A Evaluate this paper on the derivation of the Born rule

[A. Neumaier]

This I don't understand. How are expectation values, including the standard deviation (there should be square root to meet the usual definition

I corrected the formula for $\sigma_A$. I gave clear and complete mathematical definitions of all notions used (except for Hermitian quantity, or observable).
The formula is enough to define what it means in a logical sense, just as $[a,b]:=\{x\in R \mid a\le x\le b\}$ completely defined the meaning of an interval.

Note that I use the brackets simply as an abbreviation for the trace, not presuming any other meaning than the formula through which it is defined. This is the common practice in definitions that you find in all mathematically oriented texts. And I am nowhere using the statistical connotations ''expectation value'' or ''standard deviation'' but ''uncertain value'' and ''uncertainty''. These two notions are axiomatically defined by the definitions I give, and they get their informal physical meaning through the informal words used in my formulation of the uncertainty principle and the measurement rule.

This way of proceeding, using an established term to denote something different and more general is standard practice even with physicists, who talk about state vectors, not having in mind the little arrows that once defined the concept of a vector but instead thinking about a wave function behind the same term. For this it is sufficient that the same mathematical rules hold for manipulating true vectors and state vectors.

In the same way, the words ''expectation values'' are appropriate whenever a mathematical formalism (such as that of quantum mechanics) uses formulas borrowed from statistics and then generalized (in the present case from random variables to linear operators), as long as the formal rules are the same. As in the analogy between vectors described by arrows and state vectors, there is no reason to take the name ''expectation value'' any more literal than the word ''vector''. And indeed, in my formulation, i completely avoid it. (The authors of the papers discussed in the present thread do the same but rename the expectation values to q-expectation values, hoping in this way to break the connection. This is described in detail in their paper discussed in post #85 of this thread.

 

[A. Neumaier]

If I need to read an entire book for that,

You only need to read a few sections for that. The arguments are quite elementary.

 

[vanhees71]

But we discuss a theory about physics. How is this "expectation value" related to what's measured in the lab, if not in the usual way a la Born?

If I precisely measure the spin-$z$ component of an atom with spin 1/2 as in the Stern-Gerlach experiment (which I described very often in terms of standard QT, I think even within this thread), I always get either $\hbar/2$ or $-\hbar/2$. For an unpolarized beam, i.e., for the operator $\hat{\rho}=\hat{1}/2$ the "expectation value" $\langle \sigma_z \rangle=\mathrm{Tr} \hat{\rho} \hat{\sigma}_z$ is obviously $0$. This value I never find when I precisely measure $\sigma_z$.

So, how has this most simple experiment to be formulated within your interpretation?

 

[Prathyush]

But we discuss a theory about physics. How is this "expectation value" related to what's measured in the lab, if not in the usual way a la Born?

If I precisely measure the spin-$z$ component of an atom with spin 1/2 as in the Stern-Gerlach experiment (which I described very often in terms of standard QT, I think even within this thread), I always get either $\hbar/2$ or $-\hbar/2$. For an unpolarized beam, i.e., for the operator $\hat{\rho}=\hat{1}/2$ the "expectation value" $\langle \sigma_z \rangle=\mathrm{Tr} \hat{\rho} \hat{\sigma}_z$ is obviously $0$. This value I never find when I precisely measure $\sigma_z$.

So, how has this most simple experiment to be formulated within your interpretation?

In the experiment you suggested, we separate the particles that have different spins. On interaction with the experiment apparatus(suitably constructed not to destroy the particles ex. Using light photons as probes) the possiblity for the interference between the spins disappears. We understand that it happens due to the interaction with the apparatus. We want to understand why it disappears using a simple apparatus.

Now that I think about it the stage for the description of an amplification apparatus can be moved to the light photon. So long as we don't cause an interaction between the photon and the spin particle. The spin particle can be considered measured. And correlated with the state of the photon.

The important point is that the final stage of amplification can be separated from the interaction with the probe particle.

 

[Prathyush]

Or one needs to show that the probability for both detectors clicking is zero. It can be recast into a question about probes.

 

[PeterDonis]

the expectation value is

I don't think anyone is disputing that this is the mathematical formula for an expectation value. The question is whether the ordinary language term "Born's rule" has anything to do with this mathematical formula. You say yes, @A. Neumaier and I say no. It doesn't seem like we're making any progress on deciding that question.

 

[PeterDonis]

Of course, expectation values need not be eigenvalues of the corresponding operator. How do you come to that idea?

I didn't say they had to be. You don't appear to understand the point I'm making. I'll try once more. Here is a statement of Born's rule, from your own post:

The possible outcome of measurements of $A$ are the eigenvalues of the operator
$\hat{A}$. Let $|a,\beta \rangle$ denote a complete set of orthonormalized eigenvectors of eigenvalue $a$, then the probability to measure the value $a$, if the system is prepared in the state described by $\hat{\rho}$ is...

Where does it say anything there about expectation values? Nowhere. It only talks about eigenvalues and probabilities, neither of which are expectation values. So expectation values have nothing to do with Born's rule. That is the point I'm making. Is it clear?

 

[vanhees71]

Come on, this formula implies, how expectation values have to be evaluated (as long as you allow me to use the usual definitions of usual probability theory a la Kolmogorov). I've shown all this in my previous postings (as well as the fact that you can define probabilities as expectation values of particular observables, which is used by any Monte-Carlo simulation).

 

[PeterDonis]

this formula implies, how expectation values have to be evaluated...

Let me rephrase this: Born's rule implies how expectation values have to be evaluated, given some other assumptions. Fine. That's not the same as saying Born's rule is expectation values. It's just an implication given some assumptions.

You appear to think the assumptions are obvious, but @A. Neumaier has repeatedly given examples of single measurements (e.g., a single measurement of the mass of an iron brick, or a single measurement of the total energy of a macroscopic object) that do not even appear to match the basic statement of Born's rule, let alone any implications from it. If an expectation value is obtained as the result of a single measurement, that does not appear to be consistent with the statement of Born's rule that you yourself gave, nor with the additional assumptions you state in deriving how expectation values have to be evaluated. So it seems evident that Born's rule and those additional assumptions do not apply to all measurements.

 

[A. Neumaier]

If I precisely measure the spin-$z$ component of an atom with spin 1/2 as in the Stern-Gerlach experiment (which I described very often in terms of standard QT, I think even within this thread), I always get either $\hbar/2$ or $-\hbar/2$. For an unpolarized beam, i.e., for the operator $\hat{\rho}=\hat{1}/2$ the "expectation value" $\langle \sigma_z \rangle=\mathrm{Tr} \hat{\rho} \hat{\sigma}_z$ is obviously $0$. This value I never find when I precisely measure $\sigma_z$.

Neither do you find precisely the value $\pm \hbar/2$ claimed to be measured by Born's rule. For in spite of many thousands of measurements of Stern-Gerlach type, Planck's constant $\hbar$ is still known only to an accuracy of 9 decimal digits.

Thus Born's rule is a fiction even in this standard textbook example!

 

[RockyMarciano]

A. Neumaier has repeatedly given examples of single measurements (e.g., a single measurement of the mass of an iron brick, or a single measurement of the total energy of a macroscopic object) .

Unfortunately calling these single measurements outcomes "expectation values" is a source of confusion when it has little to do with what is usually understood by "expectation value" in QM that implies repetition of measurements.

I think the limitations of the Born rule pointed out by Neumaier are fair but taking the uncertainty to the classical realm solves nothing, it just confirms what most knew, that the problem lies in the leaving measurements out of the formalism.

 

[PeterDonis]

Unfortunately calling these single measurements outcomes "expectation values" is a source of confusion

Yes, the term "expectation value" is ambiguous, since it can refer either to the result of applying a mathematical formula, or to a particular physical interpretation of that result. In the cases @A. Neumaier describes, the former applies (since we can always compute a mathematical formula), but the physical interpretation is different from the usual one. As I think was mentioned in one of his posts, at least one paper adopts the term "q-expectation value" to deal with this issue.

 

[A. Neumaier]

that the problem lies in the leaving measurements out of the formalism.

No, the main problem lies in having measurement (which is a poorly defined notion) in the formalism, in the form of Born's rule.

 

[vanhees71]

Let me rephrase this: Born's rule implies how expectation values have to be evaluated, given some other assumptions. Fine. That's not the same as saying Born's rule is expectation values. It's just an implication given some assumptions.

You appear to think the assumptions are obvious, but @A. Neumaier has repeatedly given examples of single measurements (e.g., a single measurement of the mass of an iron brick, or a single measurement of the total energy of a macroscopic object) that do not even appear to match the basic statement of Born's rule, let alone any implications from it. If an expectation value is obtained as the result of a single measurement, that does not appear to be consistent with the statement of Born's rule that you yourself gave, nor with the additional assumptions you state in deriving how expectation values have to be evaluated. So it seems evident that Born's rule and those additional assumptions do not apply to all measurements.

Ok, from now on I stick to the narrow sense of Born's rule, giving only the probabilities of precisely measuring observables, given the state of the system (in terms of a Statistical operator, so that I do not always have to distinguish between pure and mixed states).

Now, concerning measuring the mass of a iron brick, it's very clear that you measure a very coarse-grained observable. The measurement apparatus, a usual balance, does the averaging implied by the coarse graining for you, as any macroscopic body does leading to classical behavior of the coarse-grained macroscopic variables you measure in such cases.

If I'd be allowed to read Arnold's symbols in the usual way, that's also what he is saying when he associates the expectation values with what's measured in such cases, but I am not allowed to use the usual probabilistic interpretation and I don't understand the meaning of the symbols he is using. That's the problem, not that quantum theory would in any sense be invalid to describe macroscopic coarse-grained observables. That the whole point of statistical physics since Boltzmann: To understand the macroscopic observables from the underlying microscopic fundamental theory.

 

[vanhees71]

Neither do you find precisely the value $\pm \hbar/2$ claimed to be measured by Born's rule. For in spite of many thousands of measurements of Stern-Gerlach type, Planck's constant $\hbar$ is still known only to an accuracy of 9 decimal digits.

Thus Born's rule is a fiction even in this standard textbook example!

The uncertainty of $\hbar$ is not fundamental but a technical problem, which will be solved next year or so by fixing its value, using either a Watt balance or a silicon ball. Then $\hbar$ will be exact as is the value of $c$ already since 1983. All this has absolutely nothing to do with any interpretation issues about QT!

 

[A. Neumaier]

Then $\hbar$ will be exact as is the value of $c$ already since 1983.

So Born's rule was not valid in the past, and its validity depends on the choice of units?? This would be the only instance in physics where something depends in an essential way on units...

But there are problems with the experiment even when $\hbar$ is fixed: The measurement of angular momentum in a Stern-Gerlach experiment is a more complicated thing. One doesn't get an exact value $\pm\frac{\hbar}{2}$ even when $\hbar$ is fixed.

For in spite of what is claimed to be measured, what is really measured is something different -- namely the directed distance between the point where the beam meets the screen and the spot created by the particle on the screen (by suitable magnification). This is a macroscopic measurement of significant but limited accuracy since the spot needs to have a macroscopic extension to be measurable. From this raw measurement, a computation based on the known laws of physics and the not (or not yet) exactly known value of $\hbar$ is used to infer the value of the angular momentum a classical particle would have so that it produces the same spot. This results for the angular momentum in a value of approximately $\pm\frac{\hbar}{2}$ only, with a random sign; the accuracy obtainable is limited both by the limited accuracy of the distance measurement and (at present) the limited accuracy of the value of $\hbar$ used.

Thus for a realistic Stern-Gerlach measurement, Born's rule is only approximate, even when $\hbar$ is exactly known.

Only the idealized toy version for introductory courses on quantum mechanics satisfies Born's rule exactly since the two blobs at approximately the correct position and the assumed knowledge of exact 2-valuedness obtained from the quantum mechanical calculation count for demonstration purposes as exact enough. If the quantization result is not assumed and a true measurement of angular momentum is performed, one gets no exact numbers!

 

[vanhees71]

I think, it's non-sensical to discuss further. I'm out at this point, to prevent provoking more off-topic traffic.

 

[PeterDonis]

concerning measuring the mass of a iron brick, it's very clear that you measure a very coarse-grained observable.

Which observable, and what are its eigenvalues?

 

[vanhees71]

The observable is $m$. In non-relativistic physics the possible values are $\mathbb{R}_{>0}$; in relativsitic QFT $\mathbb{R}_{\geq 0}$.

 

[PeterDonis]

The observable is $m$.

What self-adjoint operator is this?

 

[vanhees71]

The mass operator. The answer depends on, whether you work in relativistic or non-relativistic QT.

In relativistic QT it's more easy. The mass of a quantum system is defined by $\hat{M}^2=\frac{1}{c^2} \hat{P}^{\mu} \hat{P}_{\mu}$, where $\hat{P}$ is the total four-momentum operator of the system.

In non-relativistic QT, it's a bit more complicated to define, what mass is. When investigating the unitary ray representations of the Galilei group's Lie algebra, it turns out that it has a non-trivial central charge, which turns out to be mass. For details, see Ballentine, Quantum Mechanics - A modern Development.

 

[PeterDonis]

where $\hat{P}$ is the total four-momentum operator of the system

Is this just the sum of the 4-momentum operators for each particle? (For each iron atom in the brick, for example?)

 

[vanhees71]

No, it's the total momentum of the entire system. For the iron brick it's a lot of atoms bound together to a solid body.

 

[PeterDonis]

it's the total momentum of the entire system

Ok, so how do we construct it out of operators we already know?

 

[A. Neumaier]

I think, it's non-sensical to discuss further. I'm out at this point, to prevent provoking more off-topic traffic.

I find it strange that you lengthily contribute to the discussion but when defeated, suddenly declare the problem to be off-topic.

The Born rule is in the title of the thread, which is about evaluating a paper that derives in a special (but representative) case the Born rule - based (as the authors say in another, closely related paper cited in post #85) among others on the alternative assumption (essentially that of my thermal interpretation) that when the uncertainty is small enough, $\langle A\rangle$ (rather than any condition based on eigenvalues) is essentially the value measured in each single case. Thus clarifying the relation between this rule and Born's rule, which appear conflicting, is a central part of the evaluation.

But because of your complaint I continue my discussion of Born's rule and the the Stern-Gerlach experiment here, in a thread exclusively devoted to the limits of Born's rule.

 

[vanhees71]

Come on, the uncertainties in the value of $\hbar$ in the SI is really off-topic in a thread about the foundations of QT, and you have not given a clear explanation for your very bold claim that Born's rule, one of the very foundations of QT is invalid. You simply rename the formalism to take expectation values by not telling it taking the average given a probability distribution but something else, you don't clarify.

 

[A. Neumaier]

Born's rule, one of the very foundations of QT

As the paper under discussion in this thread shows (in conjunction with the analysis cited in my post #85) , it is not a necessary foundation for QT, since the authors give alternative foundations where it is not needed, but special instances of it are derived.

 

[bhobba]

I don't think any new postulate is required, I think it would only require a careful analysis of what we mean by measurement.

Have you studied Gleason:
http://kiko.fysik.su.se/en/thesis/helena-master.pdf

Weinberg is indeed correct - an extra assumption is required and since Gleason it's well known what that extra assumption is - its non-contextuality. There are a couple of others such as the strong principle of superposition but that's the main one.

When going through the theorem its so beautiful and elegant you are inclined to forget its implicit assumption - the measure is basis independent because basis are usually something you simply choose to make a problem easier - not of fundamental importance - but in this case it is. That it not to diminish Gleason - its one of my favorite QM results and I have created some versions of my own it fascinates me so much. But the assumption is there and can't be ignored.

Be very very careful with mathematical proofs of physical things - you must always look at what's really going on physically. My background is math - not physics and I just love some of these mathematical derivations. But while physics is written in the language of math its not math.

As another example see the post I did about Feynman's proof of Maxwell's equations. Dyson put a challenge out there - since it only uses classical assumptions where did the relativity of Maxwell's equations come from. There were a few conjectures put forward, some I agreed with but I wanted my own and came up with it. But someone else posting in that thread saw the rock bottom reason - it assumes C=1 which is only true in all frames relativistically - relativity in - relativity out.

It's the same reason for the Kaluza-Klein miracle that fascinated me in my GR days until I realized what was going on. The foundation of EM is U(1) gauge invariance ie the symmetry of a circle - but that's what you do in Kaluza-Kelin - you assume the equations are not dependent on the 5th dimension which physically was enforced by the 5th dimension being curled up in a circle - so we have EM in (ie U(1) invarience) so its no surprise you get EM out.

Thanks
Bill

 

[vanhees71]

As the paper under discussion in this thread shows (in conjunction with the analysis cited in my post #85) , it is not a necessary foundation for QT, since the authors give alternative foundations where it is not needed, but special instances of it are derived.

Already in the Abstract of the paper in #1 you can read

Finally the field induced by S on M, which
may take two opposite values with probabilities given by Born’s rule, drives A into its up or down
ferromagnetic phase. The overall final state involves the expected correlations between the result
registered in M and the state of S. The measurement is thus accounted for by standard quantum
statistical mechanics and its specific features arise from the macroscopic size of the apparatus.

[emphasis mine]

 

[A. Neumaier]

Already in the Abstract of the paper in #1 you can read

Yes. They explicitly explain what they mean in the follow up paper, mentioned already in post #85:

A.E. Allahverdyan, R. Balian and T.M. Nieuwenhuizen,
A sub-ensemble theory of ideal quantum measurement processes,
Annals of Physics 376 (2017): 324-352.
https://arxiv.org/abs/1303.7257

They mean with standard quantum statistical mechanics the mathematical formalism, with a physical interpretation not based on Born's rule but on their alternative interpretive rules. Otherwise their derivation of Born's rule would be circular. The first of their interpretive rules effectively replaces Born's rule in their view of statistical mechanics. This rule is is precisely the thermal interpretation:

Interpretative principle 1. If the q-variance of a macroscopic observable is negligible in relative size its q-expectation value is identified with the value of the corresponding macroscopic physical variable, even for an individual system.