmian mechanics for measurements not falling into this special class.

 

[Demystifier]

Nondemolition measurements are well-known to satisfy the condition of the definition in post #108.

No, your definition of non-demolition measurements is wrong. Where did you get this definition from?

A nondemolition measurement is a transition of the form
$$|k\rangle |A_0\rangle \rightarrow |k\rangle |A_k\rangle \;\;\; (1)$$
But a transition
$$|k\rangle |A_0\rangle \rightarrow |\Psi_k\rangle \;\;\; (2)$$
is, in general, not a nondemolition meaurement. Instead it is the most general transition possible, where the label $k$ on the right-hand side only means that the final state depends on the initial measured state $|k\rangle$. It is impossible to have a unitary transition that does not have the form (2).

Besides, there is a fine difference between nondemolition measurements and projective measurements. For projective measurements the right-hand side of (1) has the form $|k\rangle |A_k\rangle$ immediately after the measurement at time $t_m$, while for nondemolition measurements it has this form during a long time after the measurement. For projective measurements the right-hand side is really $|\psi_k(t)\rangle |A_k(t)\rangle$ with $|\psi_k(t_m)\rangle=|k\rangle$. Nondemolition measurements are rare, but projective measurements are not. But this distinction is not so important here.

 

[A. Neumaier]

A nondemolition measurement is a transition of the form
$$|k\rangle |A_0\rangle \rightarrow |k\rangle |A_k\rangle \;\;\; (1)$$
But a transition
$$|k\rangle |A_0\rangle \rightarrow |\Psi_k\rangle \;\;\; (2)$$
is, in general, not a nondemolition meaurement. Instead it is the most general transition possible,

I didn't claim anything else than what you just wrote. But the question is whether a physically meaningful class of non-nondemolition measurement in this agreed sense actually satisfies the criterion in post #108 (which I abstracted from your treatise by dropping irrelevant calculations). You just assume that any reasonable measurement satisfies this criterion, while my reading of Wigner suggests that only the very restricted class of nondemolition measurements can satisfy it.

 

[Demystifier]

while my reading of Wigner suggests that only the very restricted class of nondemolition measurements can satisfy it.

Well, my problem is that I still don't have a clue why do you read Wigner that way. It certainly isn't what Wigner explicitly said, is it?

 

[Demystifier]

Anyway, my reading of Wigner is different. He objects that a realistic measurement takes a finite time, so it's not clear to what value of the measured observable the result of measurement refers, unless the measured observable is a conserved quantity. That indeed is a valid objection, but can be easily resolved. Let the duration of the process of measurement be $\tau_m$ and let $\tau_i$ be the characteristic intrinsic time during which the measured observable significantly changes. Then if the process of measurement is sufficiently fast so that $\tau_m \ll \tau_i$, the objection by Wigner is resolved for all practical purposes. What was not known in the Wigner's time is that duration time of measurement $\tau_m$ is indeed typically very short, due to the fast decoherence caused by a large number of the apparatus degrees of freedom. @A. Neumaier I hope it makes sense to you.

 

[A. Neumaier]

Well, my problem is that I still don't have a clue why do you read Wigner that way. It certainly isn't what Wigner explicitly said, is it?

Independent of what Wigner says, the only class of processes where it is known that the postulate I stated in post #108 (which I abstracted from your treatise by dropping irrelevant calculations) is satisfied is the class of nondemolition measurements. You have not shown that there are other experiments that fit this definition but simply assumed that all meaningful measurement settings belong to this class.

On the other hand, Wigner said (p.298 in Wheeler/Zurek) that ''only quantities which commute with all additive conserved quantities are precisely measurable''. Here ''precisely measurable'' means that they satisfy the Born rule to in principle arbitrary accuracy when the detector is constructed appropriately (depending on the requested precision). This precludes the derivation of the Born rule to arbitrary accuracy for quantities that do not commute with all additive conserved quantities. But these are precisely the non-nondemolition measurements.

 

[Demystifier]

This precludes the derivation of the Born rule to arbitrary accuracy for quantities that do not commute with all additive conserved quantities.

Maybe you are right about that, but I don't think that we need a derivation of it to arbitrary accuracy. All we need is a derivation to an accuracy that matches the accuracy in actual experiments. Note that quantum theory is tested with a great accuracy only for some conserved quantities (most notably $g-2$ in QED), while for other quantities it is good but not so great. A typical match between ideal theoretical Born rule and actually measured frequencies looks something like this:

 

[A. Neumaier]

Maybe you are right about that, but I don't think that we need a derivation of it to arbitrary accuracy. All we need is a derivation to an accuracy that matches the accuracy in actual experiments.

Yes. But still it must be shown that this is the case for a nontrivial class of measurements of nonconserved observables. Simply claiming that it must be the case by definition is not enough.

 

[Demystifier]

Yes. But still it must be shown that this is the case for a nontrivial class of measurements of nonconserved observables. Simply claiming that it must be the case by definition is not enough.

Fair enough, but I think in #116 I gave some additional heuristic arguments. If I find a reference where a more serious quantitative analysis is done, I will let you know.