Errors in Ballentine (QM Textbook)?

[vanhees71]

Since your post took this question far beyond just a question about Ballentine specifically, and well over the line into interpretation, I have moved it to the other thread in the interpretations forum where collapse is being discussed:

https://www.physicsforums.com/threads/difference-between-collapse-and-projection.998545/post-6445336

Fine with me, but if this splits in zillions of subthreads it's hard to follow. I think the claim that Ballentine's book is "wrong" is just the claim that the ensemble interpretation is "wrong". So why not keeping the postings in one thread such that the context of the arguments is clear.

But I think, I made my argument now several times, and don't need to repeat it further anyway.

 

[PeterDonis]

I think the claim that Ballentine's book is "wrong" is just the claim that the ensemble interpretation is "wrong".

And that discussion belongs in the thread in the interpretations subforum, which is where I have moved all posts along those lines.

why not keeping the postings in one thread such that the context of the arguments is clear.

Because interpretation discussions are always matters of opinion. If Ballentine prefers the ensemble interpretation, that's his business. No one can say he's "wrong" for doing that; doing it is not an "error" in his textbook. But if there are errors in Ballentine about minimal QM itself, apart from any interpretation, that is what this thread, which is in the regular QM forum, is about.

 

[atyy]

There are ways to derive the state reduction rule without hidden variables. They involve using a simultaneous measurement (for which the Born rule applies) to define a sequential measurement (which requires state reduction), and requiring consistency between the simultaneous and sequential calculations.

Heuristically, one can see this in Bell tests, where if the measurement is simultaneous in one frame, it is sequential in another, then it can be seen that state reduction is required for consistency.

Another example is found in using an instrument to define state reduction, where a simultaneous measurement on the apparatus and the system is considered to be equivalent to sequential measurements on the system, eg. section 6.2.3 on Conditional output states in
https://arxiv.org/abs/0810.3536
Guide to Mathematical Concepts of Quantum Theory
Teiko Heinosaari, Mario Ziman

However, these are not compatible with Ballentine's assertion that unitary evolution alone should disallow state reduction and they are not compatible with Ballentine's criticism of Messiah's statement that the measurement unpredictably disturbs the system. In the above notes by Heinosaari and Ziman, section 6.3.1 says No information without disturbance. If one is not continually enlarging the Hilbert space with each measurement, then the state reduction postulate does give correct quantum mechanics.

 

[A. Neumaier]

@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. Neumaier 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.

See my comments here.

 

[PeterDonis]

Moderator's note: A discussion about the ensemble interpretation and the PBR theorem has been moved to a new thread in the interpretations subforum:

https://www.physicsforums.com/threa...ion-inconsistent-with-the-pbr-theorem.998624/

 

[atyy]

See my comments here.

Thanks. I think the revised comments have essentially the same meaning as the original comments.

With respect to the subject of the OP of this thread, I think Ballentine's derivation in that section is problematic, as it
(i) is in the context of wrongly assuming that the standard interpretation has a state reduction, where the standard interpretation has none.
(ii) on p244, Ballentine says about his derivation that "This “reduction” of the state is not a new fundamental process, and, contrary to the impression given in some of the older literature, it has nothing specifically to do with measurement."

Well, perhaps it is not a new fundamental process (we don't care about that in the orthodox interpretation, as the state is just a way of calculating probabilities of measurement outcomes), but given that Nielsen and Chuang still state reduction as a postulate, explicitly acknowledging that its derivation is controversial, it still remains correct to state it as a postulate. And even if one derives state reduction in the orthodox interpretation by defining it via consistency of simultaneous and sequential measurements (reference in post #93), there the state reduction is specifically related to measurement, and specifically with the measurement outcome.

 

[strangerep]

Um, his entire textbook? [...]

Oh, I see now what you meant. I misunderstood you before.

(It's probably time for me to exit this thread, take a Bex, and have a good lie down.)

 

[A. Neumaier]

Thanks. I think the revised comments have essentially the same meaning as the original comments.

With respect to the subject of the OP of this thread, I think Ballentine's derivation in that section is problematic, as it
(i) is in the context of wrongly assuming that the standard interpretation has a state reduction, where the standard interpretation has none.
(ii) on p244, Ballentine says about his derivation that "This “reduction” of the state is not a new fundamental process, and, contrary to the impression given in some of the older literature, it has nothing specifically to do with measurement."

Well, perhaps it is not a new fundamental process (we don't care about that in the orthodox interpretation, as the state is just a way of calculating probabilities of measurement outcomes), but given that Nielsen and Chuang still state reduction as a postulate, explicitly acknowledging that its derivation is controversial, it still remains correct to state it as a postulate. And even if one derives state reduction in the orthodox interpretation by defining it via consistency of simultaneous and sequential measurements (reference in post #93), there the state reduction is specifically related to measurement, and specifically with the measurement outcome.

I agree.

 

[vanhees71]

Heuristically, one can see this in Bell tests, where if the measurement is simultaneous in one frame, it is sequential in another, then it can be seen that state reduction is required for consistency.

This is self-contradictory: According to local relativistic QFT (in this case particularly QED) describes all findings correctly. This implies that there can be no causal effect between measurement events that are spacelike separated (that's a mathematical statement!). So there can be no state reduction through the measurement at one place affecting causally the outcome of the (in some frame) later measurement at the other place.

Since nature is frame-independent if there's no state reduction in one frame, there cannot be one in any other.

 

[atyy]

This is self-contradictory: According to local relativistic QFT (in this case particularly QED) describes all findings correctly. This implies that there can be no causal effect between measurement events that are spacelike separated (that's a mathematical statement!). So there can be no state reduction through the measurement at one place affecting causally the outcome of the (in some frame) later measurement at the other place.

Since nature is frame-independent if there's no state reduction in one frame, there cannot be one in any other.

Quantum mechanics is not about cause and effect. It is only about predicting the probabilities of measurement outcomes.

 

[Demystifier]

Quantum mechanics is not about cause and effect. It is only about predicting the probabilities of measurement outcomes.

No, standard (orthodox) quantum mechanics is about proving locality, whatever it takes. Sometimes it takes cause and effect, sometimes it takes denying cause and effect, sometimes it takes objective reality, sometimes it takes denying objective reality.

 

[martinbn]

No, standard (orthodox) quantum mechanics is about proving locality, whatever it takes. Sometimes it takes cause and effect, sometimes it takes denying cause and effect, sometimes it takes objective reality, sometimes it takes denying objective reality.

Ha! The thief is screaming "thief"!

 

[Demystifier]

Ha! The thief is screaming "thief"!

Please elaborate!

 

[martinbn]

Please elaborate!

Well, it seems that all fans of BohminanMech are only interested in proving non-locality.

 

[Demystifier]

Well, it seems that all fans of BohminanMech are only interested in proving non-locality.

Maybe, but to do it we don't need to contradict ourselves, which orthodox guys do. Sure, we must assume the existence of something not directly seen in experiments, and we are not very happy with that, but we believe it's a much smaller sin than self-contradiction.

 

[vanhees71]

No, standard (orthodox) quantum mechanics is about proving locality, whatever it takes. Sometimes it takes cause and effect, sometimes it takes denying cause and effect, sometimes it takes objective reality, sometimes it takes denying objective reality.

This confusion is by leaving the minimal physical meaning into the realm of vague philosophical speculation

 

[A. Neumaier]

Quantum mechanics is not about cause and effect. It is only about predicting the probabilities of measurement outcomes.

But even measurements respect causality. You cannot measure an effect before it was caused.

 

[vanhees71]

And of course, as any physical dynamical theory also QT must be causal. If nature weren't describable by causal laws, there'd be no natural sciences to begin with!

 

[atyy]

Yes, but assuming a measurement causes a collapse, i.e., a change of the state, implies a causal influence of the measurement on the state, and that's the problem particularly in this context.

No it does not.

It's contradicting the very assumptions you make about the dynamics of the system (microcausality condition), which by construction cannot violate causality, i.e., space-like separated events cannot be causally connected.

No it does not. Microcausality means that one cannot information faster than light. Collapse does not allow information to be sent faster than light, so it is consistent with microcausality.

 

[atyy]

But even measurements respect causality. You cannot measure an effect before it was caused.

We can use terminology in which we accept some notion of causality, but reject another form. So to use the terminology of Wiseman and Cavalcanti, we accept relativistic causality and agent causation, but reject Reichenbach's principle.
https://arxiv.org/abs/1503.06413 (Fig. 5, Operationalist Version)

When vanhees71 is talking about collapse as a cause, he is using it in the sense of (for example), collapse explaining the Bell correlations in the sense of Reichenbach's principle.

 

[vanhees71]

All I'm saying is that there is no general rule, i.e., no general postulate, to say which state a quantum system is in when doing a measurement. One has to analyze the specific experiment for this.

What's also clear is that a naive collapse assumption in the context of far-distant experiments on an entangled system (e.g., two entangled photons measured at far distant positions) with measurement events that are space-like separated, doesn't make sense and contradicts the fundamental property of microcausality which excludes the possibility that the observed correlations of the outcomes at the far distant places are due to a causal connection between these far-distant measurements.

My solution is trivial: We have prepared the entangled state, and thus the observed correlation is already there from the very beginning. Though the single-photon polarizations are totally indetermined in such an entangled state, they are strongly correlated, no matter how far away the photon measurements, but it's a correlation due to the state preparation in the beginning and not mutually caused by the two measurments made. There's thus no need for a collapse in analyzing this experiment.

 

[PeterDonis]

According to local relativistic QFT (in this case particularly QED) describes all findings correctly. This implies that there can be no causal effect between measurement events that are spacelike separated (that's a mathematical statement!).

No, it doesn't; "no causal effect" is not correct because there is no rigorous definition of "causal effect" in QFT. What QFT does say rigorously is that measurements at spacelike separated events must commute--the results must not depend on the order in which the measurements are made.

You can, of course, define "no causal effect" to mean "the measurements commute"; but then you are just inviting argument about your definition of "causal effect".

 

[PeterDonis]

a naive collapse assumption in the context of far-distant experiments on an entangled system (e.g., two entangled photons measured at far distant positions) with measurement events that are space-like separated, doesn't make sense and contradicts the fundamental property of microcausality

What, mathematically, is "the fundamental property of microcausality"? Does it just refer to the fact that spacelike separated measurements must commute? Or something else?

 

[Demystifier]

We have prepared the entangled state, and thus the observed correlation is already there from the very beginning.

This statement doesn't make sense because it doesn't say - correlation of what? In experiments we observe correlations of measurement outcomes, but certainly measurement outcomes do not exist from the very beginning. So if there is something which is correlated from the beginning, and if that something can be described by math,
then various Bell-like theorems (many of which do not assume determinism, contrary to what you repeat over and over again) show that measured correlations cannot be explained by correlations of something from the very beginning. Of course, you refuse such theorems because they introduce a mathematical symbol for that something (e.g. $\lambda$) which is not a part of the standard QM formalism. You want to use just standard QM and nothing else. Hence you are confident with saying "correlation", but not confident with saying "correlation of what". But as long as you refuse to say what is correlated from the beginning, for many of us your statement quoted above does not make sense.

 

[A. Neumaier]

What, mathematically, is "the fundamental property of microcausality"? Does it just refer to the fact that spacelike separated measurements must commute? Or something else?

Measurement is not a notion of QFT. Microcausality says by definition that field operators commute or anticommute at spacelike pairs of arguments. This implies (and is indeed equivalent to) the statement that arbitrary observables with spacelike separated support commute.

 

[bhobba]

I have read the thread, and since Ballentine is my 'Bible' on QM I believe I can make some comments.

Ballentine has a few errors, but so do many textbooks. The errors are the type you can spot with a bit of thought and in doing so actually helps in your understanding. For example he makes the mistake of in Copenhagen thinking the only version is one in which the state is real and an instantaneous collapse violates relativity. Of course that is not the case - the state can simply be something that helps in calculating things. The huge advantage of the book IMHO is he only states two actual axioms (the second - he calls the Born Rule) being at least partially derivable from the first via Gleason (which strangely he doesn't mention). There are of course more rules than just 2 but he introduces them in such a way they seem natural. That way we see the fundamental assumption of QM is Axiom 1 about observables and eigenvalues. The Born Rule is automatically true if we make a few reasonable assumptions, the main one being non-contextuality. The other is Chapter 3 which is unique in all other QM books I have read in deriving Schrodinger's Equation rather than postulating it. For me that was simply eye opening. You learn a lot from thinking about the derivation. The actual content is you notice that if you use Ehrenfest's Theorem you get the classical Hamiltonian equation so you understand why the momentum and energy operators are defined the way they are. It lies at the heart of QM how a classical system is quantized. This derivation is the reason we do it by replacing classical variables with operators - usually the momentum, position and energy operators. But we see there is an ambiguity in doing it because the operators may not commute so what order are they in? However as Ballentine comments it does not seem to cause problems in practice.

Yes it is advanced - I would study Modern Quantum Mechanics by Sakurai first, and Susskind before that (I just love Susskind's books). But if you do study it, and think about it, you will have a very good understanding of QM.

Thanks
Bill

 

[bhobba]

Measurement is not a notion of QFT.

I agree. But since QM is a limiting case of QM how does it emerge? Or is it an actual limiting case?

Thanks
Bill

 

[PeterDonis]

Measurement is not a notion of QFT. Microcausality says by definition that field operators commute or anticommute at spacelike pairs of arguments. This implies (and is indeed equivalent to) the statement that arbitrary observables with spacelike separated support commute.

Yes, agreed, this is a better way of saying what I was trying to say.

 

[atyy]

Here is a paper that includes discussion of state reduction in the context of AQFT.
https://arxiv.org/abs/1810.06512
https://link.springer.com/article/10.1007/s00220-020-03800-6
Quantum fields and local measurements
Christopher J. Fewster, Rainer Verch

"(The term ‘post-selected’ is used in various different ways in the literature – the precise meaning we have in mind, which amounts to updating the state based on the measurement outcome, will be spelled out in detail.)"

State reduction is given in Eq. 3.20.

 

[vanhees71]

No, it doesn't; "no causal effect" is not correct because there is no rigorous definition of "causal effect" in QFT. What QFT does say rigorously is that measurements at spacelike separated events must commute--the results must not depend on the order in which the measurements are made.

You can, of course, define "no causal effect" to mean "the measurements commute"; but then you are just inviting argument about your definition of "causal effect".

But that's indeed what's usually understood to be "no causal effect" and that's why you impose the microcausality condition which then leads to unitarity and Poincare covariance of the S-matrix, the cluster-decomposition principle.

It's, maybe, only a sufficient but not necessary condition for a relativistic QFT to have all these desired features, but I'm not aware of any example that's not in this sense a "local/microcausal relativistic QFT".

See Weinberg, QT of fields vol. 1 for a comprehensive treatment of these issues for fields of arbitrary spin.