I The thermal interpretation of quantum physics

[A. Neumaier]

I think Dr Neumaier has a good point - QFT may indeed be a better place for interpretations. I do not know enough of his thermal interpretation to comment on its specifics.

A complete description of the thermal interpretation of quantum physics can be found in my just finished papers (for the bare bones, see Section 2.5 of Part II)

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). The simplest quantum system, a qubit, was already described by Stokes 1852, in terms essentially equivalent to the thermal interpretation.

This is a very long thread. DarMM gave in post #268 a nice summary of the thermal interpretation. Post #484 is my very short summary and post #479 contains links to my explanations of the connections between theory and experiments. Open problems related to the thermal interpretation are discussed in post #293.

An additional fourth part (from April 27, 2019) is announced here, and a fifth part (from May 2, 2019) completing the series is here. Reviews (Part I, Part II, Part III, Part IV, Part V) are on PhysicsOverflow, together with some comments by me. The book

• A. Neumaier, Coherent Quantum Physics: A Reinterpretation of the Tradition, de Gruyter, Berlin 2019.

gives a polished exposition based on these papers.

Related threads are about the thermal interpretation of the Stern-Gerlach experiment and the double-slit experiment, about the derivation of atomic and molecular spectra in the thermal interpretation, about the relation to decoherence (which gives an averaged description of the description of measurements given by the thermal interpretation), and about an example illustrating the differences in the interpretation of measurement results in the thermal interpretation and in Born's statistical interpretation. Another thread is about an attempt to disprove a misunderstood version of the thermal interpretation.

 

[Sandeep T S]

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 will be on the arXiv tomorrow.

What is your 'author' name in arXive.
Can you share link to your early published work

 

[A. Neumaier]

Can you share link to your early published work

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.

 

[Sandeep T S]

Are you a qualified endorser in arXive Quantum physics or general Physics. Can you share any link to your arXive publication.

I couldn't find anything to arXive from your web

 

[Peter Morgan]

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!

 

[Peter Morgan]

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.

 

[Demystifier]

@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 $i=1,2,3$?

 

[A. Neumaier]

How does QT in the minimal interpretation describe the state of the solar system?

We only have a single realization of the solar system, which has been prepared once in ancient times.
Hence we cannot apply rules that require a large ensemble of similarly prepared systems.

You should refer to your own interpretation, which you call the "thermal interpretation"! It perfectly describes how quantum theory is applied by usual physicists of all kind.

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.

Your own interpretation (I'd somehow rename, because it goes far beyond thermal, i.e., equilibrium systems)

Equilibrium systems are not the only thermal systems - for example, all of fluid mechanics is thermal.

Essential use is made of the fact that everything physicists measure is measured in a thermal environment for which statistical thermodynamics is relevant. This is reflected in the characterizing adjective 'thermal' for the interpretation.

The "solar system", described by macroscopic physics [...] is an emergent phenomenon, and what we observed are grossly coarse-grained macroscopic observables which average at any macroscopic moment of time over zillions of quantum- and thermally fluctuating microscopic degrees of freedom.

But according to the minimal statistical interpretation,

a state is represented by the statistical operator and operationally as an equivalence class of preparation procedures. That's an objective notion of state since a preparation procedure is clearly defined

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

The state of a quantum system is described completely by a ray in a Hilbert space [...] The observables of the system are represented by self-adjoint operators. [...] A possible result of a precise measurement of the observable O is necessarily an eigenvalue of the corresponding operator O. [...] the probability to find the value o when measuring the observable O is given by $P_\psi(o)$. [...] This preparation of a system is possible by performing a precise simultaneous measurement of a complete complete set of observables.

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...

 

[A. Neumaier]

@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 $i=1,2,3$?

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.)

 

[Demystifier]

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.)

Can you write down the explicit expression for $p_{\mu}$, e.g. for the free particle without spin?

 

[vanhees71]

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).

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.

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.

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

Equilibrium systems are not the only thermal systems - for example, all of fluid mechanics is thermal.

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.

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.

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.

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. Thus 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...

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.

 

[A. Neumaier]

at the cost of adopting a philosophical position you may think too instrumental:

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.

"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.

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?

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.

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...

"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!

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.

"Information about noisy quantities of a system" is already too far from the experimental raw data we really have

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.

so the later measurement is formally identical to the earlier measurement.

But these measurement measure the field at a different time, hence measure different field operators.

 

[A. Neumaier]

Can you write down the explicit expression for $p_{\mu}$, e.g. for the free particle without spin?

In the momentum representation of the Fock space, it is just multiplication of $\psi(p_1,\ldots,\p_N)$ by $p_1+\ldots+p_N$.

 

[Demystifier]

In the momentum representation of the Fock space, it is just multiplication of $\psi(p_1,\ldots,p_N)$ by $p_1+\ldots+p_N$.

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.

 

[A. Neumaier]

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.

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).

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.

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.

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.

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

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

Ok, maybe that's due to my embedding in the heavy-ion community, where "thermal" means "in or at least close to (local) thermal equilibrium". Fluid mechanics in this sense is indeed thermal since it's precisely about systems that can be described as close to local thermal equilibrium.

And all measurements are done by instruments in 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.

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?

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.

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?

 

[A. Neumaier]

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.

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.

 

[DarMM]

@A. Neumaier , I have finished the papers and I am now beginning my closer second read through.

 

[vanhees71]

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?

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?

 

[A. Neumaier]

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:

Only in the sense stated in Part III on p.51, in the paragraph directly after the list of bullets:

The thermal interpretation is inspired by what physicists actually do rather than what they say. It is therefore the interpretation that people work with in the applications [...], rather than only paying lipservice to it.

The formalism predicts probabilistic properties of measurement outcomes when measuring a real object with a real measurement device.

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.

I didn't get that you have a deterministic view point as the foundation.

This is stated explicitly at the top of Subsection 3.3 of Part II, where the discussion of the statistical aspects begins.

(everywhere, where you write "hermitian", I think one should read "self-adjoint" to be on the safe side, but that's a formality):

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.

(1) There's a (separable) Hilbert space associated with the system to be described within QT. Time is a real parameter.

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.

(3) Any observable is represented by a self-adjoint operator.

In the thermal interpretation, any observable (though I avoid this word) is represented by a function of q-expectations.

(4) Possible outcomes of precise measurements of an observable are the spectral values of the corresponding self-adjoint operator.

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.

(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 probabilities (or probability distributions) for pure states, but that's semantics.

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.

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.

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.

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.

The q-expectations in the formal sense, yes, but without the interpretation as sample means.

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? [...]
can you start by some heuristic intuitive physical arguments to generalize the Lie-algebra approach of classical mechanics in terms of the usual Poisson brackets of classical mechanics?

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...

 

[vanhees71]

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.

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).

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.

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.

In the thermal interpretation, any observable (thoug I avoid this word) is represented by a function of q-expectations.

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).

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.

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 let's 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.

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 wihout this measurement interpretation, (5) implies no connection to reality,
hence is not an interpretation statement but a 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.

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.

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 deerministic.

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 general 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.

The q-expectations in the formal sense, yes, but wihout the interpretation as sample means.

Yes, yes. This will be in Part IV, which answers the remaining critique from 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...

Great! I'm looking forward to it.

 

[A. Neumaier]

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

Yes, but the expectation can still be defined via the trace. That's why the probability interpretation is secondary. In the thermal interpretation it is not assumed but (under additional assumptions) derived.

Interesting, but doesn't one then run into the trouble with Haag's theorem

It is needed in rigorous interactive relativistic QFT precisely because of Haag's theorem. One cannot use a Fock space (which is separable) but needs a nonseparable Hilbert space with superselection rules for unitarily inequivalent representations of the field algebra. I recommend studying the massless free scalar field in 1+1 dimensions, which is an exactly solvable toy example where the problem of superselection rules appears already in the free case.

But then the question is, how this representation of expectation values is defined, and this has to be also given operationally.

The definition of q-expectation is given by eq. (14) of Part II, and involves neither probabilities nor eigenvalues nor the spectral theorem. It works even for nonhermitian operators such as creation and annihilation operators, and is meaningful there. An operational meaning has to be given only for those q-expectations which actually correspond in a simple way to measurements (which is a small number of all possible ones). The operational meaning is therefore dependent on what you measure and how you do it. It is based on the general uncertainty principle (GUP) from page 15 of Part II, and made more concrete by the measurement principle (MP) on p.6 of Part III. For equilibrium thermodynamics, we may use the principle introduced by H.B. Callen in his famous textbook ''Thermodynamics and an introduction to thermostatistics'',

Operationally, a system is in an equilibrium state if its properties are consistently described by thermodynamic theory.

This implies that q-expectations of 1-particle operators and energy in QFT are measured by the established techniques of equilibrium thermodynamics. This is fully operational, using single measurements only, without any need to check whether the operator is self-adjoint (which is difficult to verify), and without ever having mentioned probabilities. Callen formulated the principle for thermodynamics, but its analogue is valid everywhere in scientific modeling:

• Callen's principle: Operationally, a system is in a given state if its properties are consistently described by the theory for this state.

This together with (GUP) and (MP) is enough to find out in each single case how to measure a q-expectation.

q-probabilities are special cases of q-expectations, namely those of self-adjoint Hermitian operators with spectrum in $[0.1]$. The weak law of large numbers, discussed in Subsection 3.3 of Part II applies to probabilities and gives operational recipes for measuring them in certain cases, see Subsection 3.5 of Part II; in particular, Born's probabilistic rule follows for ideal binary measurements; see Subsection 3.4 of Part II.
Thus, in the thermal interpretation, Born's rule is not assumed but derived (where appropriate)!

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.

See Subsection 2.5 of Part III for the ease with which POVMs arise in the thermal interpretation. Instead, in an approach based on statistical mechanics lecture notes (where POVM do not even appear, though they account for many more measurements than the ideal ones you formalize with your version of Born's rule!) one would need for the justification of POVMs as measurements a very unnatural exptension of the physical Hilbert space by adding a fictitious ancilla degrees of freedom.

The Stern-Gerlach experiment is a very good example for that. [...] So, how would the analogous calculation work with the thermal representation.

Maybe I'll discuss this in Part IV.

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.

For how the thermal interpretation views coarse-grained descriptions see Section 4.2 of Part III. One takes a subspace of q-expectations - in case of the Kadanoff-Baym equations the subspace of field expectations and pair correlation functions - and derives an approximate closed dynamics for these.
The approximation is in the equations of motion, not in the meaning of the expectations! Nowhere in the derivation of the Kadanoff-Baym equations is a need to interpret the q-expectations as ensemble averages. [Over which ensemble?? People successfully apply the equations to the early universe, of which there is only one observable case, not an ensemble; the ensembles are pure imaginations without any operational content, as discussed in Subsection 2.4 of Part II!]
Neither do the derivations (see, e.g., the slides of your 2002 lecture) invoke any observations or any average over fluctuations. Instead just a gradient expansion! Everything you do in your slides is shut-up-and-calculate - no probabilities, no observations, no measurements, no operational recipes for observing the bilocal field correlations. You just worked out approximate formulas for the dynamics of some q-expectations, without ever entering the interpretation of these q-expectations! And you end up with something where the q-expectations have a semiclassical, nonstatistical interpretation - as the thermal interpretation requires, without invoking any of the ghosts of the past that you invoked!

Yes, and obviously I misinterpreted this section in thinking that, despite the somewhat unusual notation, you just describe usual quantum-theoretical averages.

I describe indeed usual q-expectations, except that I don't interpret them probabilistically as averages, since this is unnecessary baggage that caused nearly a century of unsettled dispute. This makes all the difference:

Probability is not an irreducible input to quantum mechanics but a consequence of not being able to make observations at arbitrary scales: one needs to introduce approximations, and in many (but not all) cases, these approximation introduce stochastic aspects. Just as the classical dynamics of a bistable system coupled to a bath becomes in some approximation a binary stochastic hopping process. That this analogy is not far-fetched is discussed in Section 5.1 of Part III.

 

[akhmeteli]

@A. Neumaier: Just a dumb question: in your references, what does the number in the square brackets at the end of a citation mean (for example:
[1] A.E. Allahverdyan, R. Balian and T.M. Nieuwenhuizen, Understanding quantum measurement from the solution of dynamical models, Physics Reports 525 (2013), 1–166. https://arxiv.org/abs/1107.2138 [16])? So what does this [16] mean? Maybe you explain it somewhere, but I have not found such an explanation so far.

 

[A. Neumaier]

in your references, what does the number in the square brackets at the end of a citation mean.

It refers to the page(s) where the reference is cited.

 

[akhmeteli]

It refers to the page(s) where the reference is cited.

Thank you very much!

 

[akhmeteli]

@A. Neumaier: A quote from your work: " When a particle has been prepared in an ion trap (and hence is there with certainty), Born’s rule implies a tiny but positive probability that at an arbitrarily short time afterwards it is detected a light year away"

I guess this is only true if one assumes nonrelativistic equation of motion? (The argument about Newton-Wigner does not seem clear).

 

[A. Neumaier]

@A. Neumaier: A quote from your work: " When a particle has been prepared in an ion trap (and hence is there with certainty), Born’s rule implies a tiny but positive probability that at an arbitrarily short time afterwards it is detected a light year away"

I guess this is only true if one assumes nonrelativistic equation of motion? (The argument about Newton-Wigner does not seem clear).

It is true in quantum mechanics, not in quantum field theory. Note that quantum mechanics has no consistent relativistic particle picture, except in the free case. Thus an atom in an ion trap cannot be consistently modeled by (fully) relativistic quantum mechanics.

But for a free particle, if one would know the position at one time to be located in a small compact region of space, it could be the next moment almost everywhere with a nonzero probability.

 

[akhmeteli]

It is true in quantum mechanics, not in quantum field theory. Note that quantum mechanics has no consistent relativistic particle picture, except in the free case. Thus an atom in an ion trap cannot be consistently modeled by (fully) relativistic quantum mechanics.

Do I understand correctly that, say, the Dirac equation is not adequate to model an atom in a trap (say, if you model both the nucleus and the electrons using the Dirac equation)? Or do you think that one needs to use the Newton-Wigner position operator in that case, rather than the standard position operator?

 

[A. Neumaier]

Do I understand correctly that, say, the Dirac equation is not adequate to model an atom in a trap (say, if you model both the nucleus and the electrons using the Dirac equation)? Or do you think that one needs to use the Newton-Wigner position operator in that case, rather than the standard position operator?

Even for the free Dirac equation one needs the Newton-Wigner position to get a valid dynamics.

Next, if you model the atom as a whole by a free Dirac equation and the ion trap by adding some potential, the positive and negative energies get strangely mixed and cause havoc with a probabilistic interpretation or with energy, which is no longer bounded below. Filling the holes, which can be done exactly in the free case is then only an approximate procedure without strict logical sense.

Finally, if you model the atom as a multiparticle system, all is lost from the start because there is no multiparticle Dirac equation with a reasonable behavior!

 

[akhmeteli]

Even for the free Dirac equation one needs the Newton-Wigner position to get a valid dynamics.

Next, if you model the atom as a whole by a free Dirac equation and the ion trap by adding some potential, the positive and negative energies get strangely mixed and cause havoc with a probabilistic interpretation or with energy, which is no longer bounded below. Filling the holes, which can be done exactly in the free case is then only an approximate procedure without strict logical sense.

Finally, if you model the atom as a multiparticle system, all is lost from the start because there is no multiparticle Dirac equation with a reasonable behavior!

So it looks like the statement I quoted is indeed true only for nonrelativistic quantum mechanics, if the atom in a trap cannot be modeled by relativistic quantum mechanics, without using quantum field theory.

Another thing. I agree that "standard quantum mechanics defined by the Schrödinger equation" violates the assumptions of Bell type theorems, but this seems obvious, as the Schrödinger equation is manifestly nonlocal, as far as "Bell type theorems" are concerned (it allows arbitrarily high velocities).

 

[*now*]

I’ve enjoyed a quick glance, look forward to a longer look and to the next part. If of any consequence a small typo was noted, “life”, last sentence of 2.2, page 10, Part II.

 

[akhmeteli]

@A. Neumaier: "Points 4 and 5 also show that at finite times (i.e., outside its use to interpret asymptotic S-matrix elements), Born’s rule cannot be strictly true in relativistic quantum field theory, and hence not in nature."

So it seems that you fault the Born's rule for what is actually the Schrödinger equation's fault. The Born's rule is no relative of mine, but this doesn't look like a strong point of your critique of the Born's rule. Maybe you should use the results of Allahverdyan et al., who show that the Born's rule is only approximately correct?

 

[Demystifier]

@A. Neumaier , I have finished the papers and I am now beginning my closer second read through.

Good, I hope you will present us an unbiased summary with advantages and disadvantges in comparison with other interpretations.

 

[A. Neumaier]

Good, I hope you will present us an unbiased summary with advantages and disadvantges in comparison with other interpretations.

My obviously biased comparison is in Section 5 of Part III.

 

[Demystifier]

My obviously biased comparison is in Section 5 of Part III.

A typo: At page 51 (part III) you have the item
"•has no split between classical and quantum mechanics – the former emerges naturally as the macroscopic limit of the latter;"
twice.

And by the way, your thermal interpretation has nothing to do with temperature, am I right?

 

[vanhees71]

Yes, but the expectation can still be defined via the trace. That's why the probability interpretation is secondary. In the thermal interpretation it is not assumed but (under additional assumptions) derived.

But expectation values are defined via the probabilities and vice versa. The probablities are also special expectation values like, e.g., for the position of a particle
$$P(\vec{x})=\langle \delta(\vec{x}-\vec{X}) \rangle.$$
To evaluate this expectation value from the statistical operator within the standard mathematical QT formalism you need the spectral theorem and the "generalized eigenvectors"
$$P(\vec{x})=\mathrm{Tr} (\hat{\rho} \delta(\vec{x} -\hat{\vec{x}}))=\rho(\vec{x},\vec{x}) \quad \text{with} \quad \rho(\vec{x},\vec{x}')=\langle \vec{x} |\hat{\rho}|\vec{x}' \rangle.$$

It is needed in rigorous interactive relativistic QFT precisely because of Haag's theorem. One cannot use a Fock space (which is separable) but needs a nonseparable Hilbert space with superselection rules for unitarily inequivalent representations of the field algebra. I recommend studying the massless free scalar field in 1+1 dimensions, which is an exactly solvable toy example where the problem of superselection rules appears already in the free case.

So far, I was very pragmatic with regard to Haag's theorem. I just introduced a quantization volume to regularize the mathematically meaningless abusive treatment of distributions and then take the limit at the appropriate point in the calculation. Physically there's anyway never an infinite space available since so far even the LHC is a device occupying lab space of finite volume ;-)).

The formal treatment of exact toy models of interacting fields in lower space-time dimensions is very interesting. I know the Schwinger model (massless scalar QED in (1+1) dimensions), but only in the usual nonrigorous physicists' treatment and without the use of superselection rules. Do you have a reference for some interacting model like that, where this is done rigorously?

The definition of q-expectation is given by eq. (14) of Part II, and involves neither probabilities nor eigenvalues nor the spectral theorem. It works even for nonhermitian operators such as creation and annihilation operators, and is meaningful there. An operational meaning has to be given only for those q-expectations which actually correspond in a simple way to measurements (which is a small number of all possible ones). The operational meaning is therefore dependent on what you measure and how you do it. It is based on the general uncertainty principle (GUP) from page 15 of Part II, and made more concrete by the measurement principle (MP) on p.6 of Part III. For equilibrium thermodynamics, we may use the principle introduced by H.B. Callen in his famous textbook ''Thermodynamics and an introduction to thermostatistics'',

But (II.14) is the standard definition with the usual trace. I still don't see, how you even evaluate this for concrete observables without the spectral theorem, let alone how you understand its application to real measurements in a lab. As I said, I cannot understand the interpretation, before this is clearly stated. It's not necessary to state it in general, concrete (preferrably the most simple) examples are enough. I'd try to mathematically and operationally to define how you interpret Stern-Gerlach experiments (as I summarized in my previous posting) as an example for non-relativistic QT for the most simple case of an (even idealized) von Neumann filter measurement

This implies that q-expectations of 1-particle operators and energy in QFT are measured by the established techniques of equilibrium thermodynamics. This is fully operational, using single measurements only, without any need to check whether the operator is self-adjoint (which is difficult to verify), and without ever having mentioned probabilities. Callen formulated the principle for thermodynamics, but its analogue is valid everywhere in scientific modeling:

• Callen's principle: Operationally, a system is in a given state if its properties are consistently described by the theory for this state.

This together with (GUP) and (MP) is enough to find out in each single case how to measure a q-expectation.

I think, the key of my trouble is that I don't understand, how to decide Callen's principle without the usual operational definitions of "properties". For a thermal system (i.e., a system in equilibrium) it's clear how to define the thermodynamical properties operationally, i.e., how to measure temperature, chemical potential(s), pressure etc. observables, using the adequate devices to do so. However, you don't give an operational definition. You give the mathematics of expectation values in terms of the state-of-the-art version of Born's rule (using the trace formula quoted above), but at the same time claim that you have a deterministic interpretation of this procedure by introducing some Lie algebra acting on an algebra built by functions on these expectation values (in analogy to the symplectic structure of classical phase space with the Poisson bracket defining Lie derivatives). This doesn't provide an operational definition of the averages though. In the standard interpretation, applied to thermal equilibrated systems, there's a clear definition: You use a thermometer in thermal contact with the measured system in equilibrium with the system, providing a reading of the temperature like the height of a Hg column in an old-fashioned thermometer. How is this simple example to be understood within your thermal interpretation?

q-probabilities are special cases of q-expectations, namely those of self-adjoint Hermitian operators with spectrum in $[0.1]$. The weak law of large numbers, discussed in Subsection 3.3 of Part II applies to probabilities and gives operational recipes for measuring them in certain cases, see Subsection 3.5 of Part II; in particular, Born's probabilistic rule follows for ideal binary measurements; see Subsection 3.4 of Part II.
Thus, in the thermal interpretation, Born's rule is not assumed but derived (where appropriate)!

As I said above, this is the very point I misunderstood in reading your papers. For me you use Born's rule to define the entire formalism and then claim you rederive it from these definitions. For me that's circular. Otherwise your thermal interpretation, is completely in accordance with how the QT formalism is used in physicists' lab practice.

See Subsection 2.5 of Part III for the ease with which POVMs arise in the thermal interpretation. Instead, in an approach based on statistical mechanics lecture notes (where POVM do not even appear, though they account for many more measurements than the ideal ones you formalize with your version of Born's rule!) one would need for the justification of POVMs as measurements a very unnatural exptension of the physical Hilbert space by adding a fictitious ancilla degrees of freedom.

Maybe I'll discuss this in Part IV.

The POVM is not in contradiction with the standard fundamentals of Born's rule, but just an extension of the description of measurement protocols generalizing the special case of idealized von Neumann measurements which are indeed only rare cases, applicable to very simple few-particle systems. It's of course of some merit to give a more natural derivation of this extension than the usual one (as e.g., given in Asher Peres's nice textbook).

For how the thermal interpretation views coarse-grained descriptions see Section 4.2 of Part III. One takes a subspace of q-expectations - in case of the Kadanoff-Baym equations the subspace of field expectations and pair correlation functions - and derives an approximate closed dynamics for these.
The approximation is in the equations of motion, not in the meaning of the expectations! Nowhere in the derivation of the Kadanoff-Baym equations is a need to interpret the q-expectations as ensemble averages. [Over which ensemble?? People successfully apply the equations to the early universe, of which there is only one observable case, not an ensemble; the ensembles are pure imaginations without any operational content, as discussed in Subsection 2.4 of Part II!]

Your notion of "ensemble" is too narrow. Particularly for this case, it's not an abstract ensemble a la Gibbs. The Kadanoff-Baym equations are exact equations derived from the 2PI formalism. There are no approximations so far. Of course, it's impossible to solve them (even to state them) without further approximations. One is to use the gradient expansion, which is a coarse-graining procedure leading to semiclassical transport equations (in the most simple case to the standard BUU equations). The "averaging" is here done quite implicitly using this coarse-graining in terms of the gradient expansion, but nowhere is the idea of a Gibbs ensemble envoked. You just interpret the coarse-grained Wigner function as a classical phase-space distribution function, which is consistent as long as the coarse graining is coarse enough to guarantee the positive semidefiniteness of the so defined phase-space distribution functions.

Neither do the derivations (see, e.g., the slides of your 2002 lecture) invoke any observations or any average over fluctuations. Instead just a gradient expansion! Everything you do in your slides is shut-up-and-calculate - no probabilities, no observations, no measurements, no operational recipes for observing the bilocal field correlations. You just worked out approximate formulas for the dynamics of some q-expectations, without ever entering the interpretation of these q-expectations! And you end up with something where the q-expectations have a semiclassical, nonstatistical interpretation - as the thermal interpretation requires, without invoking any of the ghosts of the past that you invoked!

In this talk, there's no gradient expansion. It's a formal thing how to renormalized self-consistent 2PI approximations in thermal equilibrium. Of course the observable quantities are given as the usual QFT expectation values of the corresponding correlation functions (e.g., the em. current-current correlation functions to calculate dilepton-production rates in heavy-ion collisions assuming (local) thermal equilibrium of the source (QGP, hadron-resonance gas) or the electric conductivity via Green-Kubo). Of course, I use the clearly defined notion of expectation values, which is the trace formula you simply don't call Born's rule. As I said, I still did not understand what the averages are within your thermal interpretation if there is no alternative definition to "the ghosts of the past".

I describe indeed usual q-expectations, except that I don't interpret them probabilistically as averages, since this is unnecessary baggage that caused nearly a century of unsettled dispute. This makes all the difference:

Probability is not an irreducible input to quantum mechanics but a consequence of not being able to make observations at arbitrary scales: one needs to introduce approximations, and in many (but not all) cases, these approximation introduce stochastic aspects. Just as the classical dynamics of a bistable system coupled to a bath becomes in some approximation a binary stochastic hopping process. That this analogy is not far-fetched is discussed in Section 5.1 of Part III.

You always say what the q-expectations are not. You have to say what they are, if it's forbidden to interpret them probabilistically. If there's a deterministic theory, I don't see, where I need expectation values to begin with. Of course, in practice, it's impossible to describe even in a classical picture the entire microscopic details of a real-world condensed-matter system, but within classical physics there's no need for expectation values at all in principle. That's not the case in the standard interpretation of QT, where the whole meaning of the formalism is based on probability theory.

It's of course clear that the famous Feynman-Vernon treatment of an open quantum system is also a good example, but also there they use standard probabilistic interpretations of the quantum state.

 

[A. Neumaier]

But expectation values are defined via the probabilities and vice versa.

Only in the minimal statistical interpretation. In the thermal interpretation, q-expectations are defined by $\langle A\rangle := Tr~\rho A$ for arbitrary operators $A$. Nowhere a claim (or even a hint) that this should be an average, except in the unfortunate name tradition gives to this object. The equivalence you talk about is not present, since $A$ may not even have a spectral resolution. As long as you do not realize this distinction, there is no hope to understand the thermal interpretation.

To evaluate this expectation value from the statistical operator within the standard mathematical QT formalism you need the spectral theorem and the "generalized eigenvectors"

This is not needed in the thermal interpretation. For example, if $\rho=\psi\psi^*$ is a pure state and $A=p^2-q^6$ we can calculate the q-expectation $\langle A\rangle = Tr~\rho A=\psi^* (p^2-q^6)\psi=\| p\psi\|^2-\|q^3\psi\|^2$ and evaluate this explicitly for every continuously differentiable wave function $\psi(x)$ decaying fast enough at infinity. The spectral theorem nowhere figures; in fact, you'd have a hard time to figure out whether $A$ is self-adjoint (probably it isn't).

You always say what the q-expectations are not. You have to say what they are, if it's forbidden to interpret them probabilistically. If there's a deterministic theory, I don't see, where I need expectation values to begin with.

I defined explicitly what a q-expectation is: a formal concept in the mathematical part of quantum mechanics, just like a phase space function is a formal concept in the mathematical part of classical mechanics. In both cases, these are claimed to be beables of the theory. In classical mechanics you don't give an operational definition of the concept of a phase space function - only a mathematical one. Why do you demand more in the quantum case?

If there's a deterministic theory, I don't see, where I need expectation values to begin with.

You need q-expectations (not statistical expectation values) since they are the beables of quantum physics, in the same way as you need phase space functions, since they are the beables of classical physics. In both cases, there is a deterministic dynamics for them, given in terms of a Lie algebra on the beables.

But (II.14) is the standard definition with the usual trace. I still don't see, how you even evaluate this for concrete observables without the spectral theorem, let alone how you understand its application to real measurements in a lab

I just demonstrated how to evaluate some without the spectral theorem. For some particular cases, I demonstrated how to understand their application in the lab in my previous posting #21: If $\rho$ is a grand canonical state, the expectation of any conserved quantity $A$ is an extensive thermodynamic variable and can be determined deterministically by the standard thermodynamic methods (which I don't have to repeat here); if you want to see details, read the chapter on equilibrium thermodynamics in my online book. For microscopic projection states, I referred to the discussion in Subsection 2.5 of Part III, and for probabilities I referred to the discussion in Subsection 3.5 of Part II. I find this very clear and don't know how to spell things out more clearly.

For me you use Born's rule to define the entire formalism and then claim you rederive it from these definitions. For me that's circular. Otherwise your thermal interpretation, is completely in accordance with how the QT formalism is used in physicists' lab practice.

Can we agree to call the definition $\langle A\rangle := Tr~\rho A$ the formal Born rule? Clearly, the formal Born rule is a purely mathematical statement, only introducing a symbolic abbreviation for the right hand side. Thus - unlike Born's rule, which refers to measurement - it has no interpretational content at all. That's why I cannot call it Born's rule - the latter establishes some idealized (but claimed to be universal) relation between the mathematical formalism and reality (aka measurement practice), while the formal Born rule doesn't.

What I claim is that I derived - in the contexts where it is valid - the statistical interpretation of the formal Born rule (and hence the actual Born rule) from the (GUP) of Section 2.5 of Part II and the (MP) of Section 2.1 of Part II, which are the interpretational assumptions made by the thermal interpretation.

On the other hand, you assume the statistical interpretation of the formal Born rule as a general property of measurements, and later have to correct for the idealization by admitting that there are other measurements governed instead by POVMs.

The formal treatment of exact toy models of interacting fields in lower space-time dimensions is very interesting. I know the Schwinger model (massless scalar QED in (1+1) dimensions), but only in the usual nonrigorous physicists' treatment and without the use of superselection rules. Do you have a reference for some interacting model like that, where this is done rigorously?

Please first understand the free massless field in 1+1 dimensions; e.g.,

• Dereziński, Jan, and Krzysztof A. Meissner. "Quantum massless field in 1+ 1 dimensions." Mathematical physics of quantum mechanics. Springer, Berlin, Heidelberg, 2006. 107-127.

which has a simple and fairly elegant exact description. Interacting exactly solvable fields are much more technical, for the Schwinger model, see, e.g.,

• Morchio, Giovanni, Dario Pierotti, and Franco Strocchi. "http://preprints.sissa.it/xmlui/bitstream/handle/1963/626/30_88.pdf?sequence=1&isAllowed=y." Annals of Physics 188, no. 2 (1988): 217-238.

and for more general exactly solvable models in 1+1D, the book

• Abdalla, Elcio, M. Cristina B. Abdalla, and Klaus D. Rothe. Non-perturbative methods in 2 dimensional quantum field theory. 1991.

I think, the key of my trouble is that I don't understand, how to decide Callen's principle without the usual operational definitions of "properties". For a thermal system (i.e., a system in equilibrium) it's clear how to define the thermodynamical properties operationally, i.e., how to measure temperature, chemical potential(s), pressure etc. observables, using the adequate devices to do so.

Callen's point (if you read the context of his book) is that these operational definitions are all based on the theory - for example, to measure temperature with a gas thermometer you already need to know what an ideal gas is and that the gas you use behaves like this. Thus once a theory is fully mature (as thermodynamics is) you can start with only the theory and some guesses about how to relate it to practice, and you can check whether your guesses are correct by checking whether the predicted consequences of the theory actually hold. See this toy situation for how this can work without being circular.

However, you don't give an operational definition.

I gave operational definitions for several special cases; why do you require more? In your statistical physics course you also give only very simple operational examples. For example, you claim that self-adjoint operators are observable, but you don't give an operational definition of how to measure the operator $qpqpq$ or $p+q$.

The Kadanoff-Baym equations are exact equations derived from the 2PI formalism. There are no approximations so far.

The approximations arise once you approximate the infinite series of contributions from all loop orders by the first few. But my point was that you are always using the formal Born rule; you never check that the arguments are actually self-adjoint; in fact they are not even Hermitian! Thus you only use the formal Born rule, which has no a priori relation at all to observation...

... unless you give it an interpretation. You give it the statistical interpretation, I give it the thermal interpretation, which are different in character.

But your statistical interpretation is inappropriate as you use q-expectations of nonhermitian operators and as you never consider any observation that would justify the statistical interpretation!

The "averaging" is here done quite implicitly using this coarse-graining in terms of the gradient expansion, but nowhere is the idea of a Gibbs ensemble envoked.

The Gibbs ensemble is directly encoded in your postulates, which say that the expectation of any operator (and hence in particular every two-point function $W(x,y)=\langle\phi(x)\phi(y)\rangle$ is to be interpreted as an average over repeated preparations of the system under consideration, i.e., a significant space-time region containing the collision center of the CERN accelerator, say. Thus:

• You need a large ensemble of many CERN accelerators, of which only one is realized!
• This average is undefined in terms of your postulates! (Indeed, $\phi(x)\phi(y)$ is not self-adjoint, not even in a finite lattice regularization.)
• The operational meaning of $W(x,y)$ is quite different and very indirect only! (This can be seen by considering the use made of 2-point functions in interpreting actual experiments.)

Thus the fairy story you tell fails on three different accounts to be even a meaningful proxy to what really happens.

 

[A. Neumaier]

So it looks like the statement I quoted is indeed true only for nonrelativistic quantum mechanics, if the atom in a trap cannot be modeled by relativistic quantum mechanics, without using quantum field theory.

The statement about the ion trap yes, but an analogous statement about a free relativistic particle somehow prepared at time $t$ in a small region of spacetime suffers the same problem.

@A. Neumaier:
So it seems that you fault the Born's rule for what is actually the Schrödinger equation's fault. [...] this doesn't look like a strong point of your critique of the Born's rule.

The Schrödinger equation just says that the Hamiltonian is the infinitesimal generator of time translations, and hence is always valid - even for a system described by the Dirac equation, where $H=cp_0$.

If someone finds fault with the Schrödinger equation then the whole of quantum mechanics breaks down completely!

 

[vanhees71]

Only in the minimal statistical interpretation. In the thermal interpretation, q-expectations are defined by $\langle A\rangle := Tr~\rho A$ for arbitrary operators $A$. Nowhere a claim (or even a hint) that this should be an average, except in the unfortunate name tradition gives to this object. The equivalence you talk about is not present, since $A$ may not even have a spectral resolution. As long as you do not realize this distinction, there is no hope to understand the thermal interpretation.


This is not needed in the thermal interpretation. For example, if $\rho=\psi\psi^*$ is a pure state and $A=p^2-q^6$ we can calculate the q-expectation $\langle A\rangle = Tr~\rho A=\psi^* (p^2-q^6)\psi=\| p\psi\|^2-\|q^3\psi\|^2$ and evaluate this explicitly for every continuously differentiable wave function $\psi(x)$ decaying fast enough at infinity. The spectral theorem nowhere figures; in fact, you'd have a hard time to figure out whether $A$ is self-adjoint (probably it isn't).

Ok, that's the problem. You do not tell me, how to calculate this expectation value. If I'm not allowed to use the spectral theorem, I cannot even write down what the abstract definitions mean. E.g., take a Gaussian wave packet of the traditional formalism. How do you define that without using the position or momentum basis. Then it's easy to calculate this expectation value (even without caring, whether it's self-adjoint or not). You define the state ket via your favorite representation, e.g., in the position representation
$$\psi(x)=N \exp(-x^2/(4 \Delta^2)+\mathrm{i} p x)$$
and then calculate your trace with the corresponding stat. op., leading to
$$\langle f(x,p) \rangle=\int_{\mathbb{R}} \mathrm{d} x \psi^*(x) f(x,-\mathrm{i} \partial_x) \psi(x).$$
Then you have a concrete way to calculate this expectation value. Now you forbid to use the spectral decomposition of the involved operators. So how is within the thermal representation this expectation value evaluated?

 

[A. Neumaier]

You do not tell me, how to calculate this expectation value.

I did it for the case $f(q,p)=p^2-q^6$, but you didn't take it up. So let me give a few more details: By the definition of $p$ and $q$ in the position representation and the rules for multiplying operators, you get $p\psi(x)$ by differentiation and $q^3\psi(x)$ by multiplication with $x^3$. Given a Gaussian wave package $\psi(x)=N \exp(-x^2/(4 \Delta^2)+\mathrm{i} p x)$ as you suggest, you can calculate the two norm squares in my previous post using integration by parts, and you get an explicit, exact value for the q-expectation!

This is an elementary exercise that already good high school pupils can do.

On the other hand, the spectral theorem for unbounded self-adjoint operators is heavy machinery (in Vienna 3rd year math bachelor) that takes a lot of math to prove. Thus it should not figure in a first introduction to quantum mechanics.

Essentially the same works for any expression in $p,q$ involving only sums of products of powers of $p,q$ and more generally for any expression $A$ that is polynomial in $p$ and often enough differentiable in $q$. In the latter case, to evaluate $A\psi$, one uses the product rule to move all $p$s to the right of all $q$s, and ends up with a sum of terms of the form $f_i(q)p^n\psi$ which can be evaluated directly in the position representation and then integrated numerically after multiplication with $\psi^*$. If $A$ is nonpolynomial in $p$ one can use limits, power series, etc...
Only the definition of the operators $p$ and $q$ is needed for all this, no spectral theorem!

 

[vanhees71]

I gave operational definitions for several special cases; why do you require more? In your statistical physics course you also give only very simple operational examples. For example, you claim that self-adjoint operators are observable, but you don't give an operational definition of how to measure the operator $qpqpq$ or $p+q$.

My notes describe the established theory in the shutup-and-calculate interpretation. I make no claim, I'd have a new interpretation which solves all (apparent philosophical) problems of this standard interpretation. I also assume that the reader is familiar with QM at the level of the QM 1 lecture. Perhaps at this time, I was not as careful to formulate it, but I never claim that an observable is a self-adjoint operator on Hilbert space. Observables are defined operationally by giving a measurement procedure. The position observable is defined by a reference frame implying rulers to measure distances from a given point along three perpendicular directions in space.

In QM these observables are represented by self-adjoint operators on Hilbert space, and the states have a probabilistic meaning, relating the formalism to the operationally defined observables in the lab.

Of course, you always need a theory to define the observables, and particularly Callen's book is very clear about these foundations.

As I understood only now after the discussion here, in your thermal interpretation it's not even allowed to use the spectral theorem. Then for me the formulae have no more relation to anything observable in the sense of physics. This is also formally clear since the statistical operator is even picture dependent. For itself it cannot have a physical meaning. You also need the (also picture dependent) observable operators (forming the observable algebra) and the spectral theorem to make physical sense to the formalism.

If you want an alternative physical interpretation of QT, you have to make this connection between observables and the formalism representing these observables in the mathematical constructions clear. As it stands $\mathrm{Tr} \hat \rho \hat{A}$ is not even defined in a way that I can calculate anything with it. Of course, it's my mistake to think that your text associates the usual physical (!) meaning of these symbols, but how else can I understand the text, if I don't make this assumption, if the text doesn't give a clear definition of what's meant by the symbols? This should at least be done for the most simple examples.

I also don't need to think about POVMs if I haven't even understood the most simple case of von Neumann measurements first! If I haven't even understood the most simple textbook case within the new interpretation, I don't dare to hope to understand an even more general and even more abstract concept!

The approximations arise once you approximate the infinite series of contributions from all loop orders by the first few. But my point was that you are always using the formal Born rule; you never check that the arguments are actually self-adjoint; in fact they are not even Hermitian! Thus you only use the formal Born rule, which has no a priori relation at all to observation...

Since when are position and momentum operators within non-relatvistic first-quantized QT not self-adjoint?

... unless you give it an interpretation. You give it the statistical interpretation, I give it the thermal interpretation, which are different in character.

But your statistical interpretation is inappropriate as you use q-expectations of nonhermitian operators and as you never consider any observation that would justify the statistical interpretation!


The Gibbs ensemble is directly encoded in your postulates, which say that the expectation of any operator (and hence in particular every two-point function $W(x,y)=\langle\phi(x)\phi(y)\rangle$ is to be interpreted as an average over repeated preparations of the system under consideration, i.e., a significant space-time region containing the collision center of the CERN accelerator, say. Thus:

• You need a large ensemble of many CERN accelerators, of which only one is realized!
• This average is undefined in terms of your postulates! (Indeed, $\phi(x)\phi(y)$ is not self-adjoint, not even in a finite lattice regularization.)
• The operational meaning of $W(x,y)$ is quite different and very indirect only! (This can be seen by considering the use made of 2-point functions in interpreting actual experiments.)

Thus the fairy story you tell fails on three different accounts to be even a meaningful approximation to what really happens.

We argue in circles. So far you haven't given any interpretation but a mathematical formalism which doesn't even allow me to calculate the most simple things, which are not problem in the standard interpretation. That's not a very attractive alternative.

Of course, I don't need a large ensemble of the LHC. One LHC is enough to generate zillions of pp, pA, and AA collision events to collect "enough statistics".

Averages are very well defined with the same formula you give. The difference is that it's also clearly said, how to really calculate it, namely by using the observable algebra to construct the probabilities or equivalently the expectation values.

Just take the most simple thinkable QM 1 system:

There's a separable Hilbert space, on which the Heisenberg algebra with the fundamental self-adjoint operators $\hat{x}$ and $\hat{p}$ is realized. All you need is the general Hilbert-space structure and a notion of self-adjointness. The physics is defined operationally via Noether's theorem (or even more simply and heuristically motivated as "canonical quantization", i.e.,
$$[x,p]=\mathrm{i}.$$
From this it follows
$$\langle x|p \rangle=\frac{1}{\sqrt{2 \pi}} \exp(\mathrm{i} p x),$$
and from this everything else follows in a not too complicated way. Of course, the mathematician has to specify this much more carefully, defining the domains of the self-adjoint operators, the nuclear space and its dual of the rigged Hilbert space etc. But that's not the point of the interpretational issues.

The interpretation comes in, when I interpret pure states in the usual probabilistic way, i.e., interpreting the wave functions $\psi(x)=\langle x|\psi \rangle$ (with $|\psi \rangle$ in the Hilbert space and thus $\psi(x)$ in $L^2$) via Born's rule as giving the position probability distribution $P(x)=|\psi(x)|^2$. Then expectation values of properly defined operator functions $f(x,p)$ can be evaluated since the construction of the rigged Hilbert space with the observable algebra implies that in the position representation
$$f(\hat{x},\hat{p}) \mapsto f(x,-\mathrm{i} \partial_x)$$
and in this sense
$$\langle f(x,p) \rangle = \int_{\mathbb{R}} \mathrm{d} x \psi^*(x) f(x,-\mathrm{i} \partial_x) \psi(x).$$
The generalization to general ("mixed" states) is also straight forward, as long as you allow the use of the spectral decomposition with respect to (a complete set of compatible) observable operators.

If you now claim, you have a new interpretation of expectation values underlying a new interpretation, you have to give a concrete description of how to calculate these expectation values, if it's "forbidden" to use the standard meaning of the symbols!

 

[vanhees71]

I did it for the case $f(q,p)=p^2-q^6$, but you didn't take it up. So let me give a few more details: By the definition of $p$ and $q$ in the position representation and the rules for multiplying operators, you get $p\psi(x)$ by differentiation and $q^3\psi(x)$ by multiplication with $x^3$. Given a Gaussian wave package $\psi(x)=N \exp(-x^2/(4 \Delta^2)+\mathrm{i} p x)$ as you suggest, you can calculate the two norm squares in my previous post using integration by parts, and you get an explicit, exact value for the q-expectation!

This is an elementary exercise that already good high school pupils can do.

On the other hand, the spectral theorem for unbounded self-adjoint operators is heavy machinery (in Vienna 3rd year math bachelor) that takes a lot of math to prove. Thus it should not figure in a first introduction to quantum mechanics.

Essentially the same works for any expression in $p,q$ involving only sums of products of powers of $p,q$ and more generally for any expression $A$ that is polynomial in $p$ and often enough differentiable in $q$. In the latter case, to evaluate $A\psi$, one uses the product rule to move all $p$s to the right of all $q$s, and ends up with a sum of terms of the form $f_i(q)p^n\psi$ which can be evaluated directly in the position representation and then integrated numerically after multiplication with $\psi^*$. If $A$ is nonpolynomial in $p$ one can use limits, power series, etc...
Only the definition of the operators $p$ and $q$ is needed for all this, no spectral theorem!

You told me that precisely this is not allowed to do within your thermal interpretation since it's forbidden to use the spectral theorem. Of course, I know how to calculate the said expectation values. Even a machine can do this nowadays ;-)).
I don't talk about the complicated rigorous definition in a functional-analysis course at all. For me the usual sloppy physicists approach is enough. I want to concentrate on the interpretational aspect, but if it is not allowed to use even this in principle strictly definable rules of the rigged-Hilbert-space formalism, I cannot make sense of the symbols at all. I must have something that let's me boil it down "to the numbers"! As I said in my previous posting, in standard QT there's no problem with the construction of these observables down to the level of making it computable down to the numbers, but I don't see, how this is done without the use of a concrete representation (naturally the position and/or momentum representation in this case).

 

[A. Neumaier]

You told me that precisely this is not allowed to do within your thermal interpretation since it's forbidden to use the spectral theorem.

I was not using the spectral theorem. I was only using the definition of $p$ and $q$ in the Hilbert space $L^2(R)$. This has nothing to do with the spectral theorem. Schrödinger didn't know the spectral theorem when he invented the position representation! (It was first formulated by von Neumann a few years later.)

Moreover, I do not forbid the spectral theorem as a mathematical tool to work with operators that are self-adjoint. Whenever this is the case one can of course work with the spectral theorem on the shut-up-and-calculate level (but one still need not impose a statistical interpretation). But the self-adjoint operators are not general enough - they are neither closed under addition nor under multiplication with a scalar, while these operations are ubiquitous in calculations with q-expectations!

And they are not needed for much of quantum mechanics, including your cited lecture on the Kadanoff-Baym equations.

I don't see, how this is done without the use of a concrete representation (naturally the position and/or momentum representation in this case).

Neither do I. Without giving some Hilbert space and a definition of $p,q$ satisfying the CCR one cannot do any quantum mechanics. But once this is given (as a space of sequences or functions, or in still other ways), one has a representation and can do most of the calculations without the spectral theorem. The latter may be needed when one wants to change to a different representation (even this can often be done without it), but not when working in a fixed representation.

Since when are position and momentum operators within non-relatvistic first-quantized QT not self-adjoint?

They are, but in your derivation in this lecture, you are not taking q-expectations of position and momentum operators but of (regularized) products of fields such as $\phi(x)\phi(y)$. According to you, it is an ordinary expectation, hence the argument of the expectation should (according to your definition of expectation) be self-adjoint. Thus this product should be self-adjoint - but it is not even Hermitian!

 

[vanhees71]

Well, if you always redefine standard language, it's really hard to discuss :-(. For me the position representation is an application of the spectral theorem for the position operator. For me QT is formulated representation independently by an observable algebra and its representation as self-adjoint operators on a Hilbert space. The representations are constructed from this via the spectral theorem. If you start in the position representation that's fine with me, but then what's the difference between the thermal and the standard probabilistic interpretation? It becomes more and more enigmatic to me rather than clear!

It also doesn't make sense to discuss about non-Hermitean operators which may have some applications in QT. All I want is a clear definition of the thermal interpretation, particularly how to understand the expression $\mathrm{Tr}(\hat{\rho} \hat{A})$ if not probabilistically as an expectation value of a random variable as in standard QT.

Finally, it's of course clear that you use calculational expressions which are not related to observables like propagators, i.e., something $\propto \langle \hat{\phi}(x) \hat{\phi}^{\dagger}(y)$. It's not claimed that this directly refers to observational (probabilistic) quantities. As usual in QFT the N-point functions (and building blocks as the connected and the proper vertex functions) are used as calculational tools towards directly observable quantities (if you want you can use the modern word "beables" for this too) like cross sections or expectation values of observable quantities etc.

To understand better the meaning of your formalism, it would really be helpful to stick to the most simple examples like the Stern-Gerlach experiment of non-relativistic QT first. This is so close to classical mechanics because of the linearity of the forces involved that this must be also simple in your thermal representation.

 

[A. Neumaier]

Well, if you always redefine standard language, it's really hard to discuss :-(. For me the position representation is an application of the spectral theorem for the position operator.

My language is at least as standard as yours: Before you can apply the spectral theorem in some Hilbert space to some operator, you need definitions of both! I define an inner product on $L^2(R)$ and then the operators $p$ and $q$, to get the necessary Hilbert space and two particular operators on it. Having these definitions, I don't need the spectral theorem at all - except when I need to define transcendental functions of some operator.

If you start in the position representation that's fine with me, but then what's the difference between the thermal and the standard probabilistic interpretation?

The difference, given the position representation (or any other representation), is as follows:

What you call the minimal statistical or standard probabilistic interpretations uses this representation for defining irreducible probabilities of measurement in an ensemble of repeated observations, and thus introduces an ill-defined notion of measurement (and hence the measurement problem - though you close your eyes to it) into the very basis of quantum mechanics. It is no longer clear when something counts as a measurement (so that the unitary evolution is modified) and when the Schrödinger equation applies exactly; neither does it tell you why the unitary evolution of the big system consisting of the measured objects and the detector produces definite events. All this leads to the muddy reasoning visible in the literature on the measurement problem.

The thermal interpretation uses this representation instead to define the formal q-expectation of an arbitrary operator $A$ for which the trace in the formal Born rule can be evaluated. (There are many of these, including many nonhermitian ones and many Hermitian, non-selfadjoint ones.) This is the way q-expectations are used in all of statistical mechanics - including your slides. All this is on the formal side of the quantum formalism, with no interpretation implied, and no relation to observations. This eliminates the concept of probability from the foundations and hence allows progress to be made in the interpretation questions.

Then I note that the collection of all these q-expectations has a deterministic dynamics given by a Lie algebra structure, just as the collection of phase space functions in classical mechanics. In the thermal interpretation, the elements of both collections are considered to be beables.

Then I note that in statistical thermodynamics of local equilibrium, the q-expectations of the fields are actual observables, as they are the classical observables of fluid mechanics, whose dynamics is derived from the 1PI formalism - in complete analogy to your 2PI derivation of the Kadanoff-Baym equations. In practice one truncates to a deterministic dissipative theory approximating the deterministic dynamics of all q-expectations. This gives a link to observable deterministic physics - all of fluid mechanics, and thus provides an approximate operational meaning for the field expectations. This is not worse than the operational meaning of classical fields, which is also only approximate since one cannot measure fields at a point with zero diameter.

Then I prove that under certain other circumstances and especially for ideal binary measurements (rather than assume that always, or at least under unstated conditions), Born's interpretation of the formal Born rule as a statistical ensemble mean is valid. Thus I recover the probabilistic interpretation in the cases where it is essential, and only there, without having assumed it anywhere.

calculational expressions which are not related to observables like propagators, i.e., something $\propto \langle \hat{\phi}(x) \hat{\phi}^{\dagger}(y)\rangle$. It's not claimed that this directly refers to observational (probabilistic) quantities.

What then is the meaning of the expectation in this case? It is just a formal q-expectation defined via the trace. Thus you should not complain about my notion!

Born's rule only enters when you interpret S-matrix elements or numerical simulation results in terms of cross sections.

 

[DarMM]

It also doesn't make sense to discuss about non-Hermitean operators which may have some applications in QT. All I want is a clear definition of the thermal interpretation, particularly how to understand the expression $\mathrm{Tr}(\hat{\rho} \hat{A})$ if not probabilistically as an expectation value of a random variable as in standard QT.

It's a "beable" in Bell's terminology, that is a property of the system in question no different from properties in classical mechanics. Or at least thus is my understanding so far.

 

[DarMM]

I'm currently thinking a bit about the Bell inequalities for this interpretation.

For now as a side question have you considered the SIC-POVM conjecture, i.e. that specifying the $d^2$ collection of SIC-POVMs is enough to characterize the state $\rho$ completely. If true could this be taken into the Thermal Interpretation as the SIC-POVMs being the fundamental beables/quantities?

 

[A. Neumaier]

have you considered the SIC-POVM conjecture, i.e. that specifying the $d^2$ collection of SIC-POVMs is enough to characterize the state $\rho$ completely. If true could this be taken into the Thermal Interpretation as the SIC-POVMs being the fundamental beables/quantities?

They are interesting from a combinatorial point of view but nothing fundamental. Zauner wrote his thesis on these under my supervision.

 

[Fra]

I defined explicitly what a q-expectation is: a formal concept in the mathematical part of quantum mechanics, just like a phase space function is a formal concept in the mathematical part of classical mechanics. In both cases, these are claimed to be beables of the theory. In classical mechanics you don't give an operational definition of the concept of a phase space function - only a mathematical one. Why do you demand more in the quantum case?.

I personally demand much more in the quantum case because in my vision, I understand and expect future foundations of QM to be a theory of rational expectations of observers possessing incomplete information and having limited processing resources to process the information at hand. IMO this lies at another level of scientific standard than does classical physics. Ie. introducing non-inferrable concepts are against what i thinks are constructing principles. In a sense i an holding an evolved form of logical positivism here, but motivated by contraints of the physical observing system, rather than "human empirical observation".

But I agree completely what is the key problem with QM as it stands, and that is how to interpret or attach expectations of the the P-spaces without relating to fictive infinite ensembles. Statistical ensembles from repetitive experiments of identically prepared setup are fine for the typical HEP accelerator experiments however, but the problem is the cases (QG and unification) where this breaks down.

As I understand it the thermal interpretation aims to be an effective somewhat pragmatic interpretation, is that correct? In that case i figure it may not be worse than others but i do not see what advantage it offers for attaching QG and unification? Is it supposed to? If so i probably don't understand something.

/Fredrik

 

[DarMM]

They are interesting from a combinatorial point of view but nothing fundamental.

Thanks.

Zauner wrote his thesis on these under my supervision.

I saw that under footnote 27 in the first paper, I must read the thesis.

 

[akhmeteli]

The statement about the ion trap yes, but an analogous statement about a free relativistic particle somehow prepared at time $t$ in a small region of spacetime suffers the same problem.

Could you please give a reference?

The Schrödinger equation just says that the Hamiltonian is the infinitesimal generator of time translations, and hence is always valid - even for a system described by the Dirac equation, where $H=cp_0$.

If someone finds fault with the Schrödinger equation then the whole of quantum mechanics breaks down completely!

I only had in mind the original nonrelativistic Schrödinger equation.