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I The thermal interpretation of quantum physics

  1. Feb 28, 2019 #1

    A. Neumaier

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    The official description of the thermal interpretation of quantum physics can be found in my just finished papers

    Foundations of quantum physics I. A critique of the tradition,
    Foundations of quantum physics II. The thermal interpretation,
    Foundations of quantum physics III. Measurement.

    They are also accessible through the arXiv at
    arXiv:1902.10778 (Part I), arXiv:1902.10779 (Part II), arXiv:1902.10782 (Part III).

    Three reviews (Part I, Part II, Part III) are on PhysicsOverflow, together with some comments by me.
     
    Last edited: Mar 15, 2019 at 9:50 AM
  2. jcsd
  3. Feb 28, 2019 #2
  4. Feb 28, 2019 #3

    A. Neumaier

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    Click on my picture and you'll find on the profile page (another click) an information button (another click) where you can find my web site with links to my publications.
     
    Last edited: Mar 3, 2019
  5. Feb 28, 2019 #4
    I couldn't find anything to arXive from your web
     
  6. Feb 28, 2019 #5

    Peter Morgan

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    Copy editing comment: On pages 17-18 of the first paper, there are a couple of *** notes to self ***, which perhaps you intend to be there or perhaps not.

    Your discussion of Born's rule is very interesting. I like it a lot. I have been working for some time, however, from a different version, which I suggest avoids many of the problems you list, albeit at the cost of adopting a philosophical position you may think too instrumental:
    1. An experiment or a sequence of experiments generate a (typically large or very large) list of raw experimental data that is stored in some kind of computer, which is strongly intersubjective if not objective in that if I display that experimental raw data on my computer screen I will very strongly expect to see exactly the same numbers, schematic diagrams, photographs of the apparatus, et cetera, as anyone else. A journal editor may well insist that they or their referee can read the data and confirm the claims made in a paper that describes the experiment(s).
    2. From this raw experimental data, I can construct summary statistics of any kind at all, using any mathematical operation.
    3. Given a very large list of summary statistics ##S_{ij}##, indexed by the preparation apparatus ##i## and by the measurement apparatus ##j##, I look for a set of density operators ##\hat\rho_i## and measurement operators ##\hat A_j## for which ##S_{ij}=\mathsf{Tr}[\hat\rho_i\hat A_j]##. If we choose the dimensionality of the Hilbert space large enough, we can always solve this set of linear differential equations. There may be constraints, which may be nonlinear, if, for example, a given summary statistic ##S_{ij}## is a mathematical function of other summary statistics or is a higher moment of the same raw experimental data.
    4. At any point we may introduce an idealization that converts the finite amount of data we have into effectively an infinite dataset, as would be the case if we were, say, to extrapolate a cubic spline approximation to a given data set. Some such idealizations will work better than others.
    5. There is also a need for engineering rules. That is, before we build an experiment, certainly after we have characterized its parts and built the whole apparatus, we want to predict what a given summary statistic of the raw experimental data will be. There's the problem and there's the inverse problem. This process, however, works as well as it does for pragmatic reasons and to a pragmatically decided number of standard deviations, not because QM/QFT is true, well-founded, or whatever.
    The point here is to start with the raw experimental data and work towards a model, whereas your account largely follows the common convention that the theory comes first (indeed, I discovered in Section 4.2 explicitly follows that convention). I only come to the theory at point 5, which can be whatever works, but finds something of a pragmatic justification in the previous points. If an 8-dimensional Hilbert space always works well enough for a given type of well-controlled quantum optics experiment, then we'll use it, even though we know that in the wild we might have to consider line widths, perhaps very complicated deviations of the laser beams we use away from a coherent state, et cetera.
    Perhaps I should note that this only implicitly suggests the idea that there are systems and subsystems. The engineering rules of QM may well find it worthwhile to think in such terms, but (anticipating your second paper) QFT, which I take to be much more a signal analysis formalism than is QM, is grounded in measurements associated with regions of space-time, not in there being objects occupying those regions, so it arguably has no such concept in principle.
    With a construction from the raw data of a set of operators ##\hat A_j##, we can use all the usual rules of linear algebra to derive the sample space and probability density associated with it in any given state ##\hat\rho_j##.

    Your discussion in Section 3.6 gets three cheers from me. I look forward to your second paper.

    Your Section 4.2 rather denies that I can construct QM/QFT in anything like the way I have, when it states "Thus measurement must be grounded in theory". I think I'm much more comfortable to suggest, as experimenters usually do, that there is a dialog between experiment and theory. Another difference arises in that there is no concept of nondestructive or destructive measurements: there is just experimental raw data, and, indeed, there are no systems to be destroyed.
    Section 4.3's discussion of "beables" also has a rather simple resolution: if we increase the dimensionality of the Hilbert space enough, any experimental raw data can be presented as a system of commuting measurement operators and diagonal density matrices. That's effectively to introduce ancillas and contextuality, which is not very helpful at all for engineering, but it can be done if we are determined to have beables.

    I'm looking forward to discovering what your "thermal interpretation" is, in your second paper. I decided I would comment on the first paper, then perhaps comment separately on the second and third, before I discovered I would have to wait until the second paper for that.
    I hope you won't mind me approaching your paper by introducing this contrasting discussion; if you do, say so and I won't comment further. Feel free, as well, not to respond at all to this comment: writing it out is its own reward. I'm very grateful for the impetus you gave me to write it out in response to your writing!
     
  7. Feb 28, 2019 #6

    Peter Morgan

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    I have to react immediately to this, at the head of the second paper: "Quantum physics is used to determine the behavior of materials made of specific molecules under changes of pressure or temperature, their response to external electromagnetic fields (e.g., their color), the production of energy from nuclear reactions, the behavior of transistors in the microchips on which modern computers run, and a lot more." That "determine" is too loud for me! Certainly QM is used to model all those cases, but it seems too much to prejudge whether the world is just as it is or is really determined by a few equations. This barely makes any difference to the instrumental level of physics, however, so let's suppose we don't differ over this.

    You may have noticed, but here I'll make explicit, that my discussion of experimental raw data allows summary statistics to be computed using any subset of the data. With experience we will come to know that a particular summary statistic of a particular subset of the data will be more useful or more interesting than some other statistic, or have some other merit. I see this as a plausible response, "it's whatever a physicist says it is, if they can convince other physicists to listen", to the problem you struggle with in Section 2.4 of the second paper, "What is an ensemble?", I think inconclusively.

    I find the lack of mention of the experimental raw data in your Section 3 of your second paper quite striking, which perhaps reaches its peak when you say "Statistics is based on the idea of obtaining information about noisy quantities of a system by repeated sampling from a population of independent systems with identical preparation". I would prefer to say something much more in the style of signal analysis: "Summary statistics of experimental raw data are used whenever the dynamics of a representation of the summary statistics is more tractable than the dynamics of the noisy experimental raw data". "Information about noisy quantities of a system" is already too far from the experimental raw data we really have, which is as much or more associated with a given measurement apparatus than with a measured system.

    Section 4.1:
    "2-point correlations in quantum field theory are effectively classical observables" is only true for 2-point VEVs at space-like separation. At time-like separation, there is an imaginary component.
    "it is impossible to repeat measurements" is not true in general. For a massive free quantum field, ##\hat\phi(f_1)## where the test function ##f_1(x)## has support near time ##t_1##, can be equivalent to the operator ##\hat\phi(f_2)## where the test function ##f_2(x)## has support near time ##t_2##, if the fourier transforms ##\widetilde{f_1}(k)## and ##\widetilde{f_2}(k)## have the same projection to the mass-shell, so the later measurement is formally identical to the earlier measurement. This may not be possible for interacting quantum fields, however the time-slice axiom (G in Haag's "Local Quantum Physics") is as much as to insist that it should be possible.

    Hurrah for Section 4.2!!! Oh yes, the near-field can and should be discussed!!! Furthermore, quantum non-demolition measurements can allow some aspects of the dynamics to be discussed almost as if the quantum field is classical, as I lay out in my "Classical states, quantum field measurement", arXiv:1709.06711, which was this week recommended for publication by a referee, but with small changes requested that I've now resubmitted.
     
  8. Mar 1, 2019 #7

    Demystifier

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    @A. Neumaier I have a question on paper II, page 34. What is ##p_{\nu}## in the covariant Schrodinger equation? (I didn't read the whole paper, so you can just pinpoint to the right part of the paper where it is explained.) I mean, if ##p_{0}## is the Hamiltonian, then what is the ##p_{i}## for ##i=1,2,3##?
     
  9. Mar 1, 2019 #8

    A. Neumaier

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    I know that my interpretation has no problem with the solar system (Subsection 3.4 of Part II). But my thermal interpretation is not the minimal statistical interpretation but one could call it the maximal nonstatistical interpretation since it is a completely deterministic (Subsection 2.1 of Part II) interpretation of everything (Section 5 of Part II).

    All probabilistic aspects of quantum mechanics are derived in the same way (Sections 4 and 5 of Part III) as classical probabilities are derived from classical mechanics.

    Equilibrium systems are not the only thermal systems - for example, all of fluid mechanics is thermal.
    But according to the minimal statistical interpretation,
    Thus if one takes your ''clear'' definition of the state at face value, the solar system has in the minimal interpretation no state (unless you are able to come up with an equivalence class of preparation procedures), and is thus outside the scope of minimally interpreted quantum mechanics. Only the microscopic degrees of freedom (which form a huge ensemble) are described by your minimal interpretation. Or not even these - since it would be difficult to come up with a single equivalence class of preparation procedures for these zillions of systems.

    Thus I should probably take your definition liberally and should not insist on preparation. Let me assume instead that pp.21-23 of your Lecture Notes on Statistical Physics
    are a faithful reflection of your views of the minimal interpretation. Thus the state of the solar system is supposed to be a ray in some Hilbert space, and the observables of the solar system are supposed to be certain self-adjoint operators on this Hilbert space. I have no idea how to prepare the solar system in the way you require, but let us perhaps assume that God did it. Then the sun qualifies as a quantum system according to your version of the minimal interpretation.

    Now let us consider some observable consequences. The effective temperature of the photosphere of the sun is surely an observable since Wikipedia gives for it the value of 5772 K and the value of 5777 K. They don't agree, so there seems to be a probability distribution of possible measurement results. But where is the associated self-adjoint operator that would allow me to apply your minimal interpretation? I cannot see how this squares with your version of the minimal interpretation? One needs to stretch your words quite a lot by adding much nonminimal stuff....
     
    Last edited: Mar 1, 2019
  10. Mar 1, 2019 #9

    A. Neumaier

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    They are defined on the page before as the generators of space translations. (I missed a factor of ##c## in the second displayed formula of p.34.)
     
  11. Mar 1, 2019 #10

    Demystifier

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    Can you write down the explicit expression for ##p_{\mu}##, e.g. for the free particle without spin?
     
  12. Mar 1, 2019 #11

    vanhees71

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    Obviously I haven't understood your interpretation then as you seem to mean it, as you start with standard notions of QT in the Hilbert-space formulation and then everything macroscopic is defined also in the usual way of quantum many-body physics. That's at least how I understand your bullet list at the beginning of this very Sect.

    For me this IS the minimal interpretation, and the more I think about the foundations, I come to the conclusion that this is all there is to QT. As long as there's no other more comprehensible theory than QT we have to live with this irreducible statistical aspects of nature as we preceive and comprehend it.

    Which just underlines, how I understood it. So I don't see where my mistake should be.

    Ok, maybe that's due to my embedding in the heavy-ion community, where "thermal" means "in or at least close to (local) thermal equibrium". Fluid mechanics in this sense is indeed thermal since it's precisely about systems that can be described as close to local thermal equilibrium.
    The complete state of the solar system is undescribable. That's why we use the corresponding statistical operator appropriate for the relevant macroscopic observables as, e.g., the classical positions and momenta of the planets and moons etc. as described by (post-)Newtonian celestial mechanics.
    Well, this is a very nice example. Of course, it's impossible to describe the Sun in all microscopic detail. Rather it's a good assumption to use a quantum-statistical description of the Sun in thermal equilibrium with the radiation pressure counterabalancing the gravitational force.
     
  13. Mar 1, 2019 #12

    A. Neumaier

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    I view any interpretation as inadequate that cannot account for the meaning of quantum physics at a time before any life existed.
    The universe exists for billions of years - in a sense to be explained by any interpretation that allows for quantum cosmology.
    But experimental data exist for a few thousands of years only.

    Well, how can you sensibly assert that a silver speck at a screen in a Stern-Gerlach experiment is the measurement of a particle with spin up, without having first a theory of how such a particle behaves when passing through a magnetic field?

    Well, already the Hilbert space of a harmonic oscillator has countably infinite dimension, Fock space too. Thus unless you want to work with nonseparable Hilbert spaces, you cannot increase the dimensiononality....
    Determine - this is the goal of ab initio quantum chemistry, realized to a large extent. One can determine ab initio the color of gold, the melting point of mercury, the equation of state of small molecules, etc., and only computer power seems to limit the extent and accuracy with which this can be done.
    The data produced in scattering experiments or experimental checks of Bell inequalites are heaps of noisy statistical raw data, from which a small number of reproducible (and hence scientifically relevant) data (cross sections, coincidence probabilities) are
    produced.
    But these measurement measure the field at a different time, hence measure different field operators.
     
  14. Mar 1, 2019 #13

    A. Neumaier

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    In the momentum representation of the Fock space, it is just multiplication of ##\psi(p_1,\ldots,\p_N)## by ##p_1+\ldots+p_N##.
     
  15. Mar 1, 2019 #14

    Demystifier

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    Isn't this a bit circular? To define the momentum operator ##p_{\mu}##, you must first know the momentum representation, as you defined above. But to know the momentum representation, you must first know what is the momentum operator.
     
  16. Mar 1, 2019 #15

    A. Neumaier

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    Well, Section 5 of Part III is the summary of what emerges from the whole paper. Of course, my interpretation is consistent with the shut-up-and-calculate part (the formal core given in Part I) and with everything done in quantum many-body physics!

    But it is derived from a completely deterministic dynamics without assuming Born's rule (which has only a restricted domain of validity).

    But what you wrote as foundation on p.21-22 of your statistical physics lecture note is far from what you now say the minimal interpretation is. It seems to me that in fact you really adhere to my thermal interpretation while only paying lipservice to your own formulation of the minimal interpretation.

    If you really understand it in this way, why then do you postulate Born's rule and probabilities in the very foundations?

    And all measurements are done by instruments in local thermal equilibrium....

    I have never seen a statistical operator for (post-)Newtonian celestial mechanics, which to me is pure classical mechanics. Could you please point me to a source?

    Even there, where is the self-adjoint operator associated to temperature that would allow me to apply the minimal interpretation and get a probability distribution for the measured temperatures?
     
    Last edited: Mar 1, 2019
  17. Mar 1, 2019 #16

    A. Neumaier

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    No. One starts with a definition of the Fock space in terms of single-particle momenta, and then defines the generators of translations as operators on Fock space. This is the usual building-up procedure.

    If you prefer to work with the position representation in Fock space, you can define the generators of spatial translations instead as the sum of the 1-particle momentum operators. This is the way it is usually done in statistical mechanics, e.g., in Linda Reichl's book.
     
    Last edited: Mar 1, 2019
  18. Mar 1, 2019 #17

    DarMM

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    @A. Neumaier , I have finished the papers and I am now beginning my closer second read through.
     
  19. Mar 2, 2019 #18

    vanhees71

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    Well, perhaps I'm too biased with the traditional interpretation, but precisely from the quoted Sec. 5 I came to the conclusion that your thermal interpretation is nothing else than what every physicist using QT (however you call his/her interpretation) understands under the formalism: The formalism predicts probabilistic properties of measurement outcomes when measuring a real object with a real measurement device. Since the papers consist of a lot of text with sparse use of formulae maybe it's not precisely clear to me what you really mean, because for instance I didn't get that you have a deterministic view point as the foundation.

    So let me summarize, how I understand your concept. The only difference between standard QM1 textbook treatments and your starting point is that instead of using the special case of pure states you start right away with the general case of states ("mixed states"). This is what I always understood as the state, because also for pure states it's way more convenient to use the projection operator to represent the state in terms of a stat. op. than as a ray in Hilbert space. So after the introductory heuristics has settled, for me (and as far as I see also for you according to your 3 papers) the undisputable mathematical formalism is as follows (everywhere, where you write "hermitian", I think one should read "self-adjoint" to be on the safe side, but that's a formality):

    Kinematical part:

    (1) There's a (separable) Hilbert space associated with the system to be described within QT. Time is a real parameter.
    (2) The state of the is represented by a self-adjoint positive semidefinite operator with trace 1, ##\hat{\rho}(t)##.
    (3) Any observable is represented by a self-adjoint operator.
    (4) Possible outcomes of precise measurements of an observable are the spectral values of the corresponding self-adjoint operator.
    (5) The expectation values of any observable ##A##, represented by the self-adjoint operator ##\hat{A}(t)## is given by ##\langle A(t) \rangle=\mathrm{Tr}[\hat{\rho}(t) \hat{A}(t)]##.

    BTW. I'd call (5) Born's rule, while you seem to restrict the notion to apply only to the special case and the proabilities (or probability distributions) for pure states, but that's semantics.

    Dynamical part (for not explicitly time-dependent observables as in your papers):

    (6) For each system the dynamics is governed by an observable ##H##, the Hamiltonian of the system.
    (7) For any observable operator ##\hat{A}(t)## the operator describe the time derivative of this observable is given by
    $$\mathring{\hat{A}}(t)=\frac{1}{\mathrm{i} \hbar} [\hat{A}(t),\hat{H}(t)].$$

    Note that in (7) ##\mathring{\hat{A}}## is not the time derivative of ##\hat{A}## (except in the Heisenberg picture).

    The rest should follow from these axioms, among other things also Ehrenfest's theorem, which seems to be the key to your interpretation, and maybe that's the point, I don't understand correctly. Obviously you define a Lie algebra implying Lie derivatives on an abstract algebra of observables and then reconstruct the above postulates from them. On the other hand at least your notation suggests that what's meant as observables (or in Bell's language "beables") are the expectation values in the above QT sense.

    Do you have a paper with more math and less text that shows the derivation from the Lie algebra to the Hilbert space formulation, so that I can follow the logic better?

    Another, maybe much more difficult, question is, whether one can use these concepts to teach QM 1 from scratch, i.e., can you start by some heueristic intuitive physical arguments to generalize the Lie-algebra approach of classical mechanics in terms of the usual Poisson brackets of classical mechanics? Maybe that would be an alternative approach to QM which avoids all the quibbles with starting with pure states and then only finally arrive at the general case of statistical operators as description of quantum states?
     
  20. Mar 2, 2019 #19

    A. Neumaier

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    Only in the sense stated in Part III on p.51, in the paragraph directly after the list of bullets:
    But the formalism of the thermal interpretation is completely deterministic, with a conservative dynamics for the collection of all q-expectations. Itproduces statistical results only in its coarse-grained approximations where the dynamics is (as in all practical applications) reduced to a collection of relevant q-expectations.
    This is stated explicitly at the top of Subsection 3.3 of Part II, where the discussion of the statistical aspects begins.
    n
    No. q-expectations are defined and real for any Hermitian linear operator. Self-adjointness is needed only for the spectral theorem, i.e., when referring to the spectrum or spectral projections.
    Separability is nowhere needed. In fact, interacting quantum field theories typically need nonseparable Hilbert spaces. This can be most easily seen for a simple model, the relativistic massless scalar field in 1+1 dimensions.
    In the thermal interpretation, any observable (though I avoid this word) is represented by a function of q-expectations.
    In the thermal interpretation, this holds only for exact measurements of observables defined by self-adjoint operators, i.e., those where the theoretical uncertainty vanishes.
    In Part I, five different forms of Born's rule are distinguished. The universal from looks like (5), but is explicitly related to a mean of measurement results over a large sample. For q-expectations without this measurement interpretation, (5) implies no connection to reality, hence is not an interpretation statement but a formal definition of what to call a q-expectation. The thermal interpretation of these is as beables,
    that can be approximated by measurement results within the limits given by the uncertainty, as defined in eq. (15) of Section 2.4 of Part II. On the other hand, Born's rule in the minimal statistical interpretation requires (by its derivation from your postulates in your statistical mechanics lecture notes) that the q-expectation is a mean of a large number of actual measurement results.
    Section 2.1 has a large ratio of formulas to text, and explains the Ehrenfest picture in full detail. The Ehrenfest dynamics for expectations is clearly deterministic.
    The q-expectations in the formal sense, yes, but without the interpretation as sample means.
    These are questions quite different from the ones an interpretation of quantum mechanics has to address. The answer to both questions is yes. This (among other things) will be discussed in Part IV, which answers the critique from Section 5.2 of Part I and gives a coherent synthesis. It exists in draft form but is not yet ready for making it public. Please wait a few more weeks....
     
    Last edited: Mar 3, 2019
  21. Mar 3, 2019 #20

    vanhees71

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    Well, that's the important point, why I think it should be "self-adjoint". If the spectral theorem is not valid, there's no sensible probability interpretation, at least not in the usual sense, and not considering the stricter condition of self-adjointness and sloppily dealing with Hermitean operators as if they were self-adjoint leads to misconceptions and misunderstandings (e.g., in the apparently simple infinite-potential-well model when momentum instead of energy eigenvectors are discussed although there's no self-adjoint momentum operator defined).

    Intersting, but doesn't one then run into the trouble with Haag's theorem, which however is of little practical relevance since it only occurs if not regularizing the model by introducing a finite quantization volume. I'm well aware of the fact that from a strictly mathematical point of view there's no proof for the existence of realistic QFTs. For (resummed) perturbative physicists' prescriptions, it's however enough to deal with the models in this non-strict way.

    But then the question is, how this representation of expectation values is defined, and this has to be also given operationally. At least this point has been clarified a lot in recent years concerning the standard interpretation of QT with generalizing the idealized von Neumann measurements to the description of real-world experiments in terms of the POVM formalism.

    For the same reason, I've still no clue what's behind the q-bism interpretation. They claim that the probabilities of QT have meaning for single realizations of an experiment but never give an operational definition of what's measured if nothing is averaged (neither in Gibbs's sense nor in the coarse-graining sense discussed above).

    Yes, indeed. That's also the case for traditionally minimally interpreted QT, and it's the starting point for understanding the theory as a physical theory to begin with. First one has to understand the most simple cases to understand the meaning of an interpretation.

    The Stern-Gerlach experiment is a very good example for that. It can be treated analytically and exactly for Gaussian wave packets with using the approximate Hamiltonian
    $$\hat{H}=\frac{\hat{\vec{p}}^2}{2m} + g_s \mu_B (\vec{B}_0 + \beta z )\hat{s}_z$$
    which leads via the dynamics of the Ag atom to an strictly space-##s_z##-entangled state which lets you filter out the definite ##s_z=\pm \hbar/2##-states.

    This is however approximate since the magnetic field close to the beam in fact is approximated by $$\vec{B}=\vec{B}_0 + \beta (z \vec{e}_z-y vec{e}_y)$$, and so far I could treat the "perturbation"
    $$\hat{V}=-g_s \mu_B \beta y \hat{s}_y$$
    only perturbatively, which leads of course to (small) mixing of the "wrong" ##s_z##-states into the regions which are however still approximately pure ##s_z## states.

    I don't see, what's lacking with the standard minimal interpretation in this case since it predicts the outcome of measurements, and to do the experiment properly you need some amount of Ag atoms in the beams to accumulate "enough statistics" to be able to see the splitting at all.

    So, how would the analogous calculation work with the thermal representation. Since the Hamiltonian is only maximally quadratic in the observables, this must be a pretty simple thing for the thermal interpretation since the Ehrenfest equations of motion for the expectation values are of course just the classical equations of motion for the classical Hamiltonian motion of an uncharged particle with magnetic moment in an as simple as possible approximate magnetic field applicable to the fine beams prepared in the typical textbook experiment.

    Yes, 2.4 is precisely why I was misunderstanding your interpretation as being in fact the usual (minimal) interpretation, because you argue with classical phase-space distributions. For me that's already a coarse-grained description, approximating the one-body Wigner functions of many-body systems via the gradient expansion of the corresponding Kadanoff-Baym equations. This is the formal description of an "ensemble average" in the sense that one averages over the mircocopic fluctuations by just "blurring" the observation to the accuracy/resolution of typical macroscopic time and space scales, and thus "averaging" over all fluctuations at the microscopic space-time scales. Of course you don't need to take "ensemble average" in Gibbs's sense literally here. Otherwise we'd never ever have observed classical behavior of single macroscopic (many-body) systems to begin with.
    Yes, and obviously I misinterpreted this section in thinking that, despite the somewhat unusual notation, you just describe usual quantum-theoretical averages. There are however no details given, how one deals with the fact that of course for functions of averages in genral you have ##f(\langle A \rangle) \neq \langle f(A) \rangle##. Maybe that's the reason, why I didn't understand the fact that you consider this Lie-algebra formalism for expectation values as the fundamental set postulates, because I always thought you'd need the quantum formalism to define expectation values to begin with. For me expectation values are given by the above quoted trace formula, and as you say, that's not different in your paper I.
    Great! I'm looking forward to it.
     
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