Errors in Ballentine (QM Textbook)?

[PeterDonis]

suppose you measure the particle's position to be $x = x_0$.

You can't. The mathematical operator that would do that is not physically realizable.

By the postulate, the state after the measurement must (supposedly) be the position eigenstate corresponding to eigenvalue $x=x_0$.

Which is impossible; the mathematical state in question is not physically realizable.

Possibly there is an argument that can be made that applies to operators that are actually physically realizable (perhaps such an argument is in the paywalled paper), but I don't agree that it's just "obvious", since the "obvious" argument you have stated is open to the obvious objections that I have just given.

the system's state post-measurement is not, in general, the eigenstate of the observable that was just measured -- as explained by Ballentine in much more detail in his sect 9.2 et seq.

The argument given there (which I agree with) appears to me to be addressing a different issue: the issue that, since the process of measurement (and indeed more generally any interaction, as Ballentine points out) entangles the measured system with the measuring device, after the measurement neither of those subsystems by itself has any definite state at all: only the joint system containing both measured system and measuring device does. So obviously the measured system can't be in an eigenstate of the measurement operator, since it isn't in any definite state at all.

Ballentine's solution to that problem is to adopt an ensemble interpretation. But that, by itself, doesn't address the "watched pot" issue, because the computation of the joint probability of successive measurement results being the same on a given state, given in section 12.2, is not interpretation-dependent; it's a straightforward application of the math of QM. So it should be valid under the ensemble interpretation. And the handwaving claim that the simple conclusion reached based on this simple computation must be false because "continuous observation does not prevent motion" doesn't help, because that claim is not a claim about ensembles, it's a claim about individual objects. But Ballentine's whole point is supposed to be that QM doesn't tell us about individual objects, it tells us about ensembles.

So no, I don't agree that his claim is just "obvious". Again, possibly these gaps are filled in in the paper that is unfortunately paywalled. But it seems clear to me that there are gaps.

 

[strangerep]

So obviously the measured system can't be in an eigenstate of the measurement operator, since it isn't in any definite state at all.

... which is the crucial take-home message.

The rest of what you said just shows that the usual naive mathematical machinery motivating Zeno's paradox is not valid for continuous processes when examined more carefully. More sophisticated setups involving POVMs and measurements-with-uncertainties must be carefully analyzed.

 

[atyy]

Ballentine does not have to use the claim that a pure state can be associated with an individual quantum system. However, his error is that he claims that this leads to a form of state reduction that is not supported by experiment. In other words, his criticism of the interpretation is wrong.

To add to this point, it is not only textbooks of quantum mechanics that use the sort of interpretation that Ballentine criticizes. Standard texts of quantum statistical physics also use the interpretation.

Reif Section 2.1: "Specifically, the system can be described by a wave function"
Kardar (not his book, but the lecture notes on which the book is based: "The (micro-) state of a quantum system is completely specified by a unit vector |Ψ〉, which belongs to an infinite dimensional Hilbert space."

So overall, I dislike Ballentine for criticizing textbook physics. If one reads it, one may get the opinion that Ballentine is the first person to have properly understood quantum mechanics, and learn false rejections of the orthodox interpretation, which is a statistical interpretation. Messiah explicitly identifies "Copenhagen" with the "Statistical Interpretation". What Ballentine has failed to grasp is that in the orthodox interpretation, the quantum state is not necessarily real, and just a way of calculating the probabilities of measurement outcomes. So if we can notionally assign a pure state to a single system, that is simply a way of calculating that yields correct predictions.

 

[PeterDonis]

The rest of what you said just shows that the usual naive mathematical machinery motivating Zeno's paradox is not valid for continuous processes when examined more carefully. More sophisticated setups involving POVMs and measurements-with-uncertainties must be carefully analyzed.

Does a more careful analysis show that there are some cases where "watched pot" experiments are predicted to work? Because such experiments have been done, and do work.

In other words, the issue is more complicated than just "well, this naive analysis makes it seem like a watched pot experiment should work, but that's obviously false because continuous observation does not prevent motion". Yes, it's true that continuous observation doesn't prevent motion, but it's also true that, at least under some conditions, you can make a watched quantum pot never boil. So there must be some cases where the "naive" analysis does give the right answer, which means that we need to understand what makes those cases different from a case like simple particle motion.

 

[atyy]

You can't. The mathematical operator that would do that is not physically realizable.

Minor point, but in some sense an exact position measurement is possible. For the measurement, we can use the Born rule. If we need to apply state reduction, the state of the system after the measurement is not a position eigenstate, but a normalizable state.
https://arxiv.org/abs/0706.3526 (Section 2.3.2)

 

[atyy]

Not my main complaint, but Ballentine's comment on renormalization and the quote from Dirac about infinities (section 19.4) is also out of date. Here a modern text would point the reader to notions of effective field theory.

Ballentine's argument is about zero point energy (section 19.4) is also suspect.
https://arxiv.org/abs/hep-th/0503158

But as I said, these are minor (you can find similar comments quite widely in older literature). My main complaint is that Ballentine omits the state reduction postulate. Certainly there are more general forms of state reduction than projection, but they all involve a change in state due to measurement that is different from unitary evolution by a Schroedinger equation, so would not escape his erroneous main objections to textbook physics in Chapter 9.

 

[Demystifier]

Rule 7 in the Insights article is not a "collapse" postulate. It's the projection postulate.

What's the difference? (Except that the word "projection" sounds more technical and hence lacks a mystic aura.)

 

[Demystifier]

Ballentine's argument is about zero point energy (section 19.4) is also suspect.
https://arxiv.org/abs/hep-th/0503158

That can be said for 90% texts (in both books and research papers) on Casimir effect.

 

[PeterDonis]

What's the difference?

"Collapse", as @vanhees71 was using the term in the post I was responding to in what you quoted from me, is an interpretation-dependent concept. "Projection" is just the basic mathematical operation described in Rule 7.

 

[strangerep]

Does a more careful analysis show that there are some cases where "watched pot" experiments are predicted to work? Because such experiments have been done, and do work.

I vaguely recall reading such claims, but it's a long time ago now, and I don't remember the references. You'll have to remind me...

I also vaguely recall rebuttals along the lines that the apparatus was explicitly contrived to regenerate the measured state, but I don't remember those references either. Maybe I can find time later next week.

[Also, check your PMs in the next few minutes...]

 

[atyy]

I vaguely recall reading such claims, but it's a long time ago now, and I don't remember the references. You'll have to remind me...

I also vaguely recall rebuttals along the lines that the apparatus was explicitly contrived to regenerate the measured state, but I don't remember those references either. Maybe I can find time later next week.

One doesn't have to use state reduction to produce a change in dynamics that causes a quantum Zeno effect, but it doesn't mean that the analysis involving state reduction is incorrect. One of the features of quantum mechanics is that what is considered a measurement outcome is observer dependent, so if there is no measured outcome, the process does not have to be modeled by a measurement. This is not unique to the quantum Zeno effect. It is also seen in the indirect measurement formalism, where the identity and timing of the outcome is dependent on the observer's assessment. It is also seen, for measurements restricted to a subsystem, in the consistency between the density matrix obtained with decoherence without any measurement, and that obtained when a measurement is performed and information about the outcomes is discarded.

https://arxiv.org/abs/quant-ph/9512012
Projection Postulate and Atomic Quantum Zeno Effect
Almut Beige, Gerhard C. Hegerfeldt

 

[martinbn]

@vanhees71 @Demystifier

Perhaps a new thread?

 

[Demystifier]

I thought we talk about Ballentine's book and I wanted to know, to which section @atyy is referring to. Rule 7 is indeed not a collapse postulate but is part of the minimal postulates you need to work with the formalism.

Where is the Rule 7 in the Ballentine's book? I cannot find it written down explicitly. (The Rule 7 is the same as the rule (ii) in my #49 above.)

 

[PeterDonis]

@vanhees71 @Demystifier

Perhaps a new thread?

Indeed. The new thread is here:

https://www.physicsforums.com/threads/difference-between-collapse-and-projection.998545/

Discussion of "collapse" vs. "projection" should take place in that thread.

 

[PeterDonis]

Where is the Rule 7 in the Ballentine's book? I cannot find it written down explicitly.

As discussed (and as noted in the 7 Rules Insights article), Ballentine does not include Rule 7 in his axioms. His equation 9.9 is more or less equivalent to Rule 7, but as the Insights article notes, Ballentine does not accept that equation as fundamental. He derives it as an effective rule in his equation 9.28.

 

[PeterDonis]

(The Rule 7 is the same as the rule (ii) in my #49 above.)

Which is now in the new thread, here:

https://www.physicsforums.com/threads/difference-between-collapse-and-projection.998545/post-6444527

 

[atyy]

As discussed (and as noted in the 7 Rules Insights article), Ballentine does not include Rule 7 in his axioms. His equation 9.9 is more or less equivalent to Rule 7, but as the Insights article notes, Ballentine does not accept that equation as fundamental. He derives it as an effective rule in his equation 9.28.

But the question is whether the derivation is correct.

It is not generally accepted that postulate 7 can of the Insights article can be derived from the first 6 postulates alone, so if Ballentine is doing that, it is not consensus physics.

Nielsen and Chuang state in their textbook published in 2000 & 2010 (p85) "The status of Postulate 3 as a fundamental postulate intrigues many people. Measuring devices are quantum mechanical systems, so the quantum system being measured and the measuring device together are part of a larger, isolated, quantum mechanical system. (It may be necessary to include quantum systems other than the system being measured and the measuring device to obtain a completely isolated system, but the point is that this can be done.) According to Postulate 2, the evolution of this larger isolated system can be described by a unitary evolution. Might it be possible to derive Postulate 3 as a consequence of this picture? Despite considerable investigation along these lines there is still disagreement between physicists about whether or not this is possible. We, however, are going to take the very pragmatic approach that in practice it is clear when to apply Postulate 2 and when to apply Postulate 3, and not worry about deriving one postulate from the other." [Here both the Born rule and state reduction are included in their postulate 3, so it is slightly different from the case where Ballentine states the Born rule but omits state reduction as a postulate.]

It is possible to derive postulate 7 as an effective rule with other assumptions, eg. hidden variables, if so, what additional assumptions has Ballentine used?

Other possibilities are that Ballentine is unaware that he has used the postulate here, since he rejects it in his criticism of Interpretation A and the watched pot experiment, and thus has made a double error of rejecting the postulate, and using it.

 

[PeterDonis]

It is not generally accepted that postulate 7 can of the Insights article can be derived from the first 6 postulates alone

The term "derived" is ambiguous. Ballentine does not claim to derive his version of the projection postulate (his equation 9.28) as a rigorous mathematical theorem valid in all cases. He only derives it as an "effective rule" (to use the term used in the Insights article) applying to certain particular cases. Doing that is not inconsistent with it being impossible to derive it as a rigorous mathematical theorem valid in all cases.

That said, from what I can see, Ballentine's derivation of his effective rule appears to assume the ensemble interpretation; if that is the case, then that would be an additional assumption.

 

[strangerep]

As discussed (and as noted in the 7 Rules Insights article), Ballentine does not include Rule 7 in his axioms. His equation 9.9 is more or less equivalent to Rule 7, but as the Insights article notes, Ballentine does not accept that equation as fundamental.

Yes, that matches my reading of Ballentine.

He derives it as an effective rule in his equation 9.28.

No, this does not match my reading of Ballentine. (Where does he use the phrase "effective rule" or equivalent? I don't see that.)

Also, (9.28) and hence (9.29) are specific to filtering-type measurements.

[...] from what I can see, Ballentine's derivation of his effective rule appears to assume the ensemble interpretation;

He derives the formula (9.28) in the context of filter-type measurements only, and if you're going to do filtering, of course the experimental preparation must supply an ensemble to the filter's input.

But the question is whether the derivation is correct.

If there is no "derivation" of such kind in the first place, but rather a derivation of a specific formula applicable to filter-type measurements only, then this is not the question, but rather a straw man.

Other possibilities are that Ballentine is unaware that he has used the postulate here, since he rejects it in his criticism of Interpretation A and the watched pot experiment, and thus has made a double error of rejecting the postulate, and using it.

Straw man again.

Indeed, another possibility is that you haven't studied carefully what Ballentine has actually written.

 

[atyy]

That said, from what I can see, Ballentine's derivation of his effective rule appears to assume the ensemble interpretation; if that is the case, then that would be an additional assumption.

How would the ensemble interpretation allow the derivation? This is the assignment of a state to a subensemble - but at this point, has Ballentine given any postulates that allow the assignment of a state to a subensemble?

One can assign subensembles as in Bohmian Mechanics with the addition of hidden variables. However, it doesn't seem that Ballentine has stated any clear statement of hidden variables. Ballentine does mention Einstein's Ensemble Interpretation, which does have hidden variables. However, one needs to define the variables and their dynamics for a derivation, as in Bohmian Mechanics.

Possibly another way to to define subensembles would be to define the quantum state(s) of subensembles before the measurement. However, we know that in general there is not a unique assignment of the state of subensembles, and a measurement is still needed to pick out the relevant subensemble. That would again require a postulate equivalent to the postulation of state reduction, which has been rejected by Ballentine.

 

[PeterDonis]

How would the ensemble interpretation allow the derivation?

See @strangerep's response in post #54.

 

[atyy]

See @strangerep's response in post #54.

Strangerep says "He derives the formula (9.28) in the context of filter-type measurements only, and if you're going to do filtering, of course the experimental preparation must supply an ensemble to the filter's input." It's hard to see why this is different from postulating state reduction, which has been rejected. Does the "of course" correspond to obvious but unstated steps in a derivation, or is the "of course" a postulate from physical intuition?

The problem here is to make sense of Eq 9.28 given that state reduction has been rejected, and only unitary evolution of the quantum state asserted. A correct derivation consistent with Ballentine must be consistent with his criticism of Interpretation A, otherwise it will also be subject to those (wrong) criticisms.

 

[atyy]

Agreed. I had raised the possibility earlier that something might have changed between editions of Ballentine, but that turned out not to be the case. So it looks like we'll need to make some corrections to the article.

@A. Neumaier will have to clarify that part, as I think it wasn't in the drafts I read, or I missed it. However, the possibility is the edition and page numbers are correct, and that A. Neurmaier read that as a derivation of effective state reduction, because that is what Ballentine intends 9.21 to be. At this point, Ballentine believes that Interpretation A has a state reduction, and he is trying to explain why Interpretation A seems to work most of the time.

However, this cannot be taken to be a correct derivation of collapse for 9.28, since Interpretation A in fact does not have a state reduction at that point in the experiment being discussed. Only Ballentine's wrong conception of Interpretation A has a state reduction.

 

[atyy]

It's hard to say, what Bohr really meant ;-)). I'm not sure whether or not he proposed a state reduction.

Interestingly, Weinberg's QM text (p82) says "As Bohr acknowledged, in the Copenhagen interpretation a measurement changes the state of a system in a way that cannot itself be described by quantum mechanics. [footnote 3] This can be seen from the interpretive rules of the theory. If we measure ... then the state will collapse ..." The footnote 3 he gives says "There are variants of the Copenhagen interpretation sharing this feature, some of them described by B. S. DeWitt, Physics Today, September 1970, p. 30."

 

[vanhees71]

The term "derived" is ambiguous. Ballentine does not claim to derive his version of the projection postulate (his equation 9.28) as a rigorous mathematical theorem valid in all cases. He only derives it as an "effective rule" (to use the term used in the Insights article) applying to certain particular cases. Doing that is not inconsistent with it being impossible to derive it as a rigorous mathematical theorem valid in all cases.

That said, from what I can see, Ballentine's derivation of his effective rule appears to assume the ensemble interpretation; if that is the case, then that would be an additional assumption.

He simple defines what is understood as a projective or von Neumann filter measurement. It's not a general rule or postulate but just a definition of a special type of experiment, which an be formulated in terms of the postulates of the minimal interpretation (as described in our Insights article). That such types of experiments are feasible in the real world is also evident from the many real-world experiments done with all kinds of systems in the labs where QT is investigated (e.g., quantum optics, AMO, HEP, condensed matter...).

 

[atyy]

He simple defines what is understood as a projective or von Neumann filter measurement. It's not a general rule or postulate but just a definition of a special type of experiment, which an be formulated in terms of the postulates of the minimal interpretation (as described in our Insights article). That such types of experiments are feasible in the real world is also evident from the many real-world experiments done with all kinds of systems in the labs where QT is investigated (e.g., quantum optics, AMO, HEP, condensed matter...).

So it's still a postulate. It's postulate 7 in https://www.physicsforums.com/insights/the-7-basic-rules-of-quantum-mechanics/.

 

[vanhees71]

I'm not happy with calling it a postulate for the said reason. It's the definition of a special (usually idealized) kind of experiments. It's rather a question of how to apply the theory to a specific kind of preparation-observation procedures in each individual case of such a kind of experiment.

 

[atyy]

I'm not happy with calling it a postulate for the said reason. It's the definition of a special (usually idealized) kind of experiments. It's rather a question of how to apply the theory to a specific kind of preparation-observation procedures in each individual case of such a kind of experiment.

Well, that's an easily fixed reason. We can just use the more general state reduction postulate. For discrete variables, the more general state reduction postulate can be derived by using the projection postulate on a measurement model. This complaint is different from Ballentine's criticism of orthodox quantum mechanics by asserting that there is only unitary evolution of the quantum state.

 

[vanhees71]

There is only unitary evolution of the quantum state when considering a closed system.

A filter measurement necessarily involves more than the measured system, namely the filter.

 

[atyy]

There is only unitary evolution of the quantum state when considering a closed system.

A filter measurement necessarily involves more than the measured system, namely the filter.

Then after that, one needs the state reduction postulate.

 

[vanhees71]

No, you need to take a partial trace and describe the evolution by some master equation. That can be FAPP a kind of "state reduction", but it's nothing outside the dynamical laws of QT!

 

[atyy]

No, you need to take a partial trace and describe the evolution by some master equation. That can be FAPP a kind of "state reduction", but it's nothing outside the dynamical laws of QT!

Well, we shall have to disagree. There is a reason state reduction is stated in many good textbooks.

 

[vanhees71]

Well, I fail to see its necessity and why one should have, in the case of local relativistic QFT (the Standard Model!), a self-contradiction between the successful formalism and an unneeded statement.

 

[Demystifier]

There is only unitary evolution of the quantum state when considering a closed system.

But unitary evolution is deterministic. Does it mean that the quantum state of a closed system evolves deterministically and that there is no randomness in the quantum state of a closed system? But something does change randomly in a closed system, right? So does it mean that, in a closed system, there is something which is not the quantum state?

 

[vanhees71]

There is no randomness in the evolution of the quantum state at all. Why should there be? The Schrödinger equation is a perfectly deterministic equation for the wave function.

Quantum theory is probabilistic in its notion of the meaning of the state (Born's rule). Observables don't necessarily take determined values but this depends on the state the system is prepared in. That's why measuring an observable on an ensemble of equally prepared systems, which is not determined to have a certain value, results in a random-number distribution whose statistics is described by the probabilities given by the state the system is prepared in (via Born's rule).

I don't understand your last sentence. A system is described by an observable algebra (realized usually by a set of self-adjoint operators) on an appropriate Hilbert space. The properties of the system is described by the statistical operator, representing its state. What else should there be?