I Measurement results and two conflicting interpretations

[A. Neumaier]

In which of my lecture notes and where have you interpreted this in? Of course, I never wanted to claim this nonsense. Maybe I was sloppy in some formulation that you could come to this conclusion.

In https://th.physik.uni-frankfurt.de/~hees/publ/quant.pdf from 2008 you write:

Wird nun an dem System die Observable $A$ gemessen, so ist das Resultat der Messung stets ein (verallgemeinerter) Eigenwert des ihr zugeordneten selbstadjungierten Operators $A$.

[Moderator note (for completeness, although identical with what is said anyway): If the observable $A$ is measured on the system, then the result of the measurement is always a (generalized) eigenvalue of the according self-adjoint operator $A$.]

which agrees with tradition and the Wikipedia formulation of Born's rule, and says that measurements produce always eigenvalues - hence never approximate eigenvalues. Later, on p.20 of https://th.physik.uni-frankfurt.de/~hees/publ/stat.pdf from 2019, you seem to have strived for more precision in the language and only require:

A possible result of a precise measurement of the observable $O$ is necessarily an eigenvalue of the corresponding operator $\mathbf O$.

You now 1. distinguish between the observable and the associated operator and 2. have the qualification 'precise', both not present in the German version.

Thus it was natural for me to assume that you deliberately and carefully formulated it in this way in order to account for the limited resolution of a measurement device, and distinguish between 'precise', idealized measurements that yield exact results and 'unprecise', actual measurements that yield approximate results, as you did in post #2 of the present thread:

Born's statistical interpretation does not claim that there is no measurement error but, as usual when formulating a theory, discusses first the case that the measurement errors are so small that they can be neglected. If you deal with unprecise measurements the analysis becomes much more complicated

It's of course true that any measurement device has some imprecision, but you can in principle measure as precisely as you want, and theoretical physics deals with idealized measurements. The possible outcomes of measurements are the eigenvalues of the operator of the measured observable.

Here you seem to refer again to idealized measurements when you make the final statement., as no measurement error is mentioned.

Thus you have hypothetical measurement results ('precise', idealized) representing the true, predictable possible values and actual measurement results ('imprecise') representing the unpredictable actual values of the measurements. Their relation is an unspecified approximation about which you only say

That cannot be theorized about but has to be treated individually for any experiment and is thus not subject of theoretical physics but part of a correctly conducted evaluation of experimental results for the given (preparation and) measurement procedure.

It is this postulated dichotomy that I analyzed in my post #21.

 

[A. Neumaier]

It's of course always possible to make the resolution of the apparatus so bad that you don't resolve the two lines. Then you simply have a big blurr, but then I doubt very much that you always get the q-expectation values rather than an inconclusive determination of the value to be measured.

That's irrelevant.

The thermal interpretation never claims the caricature you take it to claim, namely that one always gets the q-expectation. It only claims that the measurement result one gets approximates the predicted q-expectation $\langle A\rangle$ with an error of the order of the predicted uncertainty $\sigma_A$. When the latter is large, as in the case of a spin measurement, this is true even when the q-expectation vanishes and the measured values are $\pm 1/2$!

 

[vanhees71]

In https://th.physik.uni-frankfurt.de/~hees/publ/quant.pdf from 2008 you write:

which agrees with tradition and the Wikipedia formulation of Born's rule, and says that measurements produce always eigenvalues - hence never approximate eigenvalues. Later, in 2019, you seem to have strived for more precision in the language and only require:

Thus it was natural to assume that you deliberately and carefully formulated it in this way in order to account for the limited resolution of a measurement device, and distinguish between 'precise', idealized measurements that yield exact results and 'unprecise', actual measurements that yield approximate results, as you did in post #2 of the present threadere you seem to refer again to idealized measurements when you make the final statement., as no measurement error is mentioned.

Thus you have hypothetical measurement results ('precise', idealized) representing the true, predictable possible values of the measurements and actual measurement results ('imprecise') representing the unpredictable actual values of the measurements. Their relation is an unspecified approximation about which you only say

It is this postulated dichotomy that I analyzed in my post #21.

Ehm, where is the difference between the German and the English quote (I don't know, where the English one comes from). Of course the English one is a bit sloppy, because usually unbound operators have not only eigenvalues but also "generalized" eigenvalues (i.e., if one refers to a value in the continuous part of the spectrum of the observable).

As expected, in my writing I've not changed my opinion over the years that the established and empirically very satisfactory working QT were wrong in this point. I don't discuss instrumental uncertainties in my manuscript, because they are about theoretical physics. To analyze instrumental uncertainties cannot be done in a general sense but naturally depends on the to be analyzed individual measurement device. This in general has very little to do with quantum mechanics at all.

Of course the very definition of an observable implies that you have to be able to (at least in principle) measure it precisely, i.e., to make the instrumental uncertainties negligibly small. In this connection it is very important to distinguish the "preparation process" (defining the state of the quantum system) and the "measurement process" (defining observables). The uncertainty/precision of a measurement device is independent of the uncertainty/precision of the preparation.

E.g. if you prepare the momentum of an electron quite precisely, then due to the Heisenberg uncertainty relation its position is quite imprecisely determined. Despite this large uncertainty of the electron's position you can measure its precision as accurately as you want (e.g., by using a CCD camera of high resolution). For each electron you'll measure its position very accurately. The uncertainty of the position measurement is determined by the high resolution of the CCD cam, not by the position resolution of the electron's preparation (which is assumed to be much more uncertain than this resolution of the CCD cam). For the individual electron you'd in general you'd not measure the q-expectation value given by the prepared state (not even "approximately", whatever you mean by this vague statement).

 

[vanhees71]

That's irrelevant.

The thermal interpretation never claims the caricature you take it to claim, namely that one always gets the q-expectation. It only claims that the measurement result one gets approximates the predicted q-expectation $\langle A\rangle$ with an error of the order of the predicted uncertainty $\sigma_A$. When the latter is large, as in the case of a spin measurement, this is true even when the q-expectation vanishes and the measured values are $\pm 1/2$!

But this is what you said! The "beable" is the q-expectation value and not the usual definition of an observable. If this is now all of a sudden not true anymore, we are back to the very beginning, since now your thermal interpretation is again undefined! :-(

 

[A. Neumaier]

Ehm, where is the difference between the German and the English quote (I don't know, where the English one comes from).

I edited my post, which now gives the source and explicitly spells out the differences. Please read it again.

 

[vanhees71]

That's irrelevant.

The thermal interpretation never claims the caricature you take it to claim, namely that one always gets the q-expectation. It only claims that the measurement result one gets approximates the predicted q-expectation $\langle A\rangle$ with an error of the order of the predicted uncertainty $\sigma_A$. When the latter is large, as in the case of a spin measurement, this is true even when the q-expectation vanishes and the measured values are $\pm 1/2$!

That's simply not true! The precision of the measurement device is determined by the measurement device, not the standard deviation due to the preparation (i.e., the prepared state) of the measured system.

If you want to establish that the standard deviation of $A$ is $\sigma_A$ you have to measure with much higher precision than $\sigma_A$, and you have to use a sufficiently large ensemble to gain "enough statistics" to establish that the standard deviation due to the state preparation is $\sigma_A$.

Again for the SGE: If your resolution of the spin-$z$-component measurement resolves the measured values $\pm 1/2$ and the state is prepared such that $\langle s_z \rangle=0$, you never find the result $\langle s_z \rangle=0$ with some uncertainty but you find with certainty either $+1/2$ or $-1/2$, and to establish that the standard deviation is $\sigma_{\sigma_z}$ you have to simply do the measurement often enough on equally prepared spins to gain enough statistics to verify this prediction (at the given confidence level, which usually is $5 \sigma_{\text{measurement}}$ (where here $\sigma_{\text{measurement}}$ is the standard deviation of the measurement, not that of the quantum state!).

 

[atyy]

Ok, fine with me, I know that you don't accept the minimal statistical interpretation, but I think it is important to clearly see that the claim that what's measured on a quantum system is always the q-expectation value is utterly wrong.

I do accept the minimal statistical interpretation :) the same as Dirac, L&L, Messiah, Cohen-Tannoudji etc, but maybe we should discuss this elsewhere.

Yes, the thermal interpretation completely baffles me (including the part about the measured result being a q-expectation), but maybe I'm missing something because I haven't spent a lot of time studying it.

 

[A. Neumaier]

But this is what you said! The "beable" is the q-expectation value and not the usual definition of an observable. If this is now all of a sudden not true anymore, we are back to the very beginning, since now your thermal interpretation is again undefined! :-(

No; this was never true. You were misreading the concept of a beable. Maybe this is the source of our continuing misunderstandings.

According to Bell, a beable of a system is a property of the system that exists and is predictable independently of whether one measures anything. A measurement is the reading of a measurement value from a measurement device interacting with the system that is guaranteed to approximate the value of a beable of that system within the claimed accuracy of the measurement device.

In classical mechanics, the exact positions and momenta of all particles in the Laplacian universe are the beables. and a measurement device for a particular prepared particle is a macroscopic body coupled to this particle, with a pointer such that the pointer reading approximates some position coordinate in a suitable coordinate system. Clearly, any given measurement never yields the exact position but only an approximation of it.

In a Stern-Gerlach experiment with a single particle, the beables are the three real-valued components of the q-expectation $\bar S$ of the spin vector $S$, and the location of the spot produced on the screen is the pointer. Because of the semiclassical analysis of the experimental arrangement, the initial beam carrying the particle splits in the magnetic field into two beams, hence only two spots carry enough intensity to produce a response. Thus the pointer can measure only a single bit of $\bar S$. This is very little information, whence the error is predictably large.

The thermal interpretation predicts everything: the spin vector, the two beams, the two spots, and the (very low) accuracy with which these spots measure the beable $S_3$.

 

[vanhees71]

I do accept the minimal statistical interpretation :) the same as Dirac, L&L, Messiah, Cohen-Tannoudji etc, but maybe we should discuss this elsewhere.

Yes, the thermal interpretation completely baffles me (including the part about the measured result being a q-expectation), but maybe I'm missing something because I haven't spent a lot of time studying it.

Well, the problem is that the "thermal interpretation" is either wrong for very obvious reasons or not clearly defined yet since now again we learned that the measured result is not the q-expectation value although we discuss precisely this earlier statement for weeks in several different forks of the initial thread. I'm also baffled, but for obviously different reasons.

 

[A. Neumaier]

you never find the result $\langle s_z \rangle=0$ with some uncertainty but you find with certainty either $+1/2$ or $-1/2$

If I find the result $+1/2$ or $-1/2$ with certainty, I can be sure that the measurement error according to the thermal interpretation is exactly $1/2$, since this is the absolute value of the difference between the measured value and the true value (defined in the thermal interpretation to be the q-expectation $0$). As a consequence, I can be sure that the standard deviation of the measurement results is also exactly $1/2$.

 

[vanhees71]

No; this was never true. You were misreading the concept of a beable. Maybe this is the source of our continuing misunderstandings.

According to Bell, a beable of a system is a property of the system that exists and is predictable independently of whether one measures anything. A measurement is the reading of a measurement value from a measurement device interacting with the system that is guaranteed to approximate the value of a beable of that system within the claimed accuracy of the measurement device.

In classical mechanics, the exact positions and momenta of all particles in the Laplacian universe are the beables. and a measurement device for a particular prepared particle is a macroscopic body coupled to this particle, with a pointer such that the pointer reading approximates some position coordinate in a suitable coordinate system. Clearly, any given measurement never yields the exact position but only an approximation of it.

In a Stern-Gerlach experiment with a single particle, the beables are the three real-valued components of the q-expectation $\bar S$ of the spin vector $S$, and the location of the spot produced on the screen is the pointer. Because of the semiclassical analysis of the experimental arrangement, the initial beam carrying the particle splits in the magnetic field into two beams, hence only two spots carry enough intensity to produce a response. Thus the pointer can measure only a single bit of $\bar S$. This is very little information, whence the error is predictably large.

The thermal interpretation predicts everything: the spin vector, the two beams, the two spots, and the (very low) accuracy with which these spots measure the beable $S_3$.

It's always dangerous to use philosophically unsharp definitions. If this is what Bell refers to as "beable" it's not subject of physics, because physics is about what's observed objectively in nature. Kant's "Ding an sich" is a fiction and not subject of physics!

In your 3rd paragraph you already redefine the word "beable" as to have the usual meaning of "observable". Why then not using the clearly defined word "observable".

All this contributes to the confusion rather than the clarification what your "thermal interpretation" really is meant to mean! If you are not able to express it in standard physics terms and always refer to unsharp notions of philosophy it's of course hard to ever come to a conclusion about it.

 

[A. Neumaier]

In your 3rd paragraph you already redefine the word "beable" as to have the usual meaning of "observable".

I don't understand. Where precisely do I do it?

Why then not using the clearly defined word "observable".

I cannot, because the word observable is loaded in quantum mechanics with the traditional interpretation.

If you are not able to express it in standard physics terms and always refer to unsharp notions of philosophy

The word 'observable' is also an unsharp and philosophical notion, with very different meanings in classical and quantum mechanics.

 

[A. Neumaier]

Kant's "Ding an sich" is a fiction and not subject of physics!

It is only as fictional as your idealized measurements from

It's of course true that any measurement device has some imprecision, but you can in principle measure as precisely as you want, and theoretical physics deals with idealized measurements. The possible outcomes of measurements are the eigenvalues of the operator of the measured observable.

 

[bobob]

I've read through your paper and it's not clear to me exactly what predictions you can make about experimental results. In particular, the thermal interpretation appears to hinge on the distinction of what you define as point causality, joint causality and extended causality.

Start with an example. Consider two antennas which the emission of radiation is spacelike and a point in the future in which I am going to measure the radiation arriving at my antenna. Do the amplitudes sum coherently or incoherently? Quantum mechanically, since phases are not measurable quantities, the difference is whether or not you can determine which wave corresponds to which source. The epr experiment is simply the same experiment except by exchanging past and future. The correlations at spacelike points only occur if the two photons have a common causal origin and behave like a single entity.

The Born rule is buried in the notion of extended causality. Since phases are not measurable quantities, you cannot specify a configuration of entities on a spacelike surface which meet the conditions required to be extendedly causal, since such a configuration would require specifying the phases of those entities. An intereference pattern that is actually deterministic would then require each point in the interference pattern (including the nulls) to be the coherent sum of of different extendedly causal configurations differing by phases. Instead of the Born rule being used to give a probability directly from the amplitudes associated with a past configuration in which all physical quantities are measurable, the Born rule in this case, gives a probability of which past configuration is being measured where each past configurtion differs by a quantity which is not measurable.

Since it's the unmeasurability of phases that gives rise to QFT, I'm not sure where that leaves the thermal interpreation in that regard. Phases are well defined classically, so I cannot see anyway around having to deal with an quantum mechanically unmeasurable quantity to obtain classical determinism unless you can define an experiment that makes that quantity measurable, in which case, you have actually surpassed quantum mechanics by making it truly classical.

The only out here for extended causality is the extended causality that results from a point causalty in the past of the extendedly causal entities, which leaves you back at standard quantum mechanics.

 

[A. Neumaier]

the thermal interpretation appears to hinge on the distinction of what you define as point causality, joint causality and extended causality.

Only with regard to explaining long-distance correlation experiments. In terms of prediction, it is identical to standard quantum mechanics, as it only imposes a different interpretation of it, rather than modifying it (as Bohmian mechanics or GWR).

 

[PeterDonis]

Consider two antennas which the emission of radiation is spacelike

How can emission of radiation be spacelike?

 

[bobob]

How can emission of radiation be spacelike?

Phased array radar. Emission of photons from sources which are spacelike seperated. I could have worded that better as emission from sources which are spacelike seperated.

 

[bobob]

Only with regard to explaining long-distance correlation experiments. In terms of prediction, it is identical to standard quantum mechanics, as it only imposes a different interpretation of it, rather than modifying it (as Bohmian mechanics or GWR).

Right, but that is point here. Those notions of causality are ultimately just pointlike causality. There is no difference since the extended causality requires a common origin in a past point causality. The difference would only be meaningful if you could specify an extended causality completely and independently from one with a common origin.

 

[PeterDonis]

Emission of photons from sources which are spacelike seperated

Ah, ok, that makes sense.

 

[A. Neumaier]

Those notions of causality are ultimately just pointlike causality. There is no difference since the extended causality requires a common origin in a past point causality. The difference would only be meaningful if you could specify an extended causality completely and independently from one with a common origin.

For independent point sources, point causality and etended causality are the same. For extended nonclassical sources not, because of the entanglement of their constituents.

 

[edmund cavendish]

Sir Roger Penrose posits a new physics in the area between classical and quantum. The penrose diosi theorum suggests observer independent gravitationally induced collapse of the wave function. Might this have some experimental validity?

 

[vanhees71]

No; this was never true. You were misreading the concept of a beable. Maybe this is the source of our continuing misunderstandings.

According to Bell, a beable of a system is a property of the system that exists and is predictable independently of whether one measures anything. A measurement is the reading of a measurement value from a measurement device interacting with the system that is guaranteed to approximate the value of a beable of that system within the claimed accuracy of the measurement device.

In classical mechanics, the exact positions and momenta of all particles in the Laplacian universe are the beables. and a measurement device for a particular prepared particle is a macroscopic body coupled to this particle, with a pointer such that the pointer reading approximates some position coordinate in a suitable coordinate system. Clearly, any given measurement never yields the exact position but only an approximation of it.

In a Stern-Gerlach experiment with a single particle, the beables are the three real-valued components of the q-expectation $\bar S$ of the spin vector $S$, and the location of the spot produced on the screen is the pointer. Because of the semiclassical analysis of the experimental arrangement, the initial beam carrying the particle splits in the magnetic field into two beams, hence only two spots carry enough intensity to produce a response. Thus the pointer can measure only a single bit of $\bar S$. This is very little information, whence the error is predictably large.

The thermal interpretation predicts everything: the spin vector, the two beams, the two spots, and the (very low) accuracy with which these spots measure the beable $S_3$.

Then "beable" is an empty phrase, because we cannot know about any property of a system without measuring (or observing) it. Physics deals with what's observable and measuarable and not philosophical fictitions that cannot be observed and measured.

In classical mechanics as well as in quantum mechanics the positions and momenta of all particles are observables that can (in principle) always be measured as precisely as you wish. There's no difference between classical and quantum mechanics here (of course with the caveat that there are no point particles existing and thus to measure the "position" must sometimes be specified more precisely, but let's keep it at the simple level).

Spin is in some sense special, because it's an observable without a classical analogon. Within non-relativistic quantum theory spin is an observable axial-vector quantity with the properties of an angular momentum. The three components are thus observables (within non-relativistic QT). I don't see, why you emphasize the semiclassical description (i.e., keeping the description of the external magnetic field classical) so much. That's unimportant for the very general issue we are discussing here. I've never thought about a quantum-field theoretical description of the SGE. It's perhaps even an interesting formal question, but it's totally irrelevant for the issues we are discussing here.

Now, in the SGE what's measured is one of the spin components (or to be very precise the component of the magnetic moment related to this spin component) wrt. the direction determined by the magnetic field. The pointer variable is the position of the particle, the inhomogeneous magnetic field designed such that it leads to a (nearly) 100% entanglement between position and this spin component. In this way it's even a preparation procedure for this spin component (or in other words the polarization state of the particle). The error is practically negligible, if the experiment is well designed.

I think, already the interpretation of the SGE in terms of the thermal interpretation is orthogonal to what's really achieved in the lab already in 1922 by Stern and Gerlach, while the minimal standard interpretation precisely describes these findings very well.

 

[vanhees71]

If I find the result $+1/2$ or $-1/2$ with certainty, I can be sure that the measurement error according to the thermal interpretation is exactly $1/2$, since this is the absolute value of the difference between the measured value and the true value (defined in the thermal interpretation to be the q-expectation $0$). As a consequence, I can be sure that the standard deviation of the measurement results is also exactly $1/2$.

This contradicts the empirical outcome of the SGE. Even in the lab at university with its rather limited budget you can get very well separated beams of Ag atoms!

 

[A. Neumaier]

Even in the lab at university with its rather limited budget you can get very well separated beams of Ag atoms!

Yes, but the spots are supposed to measure a very tiny spin of the order of $\hbar$. On a scale where the positions of the two silver spots represent the numbers $\pm \hbar/2$, these positions are approximations of any number of the order of $\hbar$ with an error of the order $\hbar$, in particular, one of the q-expectation of the spin, as the thermal interpretations claims.

This contradicts the empirical outcome of the SGE.

There cannot be a contradiction in cases where no significant relative accuracy is claimed!