QM Eigenstates and the Notion of Motion

[vanhees71]

What I expect from classical mechanics is rather described by a coherent or squeezed state than an energy eigenstate!

 

[Fra]

WernerQHs point was what I thoughtr the thread was about. What is a periodic "movement" in Q space, is stationary in the conjugate P space. That way, one can argue that what is stationary or not, depends on hte agents way of encoding things. Of course the fourier transform is good because it makes encoding periodic phenomena simple or even stationary.

/Fredrik

 

[gentzen]

This is a autocorrelation function, describing fluctuations. In which sense do you think it describes that the oscillator is "moving"?

Strange question. Doesn't $x$ usually denote the position of a harmonic oscillator?

Yes, and its expectation value is time-independent if the oscillator is in an energy eigenstate. What you considered is a autocorrelation function, describing fluctuations around this stationary expectation value.

Wow, so the disagreement really comes down to how you (vanhees71) interpret the word "moving". For me, the most important part of "moving" is that properties (like position) in a stationary state are not necessarily constant over time. But you basically seem to have no objections to this, you just want to call that "fluctuations" instead of "moving".

It seems like for you, "moving" implies a more systematic way of being not-constant, like for example changing position continuously over time with a certain velocity. I don't believe in that one either. WernerQH's example is nice, even so for me feels just as "unphysical" (only to the other extreme) as the constant non-moving trajectories of stationary states in Bohmian mechanics.

It's as with a gas at a finite temperature in its rest frame in global thermal equilibrium. Nothing moves, and on average the gas molecules are at rest, but of course they are still flucuating around. The average velocity (the expectation value of a time derivative) is 0, but its autocorrelation function or simply $\langle v^2 \rangle \neq 0$.

And here you gave a nice detailed description making it clear that you don't insist on everything being always constant over time, but that it is mostly the absence of systematic movement which is important to you.

 

[vanhees71]

For me moving means that the state of a system is time-dependent. In QT energy eigenstates are time-independent and thus the system is "not moving". It's the solution for the problem of instable atoms in the classical picture, where accelerating electrons crash into the nuclei within a very short time due to radiative energy loss. In QT that's not the case: Neglecting the quantization of the electromagnetic field you get electronic energy eigenstates which are the stationary states, and the atoms are thus stable in these states forever. The expectation values of the time-derivative of any observable is 0, but of course the observables don't take determined values and thus fluctuate around their time-independent expectation values, and thus the expectation values of squares of time-derivatives of observables are not 0, although "nothing is moving".

 

[Fra]

For me moving means that the state of a system is time-dependent.

That's how I understood you some posts up as well. No big point in arguing just about definitions.

In QT energy eigenstates are time-independent and thus the system is "not moving". It's the solution for the problem of instable atoms in the classical picture, where accelerating electrons crash into the nuclei within a very short time due to radiative energy loss. In QT that's not the case: Neglecting the quantization of the electromagnetic field you get electronic energy eigenstates which are the stationary states, and the atoms are thus stable in these states forever. '

Interestingly, in my preferred interpretation, the choice of encoding the information in the best way, also is what makes the agent fit which relates to stability as well. If the agent can find a transformation that transforms the pattern of incoming data strems into a stationary code, that must be a massive evolutioanry advantage. This is how i prefer to interpret this. The fourier transform is just one of many possible transforms.

I recall during my first introduction to QM, that I felt that the role of the "fourier transform" was suspicuously a key concept, but which was not really motivated in a deeper waya beyond the standard "wave duality stuff". I think I have come to a much deeper understanding of this over the years, and i think there is maybe yet deeper motivations for all this that is ahead of us.

/Fredrik

 

[vanhees71]

To translate all this philosophy about information into physics: The information about the system is encoded in the state of the system, i.e., the statistical operator $\hat{\rho}$. That's all information you can have about a system according to QT. You don't need agents or other fictitious elements but just the statistical operator!

 

[WernerQH]

It's the solution for the problem of instable atoms in the classical picture

It would be a rather poor solution if it relied on wording, on just avoiding the word "motion". We all know that classical theory is only "approximately" valid (think of Rydberg atoms). But it is not helpful to shun the correspondence principle altogether, and insist on a peculiar usage of the word "motion" that is at odds with how most physicists use the term.

 

[vanhees71]

One doesn't merely avoid the word "motion" but with QT has discovered a theory with a huge realm of applicability, where you have stable bound states of electrons to atomic nuclei. Indeed the classical theory is only an approximation to QT. The energy eigenstates provide solutions, where you have an electrostatic configuration, which is, in contradistinction to the classical theory, stable. Such a solution does not exist in the classical theory, and that's why in classical theory atoms couldn't exist as stable objects.

The correspondence principle is nowadays substituted by symmetry principles, which let us derive how QT looks like for specific problems. There's no need anymore for hand-waving arguments a la Bohr.

 

[gentzen]

... If the agent can find a transformation that transforms the pattern of incoming data strems into a stationary code, that must be a massive evolutioanry advantage. ...
... I think I have come to a much deeper understanding of this over the years, and i think there is maybe yet deeper motivations for all this that is ahead of us.

If you feel that your understanding is so deep now, then try to write down something self-contained. Maybe just the solution to some specific riddle (the easiest route, because you don't need to "convert" anyone), perhaps some coherent interpretation (like the thermal interpretation), or some specific theorem (like the quantum de Finetti theorem).

To translate all this philosophy about information into physics: ... You don't need agents or other fictitious elements ...

What I would find intersting is to do something with your agents. If all this talk about agents in the end just boils down to the perspective of a single agent, then the charge of solipsism sooner or later becomes quite justified. Probability is important in quantum physics, and probability is closely related to game theory, which is concerned with the interactions of "many agents" among each other. Now game theory is messy, more messy than physics in certain ways, but less messy than the actual biological and political realities out there. How can your philosophy help us with those issue related to agents and game theory?

 

[WernerQH]

The correspondence principle is nowadays substituted by symmetry principles, which let us derive how QT looks like for specific problems. There's no need anymore for hand-waving arguments a la Bohr.

In my diploma exam I was actually questioned about the correspondence principle. The examiner was a condensed matter theorist, who obviously did not consider it outdated! How do you explain to your students how quantum physics blends into classical physics? Unfortunately there is not a single established interpretation of quantum theory, so it would appear reasonable to me to expose students to a great variety of pictures (concepts, notions) so that they can hone their intuition (and discover the limits of applicability of those pictures). Rather than insisting on one "correct" picture. Aren't future physics teachers among your students? I can't help but feel pity for them and their future pupils.

 

[gentzen]

Unfortunately there is not a single established interpretation of quantum theory, so it would appear reasonable to me to expose students to a great variety of pictures (concepts, notions) so that they can hone their intuition (and discover the limits of applicability of those pictures). Rather than insisting on one "correct" picture. Aren't future physics teachers among your students? I can't help but feel pity for them and their future pupils.

Sorry, but this rant is both off-topic, nasty, and doesn't even make sense. In Germany, quantum physics is not really taught at school, so whatever physics teachers are taught in university about its interpretation should have a negligible influence on their future pupils.

And if you would want to teach different pictures to your students, then you should start with some existing expositions of those pictures. vanhees71 has no objections to presentations like section "3.7 Interpretations of Quantum Mechanics" in Weinberg's book or even entire books like "Verständliche Quantenmechanik: Drei mögliche Weltbilder der Quantenphysik" by Detlef Dürr and Dustin Lazarovici (or its english version). He even recommends those. I see no problem that he favors Ballentine's interpretation and his book, among others because that allows him to defend Einstein's position without embracing all the surrounding philosophical discussions. In the end such discussions would just drag away valuable time from his students.

Maybe one question is whether comprehensive books like "Do We Really Understand Quantum Mechanics?" by Franck Laloë should be recommended too. But really reading and understanding such a book would amount to dive into current research in quantum foundations, which might not be the best idea for a future physics teacher (or a future particle physics researcher).

 

[Fra]

If you feel that your understanding is so deep now, then try to write down something self-contained.

At some point I have in mind to publish, but I feel there is enough of interpretations so I do not want to publish just another interpretation that makes no substantial difference to the open problems. And I have no pressure to publish anything unless I feel ready. It's not near ready yet.

I do not like to read such papers myself, at least not after reading enough. I enjoyed lots of writings of the QM founders in the past but at this point, I don't want to read just another of those papers. It's as bad and empty as the other extreme - axiomatic reconstructions where the axioms are choosen without physical motivation.

After all my interpretation is I am well aware strange and complicated relative to say copenhagen interpretation (more so than standard qbism), so I expect noone to buy into until it's method can be shown to solve real problems and that is to be fair my problem. I have no intention to convince anyone, I just try to stick to what I think is rational reasoning, but to each his own.

What I would find intersting is to do something with your agents. If all this talk about agents in the end just boils down to the perspective of a single agent, then the charge of solipsism sooner or later becomes quite justified.

I agree completely. I have of course thought about this. To just end up with everything beeing arbitrary would be pointless, it's not what I seek.

Probability is important in quantum physics, and probability is closely related to game theory, which is concerned with the interactions of "many agents" among each other. Now game theory is messy, more messy than physics in certain ways, but less messy than the actual biological and political realities out there. How can your philosophy help us with those issue related to agents and game theory?

Yes, game theory is the right perspective to see what I talk about. (That's not to say one should jump into the formal "game theory" litterature and expect the exact math).

A short comment, which as always is a balance as I avoid put any details on the forums due to guidlines. Mentores are free to delete the post if I crossed some lines.

In my view the agents/obsevers are the players (and the agents are of course simply matter subsystems, no brains or physicists needed), but there is no objective agreement of the "rules of the game", the only rules is that the survivor wins, do what you can to survive. The agents set of "strategies" are constrained by it's physical limits. So the "strategy space" must necessarily scale (or rather evolve) with the complexity (mass) of the agent. So this implies an evolution of law, coined by Lee Smolin, but his ideas was specifically for example cosmological natura selection that the laws mutate at each big bang, and are frozen from there one... in principle I thinkg the same way, except I see no clear line, it's just some somewhere around the TOE energy scale, I expect that laws to be fixed enough so that this is why we don't see variations ofhte laws when looking out into space.

The conceptual quest in this perspective is simple enough to be explained like this:

After some evolution, can can we infer which population of agents encoding which strategies that are most likely to appear in the low energy limit without ending up in a similar landscape problem as string theory?

Could these things correspond to (be isomorphic to) matter and their respective interactions?

And the unification of all interacitons, should follow from how new interactions become possible as agents grow in complexity. This is a naturaly reason why the laws of physics must become simpler, the closer we get to unification. They only may LOOK complex, when seem from the fictive external asymptotic observer that is the coventional perspective in QFT. Ie. the theory when properly scaled (not just renormalized in the regular way) must become very simple. And simply enough to avoid the fine tuning problem of string theory for example.

The strategy is - formulate this in terms of mathematics, and algorithms/computations, and work it out and try to make contact to the familiar concepts, such as space, time, mass, energy, charge etc.

/Fredrik

 

[vanhees71]

In my diploma exam I was actually questioned about the correspondence principle. The examiner was a condensed matter theorist, who obviously did not consider it outdated! How do you explain to your students how quantum physics blends into classical physics? Unfortunately there is not a single established interpretation of quantum theory, so it would appear reasonable to me to expose students to a great variety of pictures (concepts, notions) so that they can hone their intuition (and discover the limits of applicability of those pictures). Rather than insisting on one "correct" picture. Aren't future physics teachers among your students? I can't help but feel pity for them and their future pupils.

For sure, I don't bother my students with fruitless philosophical speculations. I admit that I have not yet found a way of teaching QT I'm really satisfied with. So I take refuge to a blend of the "historical approach", i.e., I start with a short review about the historical development, which lead to modern quantum mechanics, emphasizing from the first moment on that everything before Heisenberg, Born, Jordan and Schrödinger and Dirac is outdated and not a consistent picture. Concerning modern QT itself, of course I treat only non-relativistic QM in terms of wave mechanics since with wave mechanics in my opinion you get the most intuitive picture which at the same time is closest to the full abstract content of the theory. You also can't help it, but QT is considerably more abstract than classical point-particle mechanics and also a bit more abstract than classical field theory, but that's how physics is in the 21st century. I also cover spin and the Pauli equation and as a final topic entanglement and the Bell inequality. Concerning interpretation, I present them with the minimal statistical interpretation, with the Born rule as the key postulate. I don't see any merit in thinking that one needs more than the minimal interpretation to do physics and to understand the phenomena related to QT.

 

[WernerQH]

Sorry, but this rant is both off-topic, nasty, and doesn't even make sense.

Thanks for trying to moderate. :-) But I can't understand why it should be off-topic to criticize what I perceive as a distortion of the term "motion" as most people use it.

In Germany, quantum physics is not really taught at school, so whatever physics teachers are taught in university about its interpretation should have a negligible influence on their future pupils.

It's a long time since I went to school, and we didn't have quantum physics then. But my impression from Physik Journal, the web site of Heisenberg-Gesellschaft, or physikerboard.de is that there is much effort to introduce elements of quantum physics (of course not quantum theory) already at school. And I think that some of the peculiar views expressed by van Hees can be detrimental to a young physics teacher who is supposed to explain these concepts.

Does a harmonic oscillator oscillate? Only if it is not in an energy eigenstate? How do you prepare it in the state $n = 3$ ? The formula in my previous post #26 is most easily derived by summing over energy eigenstates, but using coherent states you can obtain the exact same formula. I can't make sense of @vanhees71's point that one is allowed to speak of motion only when one uses coherent states.

 

[vanhees71]

Sorry, but this rant is both off-topic, nasty, and doesn't even make sense. In Germany, quantum physics is not really taught at school, so whatever physics teachers are taught in university about its interpretation should have a negligible influence on their future pupils.

Fortunately that's not entirely true. An idea about quantum theory is part of the general knowledge every high-school student should get before graduating, and indeed there is some QM in the high-school curricula. Fortunately also the didactics tends to level down the amount of "old quantum theory" to discuss, of course at a more qualitative level, adapted to the very limited level in mathematical prerequisites the German highschool system offers, but at least one covers wave mechanics, the double-slit experiment, Stern Gerlach, the particle in the infinite potential box. At my time we even had the Schrödinger equation and the harmonic oscillator. For the hydrogen atom only the ground state was explicitly treated and otherwise it was explained in a qualitative way. However, I had an exceptionally good highschool teacher.

And if you would want to teach different pictures to your students, then you should start with some existing expositions of those pictures. vanhees71 has no objections to presentations like section "3.7 Interpretations of Quantum Mechanics" in Weinberg's book or even entire books like "Verständliche Quantenmechanik: Drei mögliche Weltbilder der Quantenphysik" by Detlef Dürr and Dustin Lazarovici (or its english version). He even recommends those. I see no problem that he favors Ballentine's interpretation and his book, among others because that allows him to defend Einstein's position without embracing all the surrounding philosophical discussions. In the end such discussions would just drag away valuable time from his students.

Indeed, one should keep out all this philosophical confusion from the students. It doesn't in any way help to understand the physics. Concerning philosophy I think what QT teaches us is that we can understand to a certain extent natural phenomena which are way beyond what we directly perceive by our senses which are adapted to the macroscopic environment we have to survive in, but that this understanding is only possible to be expressed in a rather abstract mathematical way. On the other hand abstraction makes thinks simpler rather than more complicated, because it helps to get rid of all kinds of destractions and enables a presentation of the theory in terms of its "bare bones". For that the abstract rigged-Hilbert space formalism (aka Dirac's bra-ket formalism) is the most clear and simple exposition, but that's of course out of reach at the high-school level.

Maybe one question is whether comprehensive books like "Do We Really Understand Quantum Mechanics?" by Franck Laloë should be recommended too. But really reading and understanding such a book would amount to dive into current research in quantum foundations, which might not be the best idea for a future physics teacher (or a future particle physics researcher).

One should understand the physics first, before reading such a book concerned with interpretational issues. One must also not forget that, uncomprehensible to me, there seems to be no common decision on the "right interpretation" of quantum mechanics yet. So it's an open, in my opinion also quite unspecified, problem interdisciplinary research topic on the boundary between physics and philosophy, which is one more argument to keep it entirely out of the discussion at high school.

However, you can of course discuss the 2022 Nobel Prize in physics just with the well-understood established physical theory in its minimal interpretation. At least the math is pretty simple. All you need is elementary algebra in finite-dimensional Hilbert spaces for the quantum part and some basic probability theory for the local-realistic-hidden-variable theory.

What of course cannot be discussed is the here most discussed issue of locality vs. relativistic causality, which is in fact no problem but resolved by the microcausality constraint of modern relativistic QFT, i.e., there are no "actions at a distance" but only "long-ranged correlations", which are stronger than predicted by any local realistic hidden-variable theory.

 

[A. Neumaier]

Nothing moves, and on average the gas molecules are at rest, but of course they are still flucuating around.

This would mean that everything moves, but that you take a coarse-grained view only!? But what would this mean for an electron? Since the expectation refers to the ensemble, it would mean that every realization of the electron moves, but the net effect for the ensemble is zero.

... Unless you are adhering to my thermal interpretation, according to which the expectations are the real things observable, and correlations only tell about the amount of their intrinsic uncertainty!

 

[vanhees71]

So far I follow your "interpretation". ;-).

 

[A. Neumaier]

The information about the system is encoded in the state of the system, i.e., the statistical operator . That's all information you can have about a system according to QT.

But in the statistical interpretation, this would be a true statement only for an ensemble of identically prepared systems, not for a single system!

 

[A. Neumaier]

There's no need anymore for hand-waving arguments a la Bohr.

But all your arguments are pure handwaving when applied to a single system rather than to an ensemble!

So far I follow your "interpretation". ;-).

Ah, finally? Note that there is nothing more to my interpretation than that! Everything else is just apllication of this to various issues regarding single systems!

 

[vanhees71]

The only thing I do not understand concerning your interpretation is, how you define expectation values when you forbid to use Born's rule, but I don't think that we'll ever come to a consensus about this.

What I also don't understand is your narrow interpretation of the "ensemble". Of course quantum theory also applies to statistics made with a single system. E.g. a single electron in a Penning trap is used for very long times to get expectation values by measuring corresponding currents. Also a gas in a container ("canonical ensemble") consists of fixed molecules, and statistical physics describes the coarse grained macroscopic ("collective") observables for this "single system". For macrosocpic systems the fluctuations of these collective observables are tiny compared to the average values themselves, and that's why you have "classical behavior".

The only other assumption to use the maximum entropy principle you need the H-theorem according to which the constraints must be imposed by using the additive conserved quantitities, leading to the microcanonical, canonical, or grandcanonical ensembles.

One way to get off-equilibrium physics is to use the $\Phi$-derivable approximations (2PI/CJT/Luttinger-Ward/Baym-Kadanoff formalism) leading to the Kadanoff-Baym equations and "coarse graining" is done formally as the gradient expansion, which leads to quantum transport equations, and this is indeed a expansion in powers of $\hbar$. Also the H-theorem can be derived in this way.

In the toy-model cases (e.g., $\phi^4$ in (1+2) spacetime dimensions) where you can numerically solve the Kadanoff-Baym equation as well as the transport equation one finds that indeed the semiclassical transport equations describe the dynamics well, and particularly the "long-time limit", which of course leads to the usual standard Bose-Einstein or Fermi-Dirac distributions as expected.

 

[PeterDonis]

how you define expectation values when you forbid to use Born's rule

Where in the formula for expectation values does Born's rule get used?

 

[vanhees71]

You need the probability (distribution) to calculate the expectation value. Let $\hat{A}$ be the self-adjoint operator representing the observable and $|a,\alpha \rangle$ a complete orthonormal set of (generalized) eigenvectors for the observable $A$, according to Born's rule the probability to get the value $a$ when measuring $A$ and the system being prepared in the state $\hat{\rho}$
$$P(a)=\sum_{\alpha} \langle a,\alpha |\hat{\rho}|a,\alpha \rangle.$$
From this you get the expectation value
$$\langle A \rangle = \sum_{a} P(a) a = \sum_{a,\alpha} \langle a,\alpha| \hat{\rho} \hat{A}|a,\alpha \rangle=\mathrm{Tr} (\hat{\rho} \hat{A} ).$$
Of course you can calculate the trace using any complete orthonormal set you like, but to derive this formula you need the assumption about the probabilities finding the eigenvalue $a$ when measuring the observable $A$, given the state $\hat{\rho}$.

Of course, if there are continuous spectra involved, then instead of the sums you have to take the corresponding integrals.

 

[gentzen]

The only thing I do not understand concerning your interpretation is, how you define expectation values when you forbid to use Born's rule, but I don't think that we'll ever come to a consensus about this.

You need the probability (distribution) to calculate the expectation value.

I once worked-out a "simple" example of how expectation values can be used in an idealized model without any underlying stochastic features (which would give rise to probabilities):

... "The" real system could be a class of similarly prepared systems, it could be a system exhibiting stochastic features, or it could also be just some object with complicated features that we want to omit in our idealized model.

The absence of stochastic (and dynamic) features for this last case makes it well suited for clarifying the role of expectation values for the interpretation, and for highlighting the differences to ensemble interpretations.
As an example, we might want to describe the position and extent of Earth by the position and radius of a solid sphere. Because Earth is not a perfect sphere, there is no exactly correct radius. Therefore we use a simple probability distribution for the radius in the model, say a uniform distribution between a minimal and a maximal radius. We can compare our probabilistic model to the real system by a great variety of different observations, however only very few of them are "intended". (This is independent of the observations that are actually possible on the real system with reasonable effort.) ...

Of course, you will have to also read the mathematical part following the above quote, and try to understand yourself what it shows, and why I included it, i.e. which question is answered by that part, and why that question was important to me.

Skipping the "philosophy" at the begining of that post should be possible, but please still keep in mind the following disclaimer in the introduction:

My intention is to follow the advice to "use your own words and speak in your own authority". Because "I prefer being concrete and understandable over being unobjectionable," this will include objectionable details that could be labeled as "my own original ideas".

 

[Fra]

What I also don't understand is your narrow interpretation of the "ensemble".

Dito. If you want a really flexible ensemble, then you have the microstate of the agent!

It encodes the history of all it's interactions, which one can see as one long flexible "preparation" of the present. So every "now" of the agent is "prepared/forged" but it's own history.

Of course quantum theory also applies to statistics made with a single system.

Yes, as there is only now "now", and only one "tomorrow". No agent cares if it was wrong a week ago, it always focuses on the one future. And the decision about what todo tomorrow is made only once.

Indeed the narrow ensemble view makes no sense

I am slolwly making some associations to neumaiers view, where I think that instead of a fictive ensemble or microstate, he assumes(or defines) some sort of equiblirium parameters, where one assumes the single case to be in some sort of equilibrium with the total ensemble, just like a small piece of matter can be in equiblirium to bulk via tempearature etc??? This is some sort of equibliriym assumption, in the agent picture i also at some pointenvision that an agent is in "equiblirium" in some abstract sense to the envuronment, but the question is HOW, and at that configuration? Here is where Neumaier looses me, I don't think it's covered. I am symphatetic to the critique against the ensemble, but there seems to be several ways around it, but maybe the paths meet at some point.

/Fredrik

 

[vanhees71]

What's your definition of "agents"? I'm doing well with mundane measurement devices...

 

[A. Neumaier]

The only thing I do not understand concerning your interpretation is, how you define expectation values when you forbid to use Born's rule

The statistical (measured) expectation is the sample mean of measured single data instances.

The quantum (theoretical) expectation is defined in terms of the density operator $\rho$ characterizing the state by the formula $\langle A\rangle:=Tr(\rho A)$. This mathematical formula is a pure definition, and has nothing to do with measurement, hence it does not involve Born's rule (which depends on measurement).

The law of large numbers implies that under the right conditions (sufficiently large sample of uncorrelated single data instances) the statistical (measured) expectation is arbitrarily close to the quantum (theoretical) expectation.

 

[PeterDonis]

You need the probability (distribution) to calculate the expectation value.

No, you don't. You just need the wave function or density matrix. The expectation value is $\bra{\psi} A \ket{\psi}$ or $Tr \rho A$. You can calculate that however you want, by hook or by crook; there is no need to expand $\ket{\psi}$ in a basis corresponding to any particular observable, whether it's $A$ or any other.

 

[PeterDonis]

to derive this formula you need the assumption about the probabilities

Perhaps on your preferred interpretation you do; but as I understand it, the whole point of the thermal intepretation is that it doesn't require this; the expectation value formula for any operator and state is simply taken as a given and does not have the interpretation it normally has.

 

[vanhees71]

Ok, as a physicist I want to have some argument for why I choose such a definition. If you do pure math, of course you can state anything you like as "axioms". For me the relation to physics, i.e., to observations and measurements are lost.

The main point is that I don't understand the operational meaning of this "trace formula" according to @A. Neumaier, since he expclicitly doesn't want to interpret in the sense of probabilities at all. I never understood how I should think about the operational meaning of the formula instead within this new interpretation.

 

[A. Neumaier]

The main point is that I don't understand the operational meaning of this "trace formula" according to @A. Neumaier, since he expclicitly doesn't want to interpret in the sense of probabilities at all. I never understood how I should think about the operational meaning of the formula instead within this new interpretation.

I gave the following very clear operational meaning:

The law of large numbers implies that under the right conditions (sufficiently large sample of uncorrelated single data instances) the statistical (measured) expectation is arbitrarily close to the quantum (theoretical) expectation.

I elaborated on this operational meaning in much more detail in my paper

• A. Neumaier, Quantum tomography explains quantum mechanics, arXiv:2110.05294.