Insights Quantum Physics via Quantum Tomography: A New Approach to Quantum Mechanics

[A. Neumaier]

Sure, but all your words are just using Born's rule.

No; you just project your understanding of the foundations of quantum physics into my words.

None of my words or underlying concepts uses Born's rule, unless you empty it from all connections to measurements.

Indeed I mention measurement nowhere. Everything mentioned is macroscopic nonequilibrium thermodynamics together with the purely formal definition $\langle A\rangle:=\mathrm{Tr} \rho A$ (which is pure math, not physics), instantiated by taking for $A$ the components of a current to give the physical meaning it has in nonequilibrium thermodynamics. Calling a mathematical definition Born's rule is not appropriate.

I know that you deny that your expectation-value brackets have a different than the usual straight-forward meaning of Born's rule, but I don't see what's the merit should be not to accept Born's rule (of course in its general form for general states, i.e., also for mixed states).

The advantage is that eigenvalues play no role, and that nonprojective measurements are covered without any additional effort. Thus my approach is both simpler and more general than working with Born's rule, and the explanatory value is higher.

A "complete measurement" is described by measuring a complete set of independent compatible observables

Such a complete measurement cannot be done for most quantum systems (except for those with very few degrees of freedom). My approach does not need such fictions.

If you do the SGE with single silver atoms,

Most physics students in the lab will not do SGE with single silver atoms, but with continuous beams of silver. Therefore I only discussed the standard Stern-Gerlach experiment, as performed by them. This involves no ensemble but a silver field in the form of a dispersed beam.

If you do the SGE with single silver atoms, there's no other way to get from the formalism to the observations than Born's rule.

Really? I get the same result not from Born's rule but from the detector response principle DRP, without using eigenvalues or projections.

Maybe you will call the DRP Born's rule to save your view. Then we agree, except for the terminology.

In any case, introducing the DRP is much more intuitive than the introduction of Born's rule in your statistical physics lecture notes:

So let’s begin with some formalism concerning the mathematical structure of quantum mechanics as it is formulated in Dirac’s famous book.
[...]
If |o, j〉 is a complete set of orthonormal eigenvectors of O to the eigenvalue o, theprobability to find the value o when measuring the observable O is given by
$$P_ψ(o) =\sum_j |〈o, j |ψ〉|^2. ~~~~~~~~~~~~~~~~~~(2.1.3)$$

which is full of nonintuitive formal baggage that falls from heaven without any motivation. As only reference you give Dirac's famous book; I have the third edition from 1947. There he introduces eigenvectors on p.29, without any motivation, and states Born's rule in (45) on p.47, with formal guesswork as only motivation, and in a very awkward way, where one cannot recognize how it is related to your formulation. A more digestible version comes later in (51) on p.73,
but this is equivalent to yours only in the case of nondegenerate eigenvalues. The name Born's rule is nowhere mentioned in the book - so little importance does Dirac give to it!

Conclusion: In the foundations favored by you the students first have to swallow ugly toads, just based the promise that it will ultimately result in a consistent quantum theory later...

The DRP, in contrast, needs no eigenvalues at all, no separate consideration of degenerate cases, not even self-adjoint operators (themselves nontrivial to define but needed for the spectral resolution). The little stuff needed is simple and easy to motivate from Stokes' treatment of polarization in 1852.

it's one of the very few examples, where no semiclassical approximations are needed. You can solve the Schrödinger equation in this case exactly assuming a simplified magnetic field or use numerics.

The primary semiclassical approximation needed is the one that goes from quantum field theory to an ensemble of a sequence of single atoms moving along a beam and arriving at the bottle.

I don't know of a single paper explaining in detail how this transition in conceptual language can be justified from the QFT formalism. Maybe you can help me here with a reference?

 

[vanhees71]

We had this discussion over and over again. I don't think that we find a consensus here, because I don't see the connection between your formalism and the physics it's proposing to describe. I guess that's why Born's standard probabilistic interpretation has never been substituted by anything else by the majority of physicists.

I also don't understand your problem with the standard description of the SGE in QM. An example is

https://arxiv.org/abs/quant-ph/0409206

 

[A. Neumaier]

We had this discussion over and over again.

Each time I discuss it with you I add some new perspective, otherwise the discussion would not be interesting for me. At least I learn through the exchange (not about your view but more about mine), while you seem to be stuck in tradition.

I don't see the connection between your formalism and the physics it's proposing to describe.

My paper contains as much physics as Dirac's famous book! That you don't see it can only mean that you don't read it with open eyes.

I also don't understand your problem with the standard description of the SGE in QM.

On the quantum mechanical level there is no problem with it.

But a fundamental description would have to come from relativistic quantum field theory, where there are no ensembles. One cannot repeatedly prepare a quantum field extending over all of spacetime.

I don't know of a single paper explaining how the transition in conceptual language from a single quantum field to an ensemble of particles can be justified from the QFT formalism. Maybe you can help me here with a reference?

An example is

https://arxiv.org/abs/quant-ph/0409206

Sections 2-4 are pure theory without contact to experiment (i.e., before hitting the glass bottle); nothing to complain given their assumptions.
Section 5 interprets the final theoretical result as probability without explaining why they are allowed to do this.

The DRP gives this interpretation of the final theoretical result. On the other hand, Born's rule for projective measurements does not do the job: In Section VI, the authors of the paper write:

Thus, we can conclude that the Stern-Gerlach experiment is not, even in principle, and ideal experiment, which would “project” the internal state into the eigenvalues of the measurement operator. [...]
Our calculations indicate that the Stern-Gerlach experiment is not an ideal measuring apparatus, in the sense of [5].

[5] is von Neumann's 1932 book (in its 1955 English translation), where he describes measurement exclusively in terms of projection operators.

Thus projective measurements (and hence Born's rule) cannot be applied without making the additional semiclassical approximations proved invalid in the paper.

 

[gentzen]

Why should this imply any preferred basis? It's independent of the choice of basis (i.e., the choice of a complete set of compatible observables).

No, it doesn't imply a preferred basis. It is just tempting to go down that road, if you throw away the structure of the observables.

Let me expand my thoughts: If you say that any self-adjoint operator is an observable, and then use that "freedom" to use exactly one specific self-adjoint operator to describe the observable corresponding to the measurement in the experiment that you investigate, then the only other relevant self-adjoint operator seems to be the Hamiltonian of your system.

But how will you now ascribe physical meaning to your Hamiltonian and your selected observable? A tempting "cheap way" is to ascribe physical meaning directly to some preferred basis of your Hilbert space.

 

[vanhees71]

Each time I discuss it with you I add some new perspective, otherwise the discussion would be not interesting for me. At least I learn through the exchange (not your view but more about mine), while you seem to be stuck in tradition.My paper contains as much physics as Dirac's famous book! That you don't see it can only mean that you don't read it with open eyes.On the quantum mechanical level there is no problem with it.

But a fundamental description would have to come from relativistic quantum field theory, where there are no ensembles. One cannot repeatedly prepare a quantum field extending over all of spacetime.

I don't know of a single paper explaining how the transition in conceptual language from a single quantum field to an ensemble of particles can be justified from the QFT formalism. Maybe you can help me here with a reference?Sections 2-4 are pure theory without contact to experiment (i.e., before hitting the glass bottle); nothing to complain given their assumptions.
Section 5 interprets the final theoretical result as probability without explaining why they are allowed to do this.

The DRP gives this interpretation of the final theoretical result. On the other hand, Born's rule for projective measurements does not do the job: In Section VI, the authors of the paper write:[5] is von Neumann's 1932 book (in its 1955 English translation), where he describes measurement exclusively in terms of projection operators.

Thus projective measurements (and hence Born's rule) cannot be applied without making the additional semiclassical approximations proved invalid in the paper.

Dirac describes clearly the physical meaning including the probabilistic interpretation of the quantum state a la Born. You use the same symbols and forbid to interpret them in this standard way but don't give any clear physical interpretation I should use instead. That's why I'm stuck.

Within QFT you can as well prepare single Ag atoms as you can in QM. QFT is also used since it's conception to describe scattering cross sections, and that's also due to the standard interpretation of the quantum state in terms of Born's rule. It's explained in any QFT textbook, e.g., Weinberg, QT of Fields vol. 1.

The point of the paper by Potel is a complete quantum description of the SGE, and there's a probability for spin flips and that's why the SGE is not strictly an ideal von Neumann projection measurement. The point is that it's a complete quantum description in terms of the standard interpretation a la Born.

 

[A. Neumaier]

The point of the paper by Potel is a complete quantum description of the SGE, and there's a probability for spin flips and that's why the SGE is not strictly an ideal von Neumann projection measurement. The point is that it's a complete quantum description in terms of the standard interpretation a la Born.

You are asserting the opposite of the conclusion of the paper.

It is not a description in terms of the standard interpretation a la Born, since Born's rule is only about ideal von Neumann projective measurements (proof: p.20 of your statistical mechanics lecture notes), and Potel et al. proved that the SGE is not of this kind.

But it is a complete quantum description in terms of the interpretation by my detector response principle!

 

[vanhees71]

No, Born's rule gives the probability to find an Ag atom with a given spin component. It has nothing to do whether you get ideal entanglement between spin and position as stated in the simplified approximations in textbooks or the more accurate calculations in Potel's paper.

I still don't understand, what you think to achieve giving up the simple and well-established (as well as very successful) standard foundations of QT. I think that the minimal interpretation without collapse is the only interpretation you need.

You may introduce POVMs, but only if they can be made concrete for the analysis of real-world experiments. I don't see, where they are ever used in real-world applications of QT yet.

 

[A. Neumaier]

I still don't understand, what you think to achieve giving up the simple [...]

I achieve true simplicity. I find Born's rule far from simple, and very restrictive.

But it is impossible to communicate this to you. You block off all m< attempts with the mantra Born's rule explains all measurements., together with bogus arguments to justify it.

The problem is that it's not clear how to make the connection with equipment in the lab, which is not a problem in the standard description at all.

It is not a problem in my setting either. Together with the dynamics implies by the experimental setting, the DRP does it, with much less theoretical baggage than Born's rule:

(DRP): Detector response principle. A detection element $k$ responds to an incident
stationary source with density operator $ρ$ with a nonnegative mean rate $p_k$ depending linearly on ρ. The mean rates sum to the intensity of the source. Each $p_k$ is positive for at least
one density operator ρ

 

[A. Neumaier]

Within QFT you can as well prepare single Ag atoms as you can in QM. QFT is also used since it's conception to describe scattering cross sections, and that's also due to the standard interpretation of the quantum state in terms of Born's rule. It's explained in any QFT textbook, e.g., Weinberg, QT of Fields vol. 1.

One can model a single silver atom in this way. But how do you model an ensemble of 100 silver atoms moving at well separated times along a beam by quantum field theory? You cannot prepare multiple instances of a field extending over all of spacetime. The only use! of Born's rule in Weinberg's Vol. 1 (namely where he interprets the scattering amplitutes) doesn't address this issue - scattering has nothing to do with this question!

I don't know of a single paper explaining how the transition in conceptual language from a single quantum field to an ensemble of particles can be justified from the QFT formalism.

You also seem to know no place where this is done.

 

[Fra]

But a fundamental description would have to come from relativistic quantum field theory, where there are no ensembles. One cannot repeatedly prepare a quantum field extending over all of spacetime.

I don't know of a single paper explaining how the transition in conceptual language from a single quantum field to an ensemble of particles can be justified from the QFT formalism. Maybe you can help me here with a reference?

I agree these are some important points. Even if we don't agree on the solution, agreeing on what is a problem big enough to beg for a solution, and what we can sweep under the rug for a far future is a good start.

/Fredrik

 

[vanhees71]

Relativistic QFT has the same probabilistic interpretation as any QT (in fact there is only one overall conceptual framework and a non-relativistic (in both a "1st-quantization formulation" and a field-theoretical one as well as a special relativistic realization in terms of local QFTs).

Of course one can prepare ensembles within all kinds of QTs and, more importantly, in the lab. Relativistic QFT is among the best tested physical theories ever discovered. This would be impossible to achieve if it were not possible to prepare ensembles of the physical systems described by it, and these are particles and nuclei in particle accelerators, with which you can do scattering experiments with high precision. Another application is atomic physics. A specific quantum-field theoretical result is the explanation of the Lamb shift to several significant digits of accuracy, etc.

I don't understand, how one can claim that one cannot build ensembles within relativistic QFT, given all these successes. After all the first goal in all introductions to QFT is to establish the calculations of S-matrix elements, which precisely describe scattering processes, and obviously these can be realized with high accuracy in the lab.

 

[Fra]

I think the reason we get away with the principal problem here, is that the theory is corroborated for small subsystems. Where small refers to both spatially slow, well as short lived, at least compare to the scales of the inferences going on during the experiment. So the symptoms of the principal problems are IMO not to be found in atomic physics, it's to be find either when one includes also gravity, and times which are not infinitesimal compare cosmological scale, and perhaps also when trying to understand the unifcation of forces that are currenyl described at extremely wide energy ranges.

It seems quite obivous, that in order for an experiment to be able to produce alot of "statistics" of from effectively asymptotic scattering experiments, it simpy will not be possible to acquire and process that and make preparations if the significant scale of interactions was the whole universe in space and time. But it would work if it's all happening withing a laboratory building at quick pace?

I agree that this objection is almost silly for practical purposes of atomic or even macroscopic solid state physics(which is still microscopic on cosmo scale), but if one considers it a principal argument I don't see how it can not be valid?

/Fredrik

 

[vanhees71]

Well, all our observations are pretty local. In cosmology we extrapolate the local observations to "the whole universe" by assuming the cosmological principle.

 

[Fra]

Yes, we effectively assume that our - admittedly incomplete sampling and truncated postprocessing - is still a fair representation. We know why we do this and the assumption is not irrational. The only issue is if we get so used to this that we confuse this rational guess, with a fact. My first line of thought is not that I question the spatial cosmological principle, but what I think we call "perfect cosmological principle" which also says the laws of physics do not change. There is as far as I know no cosmological evidence that the laws did change, but OTOH the astronomical evidence are mainly coming from when there was light. And in the perspective of unification of forces the universe was relatively speaking OLD at this point.

Even in Smolins speculation of evolution of law, had admits that no evidence suggests the laws we can infer has changed. His idea is that laws changed during hte big bang, but I would say wether it's AT the big band, or when all forces was presumably blurred up and before the notion of spacetime was clear, we can not tell.

So I personally do not think the perfect cosmological principle is something to hold onto in the context of this discussion. And of course the whole notion of the spatial cosmological principle also becomes questionable if we thinkg that there was an early time before 4D spacetime was stable.

/Fredrik

 

[gentzen]

Because Bernhard wrote at https://www.astronews.com/community...etation-der-quantenmechanik.11402/post-137032 that he has "only" read Born's rule and measurement from A. Neumaier:

Ich muss mir dann erstmal das zitierte Paper genauer ansehen und hatte vorab schon etwas ein anderes paper von AM angesehen: Born's rule and measurement ebenfalls auf arxiv.org. Dort finden sich auch Iesenswerte Ideen und Anregungen.

I tried to remember my own impression of that paper. Somehow it felt boring to me, and I was skeptical, whether its "new approach to introductory courses on quantum mechanics" would be an improvement. But maybe this was just because I read this paper more out of a "felt duty" than out of curiousity.

The current paper feels totally different to me, even so I realize that it is somehow supposed to be a successor or elaboration of that paper. It picks my curiosity at thousand different small and bigger points, and is a complete joy to read with all its interesting quotes and motivations. But that old paper also has quotes, so that can't be the difference. Maybe it was really just my own state of mind when I read it. But I think it is more: I get many of the points that the current paper wants to achieve, but I had a hard time identifying clear goals of that previous paper.

 

[A. Neumaier]

Because Bernhard wrote at https://www.astronews.com/community...etation-der-quantenmechanik.11402/post-137032 that he has "only" read Born's rule and measurement from A. Neumaier:

Maybe you can post there a link to the new paper!

I tried to remember my own impression of that paper. Somehow it felt boring to me, and I was skeptical, whether its "new approach to introductory courses on quantum mechanics" would be an improvement. But maybe this was just because I read this paper more out of a "felt duty" than out of curiousity.

The current paper feels totally different to me, even so I realize that it is somehow supposed to be a successor or elaboration of that paper. It picks my curiosity at thousand different small and bigger points, and is a complete joy to read with all its interesting quotes and motivations. But that old paper also has quotes, so that can't be the difference. Maybe it was really just my own state of mind when I read it. But I think it is more: I get many of the points that the current paper wants to achieve, but I had a hard time identifying clear goals of that previous paper.

The difference of my 2019 paper "Born's rule and meansurement" and my new quantum tomography paper is that the former (30 pages of main text) was a programme (stated on 3 pages in Subsection 2.6) to be executed:
''The material presented suggests a new approach to introductory courses on quantum mechanics''
while the latter (90 pages of main text) is the general part of its execution. I go through many of the main quantum physics topics that must be discussed and show what they mean in terms of the new detector response principle.

Of course, to make it into a text on introductory quantum mechanics, one would need many more examples to which the concepts are applied, and an introduction into how to compute all the stuff treated abstractly in the paper.

 

[A. Neumaier]

Because Bernhard wrote at https://www.astronews.com/community...etation-der-quantenmechanik.11402/post-137032 that he has "only" read Born's rule and measurement from A. Neumaier:

Mich interessiert jetzt natürlich stark, welche Details nun im Rahmen dieser Interpretation den Startpunkt einer Nebelspur in einer Nebelkammer festlegen.
(Of course, I am now very interested in which details determine the starting point of a trail in a cloud chamber within the framework of this interpretation.)

and in a related discussion:

Wie gelangt die TI von einem sphärisch symmetrischen Zustand zu einem lokalisierten Zustand?

As a response you could point to Subsection 4.4 (What is a particle?) in my paper ''Foundations of quantum physics I. A critique of the tradition", where I discuss this question in parallel with a corresponding experiment with classical particles (4mm bullets from an air gun).

The paper by Schirber [54] discusses essentially the same phenomenon in a fully classical context, where a bullet is fired into a sheet of glass and produces a large number of radial cracks in random directions. This is shown in the first figure there, whose caption says,

”The number of cracks produced by a projectile hitting a glass sheet provides information on the impactor’s speed and the properties of the sheet.” In the main text, we read ”A projectile traveling at 22.2 meters per second generates four cracks in a 1-millimeter-thick sheet of Plexiglas. [...] A 56.7 meter-per-second projectile generates eight radial cracks in the same thickness Plexiglass sheet as above.” (See also Falcao & Parisio [20] and Vandenberghe & Villermaux [60].)

We see that the discrete, random detection events (the cracks) are a manifestation of broken symmetry when something impacts a material that – unlike water – cannot respond in a radially symmetric way. Randomness is inevitable in the breaking of a radial symmetry into discrete events. The projectile creates an outgoing spherical stress wave in the plexiglas and produces straight cracks. In fact, once initiated, the growth of a crack in a solid is not very different from the growth of a track in a bubble chamber, except that the energies and time scales are quite different. Only the initiation is random.

There we have exactly the same phenomenon: A radially symmetric source (here an impact impact center) causes a discrete number of cracks, solely through the properties of the detecting matter. The paper [54] is open access and has very nice pictures:

• M. Schirber, Focus: Windshield Cracks Hold Secrets of Impact, Physics 6 (2013), 48.

 

[A. Neumaier]

Because Bernhard wrote at https://www.astronews.com/community...etation-der-quantenmechanik.11402/post-137032

Wenn wir die Idee der Lokalisierung übertreiben, gelangen wir zu einem Bild, das Neumaier teilweise suggeriert, nämlich das der chaotischen Hydrodynamik. Aber klassische chaotische Hydrodynamik wird sicher die Bellschen Kriterien verletzen, d.h. sie ist lokal-realistisch, was wir durch andere Experimente ausschließen können.
(If we exaggerate the idea of localization, we arrive at an image that Neumaier partly suggests, namely that of chaotic hydrodynamics. But classical chaotic hydrodynamics will certainly violate Bell's criteria, i.e. it is local-realistic, which we can exclude by other experiments.)

Hydrodynamics is a coarse-grained approximation in which only 1-point functions are retained. Hence it cannot model microscopic correlation experiments. For the analysis of coincidence measurements one needs the dynamics of 2-point functions, which is not obtained correctly in the hydrodynamic approximation.
But one can work instead with the Kadanoff-Baym equations, which are the bilocal analogue of the hydrodynamic equations, and from which one can get the hydrodynamic equations through further coarse-graining.

The underlying determinstic dynamics is that of the $N$-point functions of quantum field theory, giving a coupled system for $N=0,1,2,\ldots$. This dynamics is local in the sense of quantum field theory, i.e., conforming to relativistic causality. However, the dynamics is nonlocal in the sense of Bell
since $N$-point functions for $N>1$ contain more than one spacetime argument. Therefore Bell's analysis does not apply.

 

[A. Neumaier]

https://www.astronews.com/community...etation-der-quantenmechanik.11402/post-137082 :

Das zusammen ergibt die Hypothese, dass die deterministische Zeitentwicklung des Zustandes alles enthält, was man benötigt, um alle objektiven Eigenschaften eines Systems zu berechnen, einschließlich der Ergebnisse einer Messung bzw. eines einzelnen Detektor-Ereignisses. Eine vollständige Lösung des Messproblems umfasst damit auch die Notwendigkeit, zu berechnen, dass genau eines und genau welches der Detektor-Elemente ein "teilchenartiges Detektorereignis" anzeigt.
(Altogether, this leads to the hypothesis that the deterministic time evolution of the state contains everything needed to calculate all the objective properties of a system, including the results of a measurement or a single detector event. A complete solution to the measurement problem thus also includes the need to calculate that exactly one and exactly which of the detector elements indicates a "particle-like detector event".)

Der vermeintliche Indeterminismus ist - analog zur klassische Mechanik - keine intrinsische Eigenschaft der Quantenmechanik. Heißt: wenn ich den initialen Zustand genau genug kenne und die Zeitentwicklung sowie die geeigneten q-expectations genau genug berechne, dann folgt daraus auch, welches einzelne Detektor-Element anspricht. Umgekehrt: wäre dem nicht so, gäbe es ein objektiv zufälliges Element - was Neumaier analog zur klassischen Mechanik explizit ausschließt - das nicht aus dem Zustand folgen würde - was er ebenfalls explizit ausschließt, da dieser alle objektiven Eigenschaften des Systems repräsentiert.
(The apparent indeterminism is - analogous to classical mechanics - not an intrinsic property of quantum mechanics. This means: if I know the initial state exactly enough and calculate the time development as well as the appropriate q-expectations accurately enough, then it also follows which individual detector element responds. Conversely: if this were not the case, there would be an objectively random element - which Neumaier explicitly excludes analogously to classical mechanics - which would not follow from the state - which he also explicitly excludes, since it represents all objective properties of the system.

This is a good exposition of part of my thermal interpretation.

Neumaier gelangt damit m.E. zu einem letztlich trivialen Bild: aus einem exakt bekannten Zustand und einer exakt bekannten und berechenbaren Dynamik folgt ein exaktes Ergebnis inkl. genau eines Detektor-Ereignisses.
(In my opinion, Neumaier thus arrives at an ultimately trivial picture: from an exactly known state and a precisely known and calculable dynamics, an exact result including exactly one detector event follows.)

Nontrivial and new are here indeed only the implied analysis of what precisely the objective properties are, and how they are encoded in the quantum formalism.

Das steht im Widerspruch zur Dekohärenz (This contradicts decoherence)

No - there is no contradiction with decoherence since the latter is completely silent about the behavior of single systems.

Decoherence only claims to compute ensemble properties, where the associated reduced density operator diagonalizes (in the pointer basis of selected experiments) after averaging over very high frequencies, which is a consequence of the Riemann–Lebesgue Lemma. The quantum tomography approach and the thermal interpretation refine the story told by decoherence in terms of averages to a different, more detailed story for each single case.