Evaluate this paper on the derivation of the Born rule

[A. Neumaier]

One way to argue is to use the Shannon-Jaynes maximum-entropy principle, which leads, when using the total energy as the one known variable,

Well, yes, and it does not involve Born's rule, in contrast to what you had always claimed. Moreover, you need to know the expectation of the total energy. How do you know this? By a single macroscopic measurement, not by identically preparing many cases. Thus the thermal interpretation is assumed to even make sense of the maximum entropy principle - not Born's rule!

And if you use $H^2$ as the one known variable you get from max entropy a ridiculous density operator that does not match experiment. Thus the max entropy principle depends on what you believe is measured macroscopically. The correct result only comes out if you believe that the ensemble expectation of $H$ is measured - i.e. if you believe the thermal interpretation.

 

[vanhees71]

The very formulation of the meaning of the statistical operator is based on Born's rule.

A argument for using $\langle H \rangle$ rather than $\langle H^2 \rangle$ (or any other non-linear function of $H$) might be that isolated systems should be uncorrelated, i.e., if I consider two non-interacting systems and look for equilibrium I should use additive conserved quantities in the entropy principle, e.g., energy. Then you have
$$\hat{H}=\hat{H}_1 + \hat{H}_2, \quad [\hat{H}_1,\hat{H}_2]=0,$$
and thus from the maximum-entropy principle
$$\hat{\rho}=\frac{1}{Z_1 Z_2} \exp(-\beta_1 \hat{H}_1) \exp(-\beta_2 \hat{H}_2).$$
If both systems are coupled to the same common "heat bath", of course, you have necessarily $\beta_1=\beta_2=1/T$, where $T$ is the temperature of the heat bath. This also leads to an additive entropy and additivity of all extensive thermodynamical quantities.

 

[A. Neumaier]

The very formulation of the meaning of the statistical operator is based on Born's rule.

Only in the vaguest sense, involving measurements never performed, and hence not subject to Born's rule.

But I see that your usage of the terms is so vague that it is impossible to discuss this with you. Effectively you are working in a shut up and calculate mode and invoke whatever interpretation appears to be needed to match predictions with experiment, but you use the catch word ''Born's rule'' (without actually using the rule) to justify what you do using hand-waving words - not logic, in terms of which nothing of this is justified.

With this hand-waving attitude there are no foundational problems at all, since they are all swept under the carpet of vagueness and imprecision in the usage of the language. On this level a fruitful discussion of foundations is impossible; we are just going in circles. The foundational problems appear only when each of the terms used gets a fixed meaning and arguments are based on that meaning only. Then the presence of problems that you don't see becomes obvious to anyone who cares.

 

[vanhees71]

It's you who doesn't define clearly what you mean with your "expectation values", if I'm not allowed to think in terms of probability theory, not me! The Born rule is very clear, and it has nothing to do with "shutup and calculate". It's one of the basic postulates (in my opinion indispensible) to relate the formalism of the theory to what's measured in the lab, and I use it in the usual textbook way to describe observations.

 

[A. Neumaier]

It's you who doesn't define clearly what you mean with your "expectation values",

I don't understand your criticism. A mathematical definition is the most precise definition one can give of anything. The interpretation depends on the application, and applied to macroscopic observable, it is very clearly defined that it means the actual value within its intrinsic uncertainty. This is enough to deduce the probabilistic interpretation in cases where it applies (sufficiently many independent replications of an otherwise very uncertain measurement).

The Born rule is very clear, and it has nothing to do with "shut up and calculate".

I agree. But the way you invoke the Born rule as being applied whenever the word probability or expectation appears is has nothing to do with the Born rule as given in the usual treatment, but is only camouflaged "shut up and calculate".

In particular, as discussed above, measuring energies in the lab is never done according to the description of a measurement according to Born's rule as given in the usual textbooks. But we are going again in circles...

 

[vanhees71]

I don't understand your criticism. A mathematical definition is the most precise definition one can give of anything. The interpretation depends on the application, and applied to macroscopic observable, it is very clearly defined that it means the actual value within its intrinsic uncertainty. This is enough to deduce the probabilistic interpretation in cases where it applies (sufficiently many independent replications of an otherwise very uncertain measurement).


I agree. But the way you invoke the Born rule as being applied whenever the word probability or expectation appears is has nothing to do with the Born rule as given in the usual treatment, but is only camouflaged "shut up and calculate".

In particular, as discussed above, measuring energies in the lab is never done according to the description of a measurement according to Born's rule as given in the usual textbooks. But we are going again in circles...

Maybe, I've not found it in your large book, but I haven't seen a clear definition of the meaning of expectation values, because you explicitly deny the usual probabilistic meaning. Now you say you want to derive it. Is it so difficult to give a clear definition of what your expectation value means, if not to be read in the usual probabilistic sense?

Ok, let's define one last time, what's understood as Born's rule. It's the probabilistic interpretation of the meaning of "quantum state" no more no less:

A quantum state is represented by a positive semi-definite self-adjoint operator $\hat{\rho}$ with $\mathrm{Tr} \hat{\rho}=1.$

An observable $A$ is represented by a self-adjoint operator $\hat{A}$. 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 probality to measure the value $a$, if the system is prepared in the state described by $\hat{\rho}$ is
$$P(a)=\sum_{\beta} \langle a,\beta|\hat{\rho}|a,\beta \rangle.$$

It's a simple corrollary of this postulate that expectation values are (basis-independently!) given by
$$\langle A \rangle = \mathrm{Tr}(\hat{\rho} \hat{A}).$$
Now, even if you deny the probabilistic meaning given that the expectation value (which for me implies a probablistic meaning, because where else than in probability theory does the notion of an "expectation value" make sense?), you can reconstruct the probabilities from that rule, because you can define the projection operator to the eigenspace of $\hat{A}$ for eigenvalue $a$,
$$\hat{P}(a)=\sum_{\beta} |a,\beta \rangle \langle a,\beta|,$$
as an observable, and then you have of course
$$P(a) = \langle \hat{P}(a) \rangle=\mathrm{Tr}[\hat{\rho} \hat{P}(a)].$$
It's also clear, how to generalize all this for spectral values of $\hat{A}$ in the continuum. Then the sums become integrals as usual. I'm aware that for a mathematically rigorous treatment it's not that easy, but we discuss the physics here rather than the mathematical rigorous foundation of (non-relativistic) QT.

So my question is, how do you in your interpretation make sense of $\hat{\rho}$ if not this usual probabilistic one? What, then, is the meaning of expectation value defined by the trace? Why should such a complication be necessary for an interpretation superior to the standard one?

Note that, according to the book by Peres, also the generalized "incomplete measurement protocos" in terms of POVMs are derivable from the above summarized standard Born rule. So the standard Born rule is at least sufficient to include these more general modern notions of measurements.

 

[PeterDonis]

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...

This I agree is Born's rule.

It's a simple corrollary of this postulate that expectation values are (basis-independently!) given by...

This is not. Probabilities are not expectation values, and Born's rule itself says nothing about expectation values.

Also, since you have already said that the possible outcome of measurements are eigenvalues, and the expectation value is not an eigenvalue (except in the special case that the state $\hat{\rho}$ happens to be an eigenstate of the operator $\hat{A}$), the expectation value clearly cannot be the outcome of a measurement, if we believe that Born's rule applies to all measurements--which you appear to be claiming. But @A. Neumaier has described measurements whose outcomes are not eigenvalues but expectation values. So it seems like Born's rule cannot apply to such measurements, which means statements about expectation values cannot be part of Born's rule.

 

[A. Neumaier]

So my question is, how do you in your interpretation make sense of ^ρρ^\hat{\rho} if not this usual probabilistic one? What, then, is the meaning of expectation value defined by the trace?

I had explained it multiple times:

Though traditionally called an ensemble expectation value, a more natural name - not suggesting a priori a probabilistic interpretation - for $\bar A=\langle A\rangle:=Tr \rho A$ would be the uncertain value of $A$. Quoting mostly from my web page, its physical meaning in general (without reference to probability or even measurement) is defined by the the following simple rule generalizing statistical intuition to situations where uncertainty is not required to be probabilistic:

Uncertainty principle: A Hermitian quantity $A$ whose uncertainty $\sigma_A:=\sqrt{\langle(A-\bar A)^2\rangle}$ is much less than $|\bar A|$ has the value $|\bar A|$ within an uncertainty of $\sigma_A$.

This is a very clear, practical principle. Physicists doing quantum mechanics (even those adhering to the shut-up-and-calculate mode of working) use this principle routinely and usually without further justification. The principle applies universally. No probabilistic interpretation is needed, so it applies also to single systems.

From this principle one can derive under appropriate conditions (see my online book) the following rule:

Measurement rule: Upon measuring a Hermitian operator $A$ in the state $\rho$, the measured result will be approximately $\bar A$, with an uncertainty at least of the order of $\sigma_A$. If the measurement can be sufficiently often repeated (on a system with the same or a sufficiently similar state $\rho$) then $\sigma_A$ will be a lower bound on the standard deviation of the measurement results.

Actually the above measurement rule should be considered as a definition of what it means to have a device measuring $A$. As such it creates the foundation of measurement theory. In order that a macroscopic quantum device qualifies for the description ''it measures $A$'' it must either be derivable from quantum mechanics, or checkable by experiment, that the property claimed in the above measurement rule is in fact valid. Thus there is no circularity in the foundations.

Moreover, Born's famous rule turns out to be derivable, too, (see my online book) but under special circumstances only, namely those where the Born rule is indeed valid in practice. (Though usually invoked as universally valid, Born's rule has severe limitations. It neither applies to position measurements nor to photodetection, nor to measurement of energies, just to mention the most conspicuous misfits.)

 

[vanhees71]

This I agree is Born's rule.
This is not. Probabilities are not expectation values, and Born's rule itself says nothing about expectation values.

Also, since you have already said that the possible outcome of measurements are eigenvalues, and the expectation value is not an eigenvalue (except in the special case that the state $\hat{rho}$ happens to be an eigenstate of the operator $\hat{A}$), the expectation value clearly cannot be the outcome of a measurement, if we believe that Born's rule applies to all measurements--which you appear to be claiming. But @A. Neumaier has described measurements whose outcomes are not eigenvalues but expectation values. So it seems like Born's rule cannot apply to such measurements, which means statements about expectation values cannot be part of Born's rule.

Ok, if it is not accepted here that probabilities can be defined as expectation values too, I try to forget this for a moment. It's not important for any argument. I hope we all agree that if the $P(a)$ for finding $a$ when measuring $A$ are given, the expectation value is
$\langle A \rangle=\sum_a a P(a)=\sum_{a,\beta} \langle a,\beta \hat{\rho} \hat{A} a,\beta \rangle=\mathrm{Tr}(\hat{\rho} \hat{A})$.
Of course, expectation values need not be eigenvalues of the corresponding operator. How do you come to that idea?

 

[vanhees71]

I had explained it multiple times:

Though traditionally called an ensemble expectation value, a more natural name for $\bar A=\langle A\rangle:=Tr \rho A$ (not suggesting a probabilistic interpretation a priori) would be the uncertain value. Quoting mostly from my web page, its physical meaning in general (without reference to probability or even measurement) is defined by the the following simple rule generalizing statistical intuition to situations where uncertainty is not required to be probabilistic:

Uncertainty principle: A Hermitian quantity $A$whose uncertainty $\sigma_A:=(A-\bar A)^2$ is much less than $|\bar A|$ has the value $|\bar A|$ within an uncertainty of $\sigma_A$.

This is a very clear and definite rule. Physicists doing quantum mechanics (even those adhering to the shut-up-and-calculate mode of working) use this rule routinely and usually without further justification. The rule applies universally. No probabilistic interpretation is needed, so it applies also to single systems.

This I don't understand. How are expectation values, including the standard deviation (there should be square root to meet the usual definition, i.e., $\sigma_A=\sqrt{\langle (A-\bar{A})^2}$), defined if not within probability theory? For me the notion of an expectation value is defined within some probability theory (e.g., the standard Kolomogorov axioms, which are for sure good enough for our discussion).

From this rule one can derive under appropriate conditions (see my online book) the following rule; the derivation is in my online book:

Measurement rule: Upon measuring a Hermitian operator $A$ in the state $\rho$, the measured result will be approximately $\bar A$, with an uncertainty at least of the order of $\sigma_A$. If the measurement can be sufficiently often repeated (on a system with the same or a sufficiently similar state $\rho$) then $\sigma_A$ will be a lower bound on the standard deviation of the measurement results.

I don't understand this, if I'm not allowed to think in terms of probability theory and the Law of Large Numbers, which is a one key result of probability theory. If I need to read an entire book for that, I'd like to know, which advantage it should have to redefine all the clear definitions used in the empirical sciences for centuries now!

Actually the above measurement rule should be considered as a definition of what it means to have a device measuring $A$. As such it creates the foundation of measurement theory. In order that a macroscopic quantum device qualifies for the description ''it measures $A$'' it must either be derivable from quantum mechanics, or checkable by experiment, that the property claimed in the above measurement rule is in fact valid. Thus there is no circularity in the foundations.

This I understand :-))). Of course, measurement apparati must be tested and calibrated to make sense. That's not a mathematical but an engineering task for experimentalists in the lab.

Moreover, Born's famous rule turns out to be derivable, too, (see my online book) but under special circumstances only, namely those where the Born rule is indeed valid in practice. (Though usually invoked as universally valid, Born's rule has severe limitations. It neither applies to position measurements nor to photodetection, nor to measurement of energies, just to mention the most conspicuous misfits.)

For me Born's rule very well applies to position measurements and photodetection. It's used in any book of quantum optics to describe photodetection within quantum theory. Why it shouldn't apply to position measurements, I also don't see (of course it cannot apply to photons, because you cannot even define a position observable in the usual sense). For massive particles, I don't see a problem to measure its position by simply putting a detector at a given place. Of course any such device has a finite resolution. To validate a given probability distribution for position, of course your device's resolution must be much better than the standard deviation of the probability distribution you want to measure, but I don't see a principle problem to measure position with arbitrary position.

 

[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.