The thermal interpretation of quantum physics

[A. Neumaier]

You cannot replace something with itself. The trace formula is nothing else than the Born rule, generalized to general mixed states!

The trace formula is a formal, interpretation-independent equation in the theory. You may call it a formal Born rule, but unless interpreted, it says nothing apart from giving names and symbols.

Born's rule (in its most frequent form) also claims that eigenvalues of operators are measured exactly, which is an interpretational statement completely independent of the trace formula.

The thermal interpretation rejects this part and says instead (and in direct opposition) that q-expectations of operators are measured approximately with a predictable minimal uncertainty.

This is in full agreement with the experimental record, in spite of what you say in the following quote.

It is by far not sufficient to understand how the formalism is used to the real-world observations. As I've stressed frequently in this discussion, the idea that expectation values are what describes observables is an old misconception of some of the founding fathers of QT (most prominently Heisenberg). It has been corrected, however, almost immediately by the very same founding fathers of QT (most prominently Bohr).

Instead of repeating assertions it would be better to point to original sources. I don't think you represent Heisenberg correctly, maybe you meant Schrödinger?

 

[A. Neumaier]

Even without decoherence or errors a quantum computer cannot compute NP-hard problems, so there's more to these limits than that.

Yes, but such theoretical statements about quantum computers are interpretation independent, so don't matter for the foundations.

 

[ftr]

In this case, the quantum field representing the beam is in a state with sharp Ne=1Ne=1N_e=1, but otherwise nothing changes. Unlike a particle (a fuzzy notion usually pictured as being a local object, which creates the mind-boggling situations that appear to make QM weird), a quantum field is always distributed; it has a density everywhere.

See this thread and the posts referred to there.

So are you saying the electron(in this case) is a long tube like or what?

 

[A. Neumaier]

So are you saying the electron(in this case) is a long tube like or what?

The electron field (in this case) is delocalized, concentrated along two beams, and the particle language is not applicable since it is appropriate only localized concentrations of fields.

Whenever you quote text containing formulas you need to quote the whole answer and then edit it; otherwise you get strange artifacts as those in your last post!

 

[ftr]

Should't qft had told us this long time ago ?

 

[A. Neumaier]

Should't qft have told us this long time ago ?

It could have, but people working in the foundations very rarely considered quantum field theory.

 

[zonde]

The electron field (in this case) is delocalized, concentrated along two beams, and the particle language is not applicable since it is appropriate only localized concentrations of fields.

And if you replace electron with silver atom? Would you say: The silver atom field (in this case) is delocalized, concentrated along two beams, and the particle language is not applicable since it is appropriate only localized concentrations of fields.

 

[A. Neumaier]

And if you replace electron with silver atom? Would you say: The silver atom field (in this case) is delocalized, concentrated along two beams, and the particle language is not applicable since it is appropriate only localized concentrations of fields.

Yes, of course. The same with buckyballs containing one C13 atom (to make them spin).

To have a silver atom as a particle you need to have it isolated on a surface where you can do atomic microscopy on it.

 

[vanhees71]

I discussed in detail in Section 3.5 of Part I what is wrong with the traditional textbook introduction of mixed states by Landau and Lifschitz. But he doesn't discuss the minimal interpretation as he has explicit collapse.

So please refer to a specific source that you regard as authoritative for your interpretation.

Its the difference between mathematical physicists (who want precise starting points and then refer in their further discussion only to these starting points and logic) and theoretical physicists (who want results and don't care about logical clarity as long as everything is intuitively plausible). DarMMM and I belong to the former, you belong to the latter.

Thus according to you, the state of a normal macroscopic body is definitely not a pure state, but an improper mixture? This would be in opposition to the (very widely used) setting given by Landau and Lifshitz, who claim that the true state of a normal macroscopic body is a pure state, but our lack of detailed knowledge of it requires that we treat it as a classical mixture of pure states (a proper mixture).

This is the reason why I want you to point out a definite authoritative source specifying the full set of assumptions you make on the fundamental level, and which I can use to make myself understood when discussing with you. At present it often feels like pushing a cloud since whatever I criticize you say its different, without clarifying your foundations.

The post numbers have changed since some subthreads where moved.

I think Landau Lifshitz is a pretty good textbook concerning the foundations, despite the use of the collapse doctrine, but this book is not very new. I think what comes closest to my views are the books by Ballentine and Peres.

Also I think that your formalism is in accord with it, and that's why I've been very surprised to learn that the implication of the usual probabilistic interpretation of this formalism is precisely what's forbidden to be used in your "thermal conjecture". I cannot call it interpretation, because the very thing the entire setup is lacking is "interpretation", i.e., the connection of the formalism with experiments and observation.

You may speculate about pure states of macroscopic systems, but it's simply not even testable in most cases, because you simply cannot resolve all the microscopic details (in the formalism the infinite tower of N-point functions containing all multi-particle correlations). There are of course examples for (nearly) pure states in low-temperature physics (liquid He, BECs etc.), but that's not the state of measurement devices and the matter surrounding us in every-day life.

 

[A. Neumaier]

I discussed in detail in Section 3.5 of Part I what is wrong with the traditional textbook introduction of mixed states by Landau and Lifschitz. But he doesn't discuss the minimal interpretation as he has explicit collapse.

I think Landau Lifshitz is a pretty good textbook concerning the foundations

Then what do you say to my critique mentioned above?

I cannot call it interpretation, because the very thing the entire setup is lacking is "interpretation", i.e., the connection of the formalism with experiments and observation.

Well, the interpretative core, and the essential difference to the tradition, is here:

The trace formula is a formal, interpretation-independent equation in the theory. You may call it a formal Born rule, but unless interpreted, it says nothing apart from giving names and symbols.

Born's rule (in its most frequent form) also claims that eigenvalues of operators are measured exactly, which is an interpretational statement completely independent of the trace formula.

The thermal interpretation rejects this part and says instead (and in direct opposition) that q-expectations of operators are measured approximately with a predictable minimal uncertainty.

This is in full agreement with the experimental record, in spite of what you say in the following quote.

Why is the first (traditional) statement an interpretation and the second (new) statement not?

 

[A. Neumaier]

I think what comes closest to my views are the books by Ballentine and Peres.

Peres uses 12 postulates (A-K and G$^*$) introduced during a long narrative from p.30-76. Most postulates concern the special case of pure states only; only postulates C(p.31) and K (p.76) are about mixtures.

Peres writes on p.11:

Bohr never claimed that different physical laws applied to microscopic and macroscopic systems. He only insisted on the necessity of using different modes of description for the two classes of objects. It must be recognized that this approach is not entirely satisfactory. The use of a specific language for describing a class of physical phenomena is a tacit acknowledgment that the theory underlying that language is valid, to a good approximation. This raises thorny issues. We may wish to extend the microscopic (supposedly exact) theory to objects of intermediate size, such as a DNA molecule. Ultimately, we must explain how a very large number of microscopic entities, described by an utterly complicated vector in many dimensions, combine to form a macroscopic object endowed with classical properties.

And on p.58:

The characteristic property of a genuine test is that it produces a permanent record, which can be described by our ordinary language, after having been observed by ordinary means, without the risk of being perturbed by the act of observation. [...]
The robustness of a macroscopic record—its stability with respect to small
perturbations such as those caused by repeated observations—suggests that
irreversible processes must be involved. This is a complicated issue, not yet
fully understood, which will be discussed in Chapter 11.

Having thus warned the reader of the difficulties lying ahead, I now return to the formal and naive approach where a quantum test is an unexplained event, producing a definite and repeatable outcome, in accordance with well defined probability rules given by quantum theory.

Do you agree that with the minimal interpretation there are still unsettled issues, which Peres points at?

On p.63, Peres writes:

the same word “measurement” is also used with a totally different meaning, whereby numerous quantum tests are involved in a single measurement. For example, when we measure the lifetime of an unstable nucleus (that is, its expected lifetime), we observe the decays of a large number of identically prepared nuclei. Very little information can be obtained from a single decay. Likewise, the measurement of a cross section necessitates the detection of numerous scattered particles: each one of the detection events is a quantum test, whose alternative outcomes correspond to the various detectors in the experiment.
Still another kind of scattering experiment, also called a measurement, is the use of an assembly of quantum probes for the determination of a classical quantity. For example, when we measure the distance between two mirrors by interferometry, each interference fringe that we see is created by the impacts of numerous photons. A single photon would be useless in such an experiment. These collective measurements will be discussed in Chapter 12. Here, we restrict our attention to measurements which involve a single quantum test.

In the thermal interpretation, measurement results must be reproducible to the accuracy specified by the calibration of the measurement device. Thus only the first three classes of measurements described here are valid high accuracy measurements. In contrast, the results of a single quantum test (in Peres most usually a binary test with possible answers 0,1) are usually not repeatable, unless very loose accuracy requirements (in binary tests of order $1/2$) are imposed.

Hence binary quantum tests only count as extremely low accuracy measurements in the thermal interpretation, while Peres and thus presumably you treat them as 100% exact measurements.

This is the main difference in the interpretations.

 

[vanhees71]

And if you replace electron with silver atom? Would you say: The silver atom field (in this case) is delocalized, concentrated along two beams, and the particle language is not applicable since it is appropriate only localized concentrations of fields.

It's well known since the very early days that the Stern-Gerlach experiment won't work with free electrons. It's very difficult for any charged particles.

In the SG experiment after the beam starting with indertermined $\sigma_z$ component has run through the magnet it is described by a (pure or mixed state) describing the (nearly 100%) entanglement between the $\sigma_z$ component and the position of the Ag atom.

All you know within QT given this preparation procedure are the corresponding probabilistic properties described by the (pure or mixed) state, i.e., for each individual Ag atom neither position nor $\sigma_z$ component are determined but only probabilities for finding this Ag atom at a given position and $\sigma_z$ component are known. What's known with certainty is that if the Ag atom is found at a position at one of the partial beams it also has the corresponding value of $\sigma_z \in \{-1/2,1/2 \}$.

Whether or not talk about "particles" is a matter of taste. In this case, I think it's save to talk about particles since it's just one particle and stays one particle all the time. It's just a particle which is not well localized, i.e., its position is not well determined at one place. It's hard to say, how to change common language to accurately talk about QT, but that's why we use mathematics and interpret it probabilistically since 1926, because that's the only interpretation which makes really sense so far for all cases where QT is successfully applied.

That's the paradigmatic example for a von Neumann filter measurement process with the position of the particle as pointer variable and $\sigma_z$ as the measured (or in this case even prepared) observable.

 

[stevendaryl]

On a personal note what I like about the thermal interpretation is the presence of clear open problems. Even if you don't "believe" it trying to see if discrete outcomes from experiments can be derived via open quantum systems and the structure of the device's set of slow modes is a very interesting problem.

I haven't been following the whole discussion in detail, but by "see if discrete outcomes ... can be derived", do you mean "discrete" as opposed to "continuous", or do you mean the fact that we always see definite results for measurements (the cat is alive or dead, not in some superposition)?

 

[DarMM]

I haven't been following the whole discussion in detail, but by "see if discrete outcomes ... can be derived", do you mean "discrete" as opposed to "continuous", or do you mean the fact that we always see definite results for measurements (the cat is alive or dead, not in some superposition)?

More so to check if realistic measurement devices develop the discrete aspect to their slow modes that the thermal interpretation suggests.
Closer to the first element you say, although the TI would then say the environment causes the second part.

 

[bhobba]

That's essentially the view of all Copenhagen flavors (Bohr, Heisenberg, Haag, Bub, Healey, Peres, Brukner, Zeilinger, Wheeler), Consistent Histories (Gell-Mann, Griffiths, Omnès) and QBism (Fuchs, Schack)

Although Consistent Histories is considered virtually the same as Decoherent Histories these days, I think the aim of Decoherent Histories is a bit more ambitious in that it wants to just use the formalism without explicit interpretation. They have not succeeded, even Consistent Histories has not succeeded - the proof of certain key theorems are missing. That's part of the reason I think we will eventually, one way or the other, sort out what's going on - but that's just a belief I have.

Thanks
Bill

 

[ftr]

Arnold, can you propose an experiment to prove TI since all interpretations seem plausable.

 

[A. Neumaier]

can you propose an experiment to prove TI

No experiment can prove anything true. it can only confirm it. But all experiments that probe quantum mechanics probe the thermal interpretation, since it is an interpretation of the standard theory, without any modification.

There is more theoretical work to be done to check the validity of the thermal interpretation. See posts #293 and #438.

all interpretations seem plausable.

Maybe to you. In my opinion, all other interpretations have significant weaknesses when trying to solve the measurement problem. The thermal interpretation hasn't.

 

[A. Neumaier]

Again, you repeat the WRONG statement that q-expectations are "beables" (in normal language "observables"). I don't need to expose this very early error of the founding fathers since I've done this at length at least twice within this very thread!

You asserted this WRONGly several times, without giving any rational justification. I am not doing the same as Schrödinger, who didn't think in terms of quantum fields.

 

[A. Neumaier]

The identification of q-expectations as the observables is not the common practice in the application of QT to real-world experiments

Yes, indeed. The thermal interpretation changes this identification, since this is a matter of interpetation only, not a matter of the theory itself. Nothing in the theory changes by changing this interpretation.

it was your declared goal to formulate an interpretation meeting the modern application of the QT formalism to real-world experiments!

Yes, indeed, and I achieved this goal.

If you compare cross sections to theory you compare q-expectations of the S-matrix, not single statistical events. If you compare spectra with experiments, you compare q-expectations of the spectral density functions, not single statistical events. If you compare quantum thermodynamic predictions with experiments, you compare q-expectations of internal energy, mass, etc., not single statistical events. You (consistent with the tradition) only talk differently about this, thus creating the illusion of a statistisc called the statistical interpretation. The termal interpretation talks instead about what is actually compared.

What do you expect of different interpretations if not that they talk differently about the same theory and the same experiments? If the talk is the same, the interpretation is the same. If the interpretation is different, the talk is different.

 

[ftr]

Maybe to you.

Actually I meant some semi impartial physicist. As I have said before I am very sympathetic to TI since the model in post #255 generates QM like and more suited to TI although other usual interpretations can be seen within it. However, I think(guessing from others) that TI is somewhat different in the sense that it does not look for "particle" solution but "field"(as more "real"). while all others I think they consider fields are mathematical artifact and push for the "particle" picture( because it is more intuitive classical concept and that is what people LOOK for in interpretation). So my guess is that you have to "prove" why do you think "fields" are more "real".

 

[A. Neumaier]

why do you think "fields" are more "real".

Because the standard model is a quantum field theory and not a quantum particle theory. Fields are the foundation on the theoretical level, and thus have every right to be regarded as more fundamental and more real.

Also, by treating q-expectations of fields and their products as real, the thermal interpretation explains the measurement problem in a natural way, whereas all other interpretations have difficulties.

 

[ftr]

I generally agree with you because the said model meshes qm and qft. However I guess others will have hard time based on the accepted model.

 

[vanhees71]

You asserted this WRONGly several times, without giving any rational justification. I am not doing the same as Schrödinger, who didn't think in terms of quantum fields.

My argument is very clear and rational. There's no restriction of the accuracy of a measurement apparatus because of the prepared state of the system to be measured. You don't even need any mathematics of quantum theory to understand this.

Again, you don't answer my question about the interpretation of your formalism!

 

[A. Neumaier]

My argument is very clear and rational. There's no restriction of the accuracy of a measurement apparatus because of the prepared state of the system to be measured.

I don't understand at all why this is an argument against q-expectations being beables some of which are approximately observable, as the thermal interpretation claims.

There is also no restriction of accuracy in measuring the position of a classical car to picometer accuracy, but this high accuracy is completely irrelevant because the position is intrisically inaccurate to many centimeters.

Again, you don't answer my question about the interpretation of your formalism!

I had answered your request for this a number of times and cannot answer it again in every post I write:

All q-expectations are auxiliary quantities like the n-point functions you mentioned in your post #304, except for the ones for which I gave an experimental interpretation in my posts #263, #266, and #278. These specify how some of the q-expectations give predictions for actual observations, and how these are to be interpreted.
I cannot understand at all why you think the statements there are not an answer to your request - please reply line by line to all these posts why not! Closely related material is in posts #313, #329, #456, and #474. There is also an extended discussion of the Stern-Gerlach experiment in another thread, which you haven't commented so far. If you want me to consider another specific experiment, just open a corresponding thread!

 

[vanhees71]

This is no answer to my request.

Again expectation values are not the observables. I have no clue what you (or Bell) mean with the word "beables". It's obviously something philosophical, but I want to understand the physics. Philosophy of science is some a posteriori analysis of science but not science, and I'm interested in the science.

It is very clear that nothing that depends on the state of the system, which is interpretationally defined as an equivalence class of preparation procedures can be the observable. An observable is interpretationally defined as an equivalence class of measurement procdures, and the measurement procedure doesn't depend on the state of the measured system but on the construction of the measurement appratus.

Of course, there are measurement devices which measure expectation values since they "smear" (coarse grain) over some finite spatial (or spatio-temporal region). This is not only the case in QT but also in classical physics. E.g., taking optics as an application of Maxwell's clasical electromagnetics, the intensity of light is defined as a (temporal) average of the energy density. Also the quantities in usual textbook macroscopic electromagnetics is a (spatial or spatio-temporal) average over the "microscopic fields" with a, to a certain extent arbitrary, split of the sources in intrinsic and extrinsic parts and application of linear-response theory. This is not much different using QED.

E.g., in the Stern-Gerlach experiment the preparation (state) is given by a silver-atom beam from an oven with a little hole, running through an appropriately defined magnetic field (with a large homogeneous part and a well-chosen field gradient around the location of the beam). Measured is the position of the silver atoms by "catching" them on a plate and then developing it like a photo plate. The position measurement is a paradigmatic example for a spatial coarse-graining procedure.

In the other thread, where you discuss the Stern-Gerlach experiment (SGE) separately, it is very clear that you do not provide an interpretation beyond the statistical interpretation, at least not one that is convincing for a physicist. If you cannot give a clear physical interpretation of the SGE, you have to think further about how to present your ideas to physicists. The SGE is the most simple example for a preparation-measurement procedure defining an experiment, and of course any interpretation must be able to explain what's physically going on in the relation of the formalism to this concrete and pretty simple setup. It's not of much help to relabel things with new words like "beables", "q-expectation values" and claiming to have abandoned the probabilistic meaning of the minimal interpretation without giving a positive explanation which physcial meaning the quantum state should take in the new interpretation. Also to use obviously ruled out interpretations from the early days of QT, as identifying expectation values with observables without taking the clear distinction between "state" and "observable" into account, is very unconvincing!

 

[DarMM]

Again expectation values are not the observables. I have no clue what you (or Bell) mean with the word "beables". It's obviously something philosophical

Not particularly it's just the basic fundamental things in your model/theory as opposed to the things that you observe. Like in General Relativity Riemann curvature is a beable, length in a given frame is an observable.

without giving a positive explanation which physcial meaning the quantum state should take in the new interpretation

To me it is very clear, the state $\rho$ assigns values to system properties via the map $\langle A \rangle_{\rho}$.

 

[vanhees71]

Interpretation is about the connection of the formal entities of the theory (for QT the Hilbert space, the statistical operators, and the operators representing observables) with physics. Of course, Arnold mathematically defines his symbols. I had the impression that the physical meaning behind the symbols is the usual probabilistic one. Then I learned in this discussion that Arnold wants to precisely abandon this probalistic interpretation. For me he has still not explained what the physical interpretation of the formalism (in his case obviously the q-expectation value algebra/analysis) is supposed to be.

For sure it is wrong to simply identify the expectation values with observables. This was known already since 1926 and lead Born to his probabilistic interpretation of the quantum state, and I've not seen any convincing argument, particularly no empirical one, that this is not the correct interpretation of QT.

 

[DarMM]

For me he has still not explained what the physical interpretation of the formalism (in his case obviously the q-expectation value algebra/analysis) is supposed to be.

Very simple, it's a value.

So when we write down the following in regular QM:
$$
\langle S_z \rangle_{\rho} = -\frac{1}{4}\\
\langle S_z^{2} \rangle_{\rho} = \frac{1}{16}
$$

What that means is that there are properties $A$ and $B$ with values $A = -\frac{1}{4}$ and $B = \frac{1}{16}$.

I know I've just relabeled them, but I'm trying to express that in the thermal interpretation they are simply properties like momentum, not statistical expectations despite the notation of the standard views.

However the thermal interpretation then goes on to explain why we get discrete values and not the true values of these quantities when we perform experiments.

 

[A. Neumaier]

This is no answer to my request.

Again expectation values are not the observables. I have no clue what you (or Bell) mean with the word "beables". It's obviously something philosophical, but I want to understand the physics. Philosophy of science is some a posteriori analysis of science but not science, and I'm interested in the science.

It is very clear that nothing that depends on the state of the system, which is interpretationally defined as an equivalence class of preparation procedures can be the observable. An observable is interpretationally defined as an equivalence class of measurement procedures, and the measurement procedure doesn't depend on the state of the measured system but on the construction of the measurement appratus.

In classical mechanics, particles exist. States define their properties (i.e., the beables of classical mechanics) and are given by the exact positions and momenta of the particles, some of which can be approximately measured. Fields are coarse-grained approximate concepts. This is the standard interpretation of classical mechanics. Experimental physics is about how to do the measurements, and under which conditions which measurements are how accurate.

Nothing else is needed. In particular, there is no need for pseudo-mathematical philosophy about states as equivalence class of preparation procedures or observables as equivalence class of measurement procedures. This is specific to a statistical interpretation, which has no place in the foundations.

In quantum field theory, fields exist. States define their properties (i.e., the beables of quantum field theory) and are given by the exact q-expectations of the fields and their normally ordered products, some of which can be approximately measured. Particles are coarse-grained approximate concepts. This is the thermal interpretation of quantum physics. Experimental physics is about how to do the measurements, and under which conditions which measurements are how accurate.

Nothing else is needed. In particular, there is no need for pseudo-mathematical philosophy about states as equivalence class of preparation procedures or observables as equivalence class of measurement procedures. This is specific to a statistical interpretation, which should have no place in the foundations.

 

[A. Neumaier]

For sure it is wrong to simply identify the expectation values with observables. This was known already since 1926 and lead Born to his probabilistic interpretation of the quantum state

You are completely wrong about the early history of quantum mechanics!

Please give explicitly the details of the argument that would prove that it is wrong to identify expectation values with observables.

 

[stevendaryl]

Again expectation values are not the observables. I have no clue what you (or Bell) mean with the word "beables"

Bell's intention behind introducing the word "beable" was to talk about properties that have values whether or not they are observed. So in classical physics, fields, and particle positions and momenta are beables. In QM, it seems that measurement results are beables, and that's about it.

 

[vanhees71]

You asserted this WRONGly several times, without giving any rational justification. I am not doing the same as Schrödinger, who didn't think in terms of quantum fields.

We agree to disagree. I give up. I've given a very simple reason that expectation values are not the observables in QT. This is independent of any interpretational opinion but an empirical fact.

Perhaps one more example (from QFT!) helps: If you measure the invariant-mass spectrum of a resonance (physicists slang) what you really measure are cross sections of a scattering process (since resonances are no asymptotic free states). Take e.g. the process $\pi^+ + \pi^- \rightarrow e^+ + e^-$. You find a pronounced peak around 770 MeV invariant mass (to be very precise, you find a quite broad peak of around 150 MeV width and a quite narrow one on top). Theoretically this can be very well described by a model by Sakurai, called vector-meson dominance, where you assume that the hadronic electromagnetic current is proportional to the light-vector-meson fields (the $\rho$ and $\omega$ for the peaks discussed here).

The peak has a finite width. In the formalism it's due to a complex pole in the corresponding connected four-point function of the scattering process, which is in your formalism a q-expectation value; and this is not an observable to begin with, because what's observable is the corresponding cross-section, but that's semantics in this case; the physics argument is the following: The invariant-mass width of this peak is 150 MeV. The inverse of this width is the life-time of the resonance (called $\rho$ meson). Now to measure this width the mass resolution of the detector must be much better than this width to really resolve this peak (the more for the much narrower $\omega$ meson and the $\phi$ meson around 1 GeV mass). The resolution of the measurement device is not restricted by the standard deviation of the invariant mass, i.e., it is independent of the quantum state the system is prepared in, and that's why the statement that quantum-mechanical expectation values (or, as in this case, quantities like the cross section in this example derived from it) are not the observables of the QT formalism.

 

[vanhees71]

You are completely wrong about the early history of quantum mechanics!

Please give explicitly the details of the argument that would prove that it is wrong to identify expectation values with observables.

I have given this very simple argument several times. For a real-world example from QFT see #487.

Also, where is my history wrong? There's a famous story found in at least one biography of Bohr that Schrödinger visited the Bohr institute and was staying with Bohr's family. Bohr involved him in such heated discussions about this very issue that Schrödinger got sick from the stress and Bohr's wife had to keep Bohr out of Schrödinger's bed room to give him some rest from the discussions ;-)).

 

[vanhees71]

I have given this very simple argument several times. For a real-world example from QFT see #487.

Very simple, it's a value.

So when we write down the following in regular QM:
$$
\langle S_z \rangle_{\rho} = -\frac{1}{4}\\
\langle S_z^{2} \rangle_{\rho} = \frac{1}{16}
$$

What that means is that there are properties $A$ and $B$ with values $A = -\frac{1}{4}$ and $B = \frac{1}{16}$.

I know I've just relabeled them, but I'm trying to express that in the thermal interpretation they are simply properties like momentum, not statistical expectations despite the notation of the standard views.

However the thermal interpretation then goes on to explain why we get discrete values and not the true values of these quantities when we perform experiments.

But that's obviously wrong and not how QT is used in practice. If I measure $S_z$ accurately, I don't get the value $-1/4$ but one of the possible values $\pm 1/2$ (supposed we deal with spin-1/2 particles). The measurement accuracy is not due to the state of the measured system but due to the construction of the measurement device. That's the very point I raised several times, and this is very clear from everyday practice of applications of the QT formalism to real-world experiments in the labs.

In other words the expectation values are not the "true values". That makes the probabilistic interpretation of the quantum state necessary (at least I've not seen any convincing alternative interpretation, where the quantum state is not probabilistically interpreted). The conclusion is that in this case $S_z$ is indetermined, and only probabilities can be given.

Here we even have incomplete information to begin with, i.e., we have only $\langle S_z \rangle$ and $\langle S_z^2 \rangle$. Then you can only "guess" the corresponding statstical operator. One idea to get one is to use information-theoretical arguments a la Shannon, von Neumann, and Jaynes, according to which the statistical operator with the "least prejudice" is the one that maximizes entropy. The result is the operator-valued Gaussian
$$\hat{\rho}=\frac{1}{Z} \exp(-\alpha \hat{S}_z - \beta \hat{S}_z^2), \quad Z=\mathrm{Tr} \hat{\rho}.$$
You can verify this guess by measuring (accurately!) $S_z$ on an ensemble of equally prepared particles and use statistical methods to "test this hypothesis".

 

[vanhees71]

In classical mechanics, particles exist. States define their properties (i.e., the beables of classical mechanics) and are given by the exact positions and momenta of the particles, some of which can be approximately measured. Fields are coarse-grained approximate concepts. This is the standard interpretation of classical mechanics. Experimental physics is about how to do the measurements, and under which conditions which measurements are how accurate.

Nothing else is needed. In particular, there is no need for pseudo-mathematical philosophy about states as equivalence class of preparation procedures or observables as equivalence class of measurement procedures. This is specific to a statistical interpretation, which has no place in the foundations.

In quantum field theory, fields exist. States define their properties (i.e., the beables of quantum field theory) and are given by the exact q-expectations of the fields and their normally ordered products, some of which can be approximately measured. Particles are coarse-grained approximate concepts. This is the thermal interpretation of quantum physics. Experimental physics is about how to do the measurements, and under which conditions which measurements are how accurate.

Nothing else is needed. In particular, there is no need for pseudo-mathematical philosophy about states as equivalence class of preparation procedures or observables as equivalence class of measurement procedures. This is specific to a statistical interpretation, which should have no place in the foundations.

In classical physics fields are beables too. If you use the word in this sense, it's obvious that only determinstic theories can describe beables. QT in the standard interpretation then doesn't describe beables, and you may well be motivated to find a determinstic theory as successful in describing the empirical facts as QT. What's sure is that this is not as simple as you seem to imply with your "thermal interpretation" by just identifying the expectation values with observables.

The definition of states and observables I gave are not pseudo-mathematical speculations but common practice in physics.