A Statistical ensemble interpretation done right

[Demystifier]

If that is the case, then I'm not sure why you had to ask this in your OP:

Ballentine is quite clear that the answer to this is "no". I've already quoted several passages showing that.

Some specific examples would help. The ones I gave, as I noted just now, seem to imply the opposite.

Here is one example. At page 207 of his book he writes:

"It is possible to prepare the lowest energy state of a system simply by waiting for the system to decay to its ground state."

He says "system", not "ensemble". To me, it looks as confirmation of my claim that in the case of known preparation he associates the state with a single system, not with an ensemble. But he is not perfectly clear and explicit about that, which I think creates confusion, so I wanted in this thread to make such things clear and explicit.

One should also have in mind what he says at page 46 (boldings are mine):

"The empirical content of a probability statement is revealed only in the
relative frequencies in a sequence of events that result from the same (or an
equivalent) state preparation procedure. Thus, although the primary definition of a state is the abstract set of probabilities for the various observables, it is also possible to associate a state with an ensemble of similarly prepared systems. However, it is important to remember that this ensemble is the conceptual infinite set of all such systems that may potentially result from the state preparation procedure, and not a concrete set of systems that coexist in space. In the example of the scattering experiment, the system is a single particle, and the ensemble is the conceptual set of replicas of one particle in its surroundings. The ensemble should not be confused with a beam of particles, which is another kind of (many-particle) system. Strictly speaking, the accelerating and collimating apparatus of the scattering experiment can be regarded as a preparation procedure for a one-particle state only if the density of the particle beam is so low that only one particle at a time is in flight between the accelerator and the detectors, and there are no correlations between successive particles."

Thus he makes clear that the ensemble interpretation of the state is not the only (in fact, not even the primary) interpretation of the state, and also that "system" is not the ensemble.

 

[Morbert]

But the difference is only in rhetoric, I don't see any substantial physical difference, at least in the classical case. Do you see a substantial physical difference? Or maybe, as an adherent of consistent histories, you see framing, i.e. rhetoric, as physical?

The difference would be the conceptualizations and intuitions invoked to reason about the single system. In all three statements, an ensemble is conceptualized, so they are quite similar, but only in 2. is this conceptualization insisted upon, in the sense that 1. and 3. imply we could discuss the state, divorced from the context of an ensemble. This is not necessarily a bad thing of course.

If by physical we mean real, there is no difference. What is real is the single system of interest.

As an aside: You could apply a statistical ensemble interpretation to the CH formalism. Given a set of histories, the probability of a history would be the relative frequency of that history in an ensemble of similarly prepared systems. Gell-Mann and Hartle went one step further and published an "extended probability ensemble decoherent histories" which embeds the real fine-grained history of a system in an ensemble of alternatives.

 

[Demystifier]

Gell-Mann and Hartle went one step further and published an "extended probability ensemble decoherent histories"

They lost me when they said that probability can be negative or larger than one.

 

[martinbn]

Yes, you summarized it very well. I see SEI as a rather practical approach, it always seemed to me that SEI is an attempt to formulate QM with a minimal amount of philosophy. Now I am becoming aware that not everybody sees SEI that way.

Then why did you start this thread?! You agreed with me, that you did not describe any interpretation of QM, and that what you wrote is not even specific to QM. So your view is that the ensemble interpretation is not really an interpretation of QM, it is just statistics. Now you have become aware that some people view the ensemble interpretation as an actual interprwtation!

 

[Demystifier]

Then why did you start this thread?!

See my post #47, the last paragraph. And also #49.

 

[martinbn]

Here is one example. At page 207 of his book he writes:

"It is possible to prepare the lowest energy state of a system simply by waiting for the system to decay to its ground state."

He says "system", not "ensemble". To me, it looks as confirmation of my claim that in the case of known preparation he associates the state with a single system, not with an ensemble. But he is not perfectly clear and explicit about that, which I think creates confusion, so I wanted in this thread to make such things clear and explicit.

He could have been more pedantic, but he probable thought it wouldnt cause a confusion.

One should also have in mind what he says at page 46 (boldings are mine):

"The empirical content of a probability statement is revealed only in the
relative frequencies in a sequence of events that result from the same (or an
equivalent) state preparation procedure. Thus, although the primary definition of a state is the abstract set of probabilities for the various observables, it is also possible to associate a state with an ensemble of similarly prepared systems. However, it is important to remember that this ensemble is the conceptual infinite set of all such systems that may potentially result from the state preparation procedure, and not a concrete set of systems that coexist in space. In the example of the scattering experiment, the system is a single particle, and the ensemble is the conceptual set of replicas of one particle in its surroundings. The ensemble should not be confused with a beam of particles, which is another kind of (many-particle) system. Strictly speaking, the accelerating and collimating apparatus of the scattering experiment can be regarded as a preparation procedure for a one-particle state only if the density of the particle beam is so low that only one particle at a time is in flight between the accelerator and the detectors, and there are no correlations between successive particles."

Thus he makes clear that the ensemble interpretation of the state is not the only (in fact, not even the primary) interpretation of the state, and also that "system" is not the ensemble.

Exactly! This show the difference between your view and his.

 

[martinbn]

See my post #47, the last paragraph. And also #49.

Yes, you say that, but it seems that you were cometely unaware of the interpretation itself!

 

[Demystifier]

Exactly! This show the difference between your view and his.

How? I see this as a confirmation that his view agrees with mine.

 

[vanhees71]

You are just switching the focus to the next word. Now you prefer your "what I'm investigating" over my "what you intent to measure":

By "investigating" I meant of course "doing an experiment".

Or maybe you are unhappy because I also used the word "system" in that sentence.

I'm unhappy about your formulation

The degrees of freedom and the Hilbert space you choose to describe them says something about what you intent to measure on the system.

because it seems to state a very common misconception about quantum mechanics. It seems as if you have an interpretation of the Heisenberg uncertainty relations in mind as if it would prevent from meausring the one or the other observable with arbitrary precision. This is, however, in no way what's implied by the uncertainty relation. I can always measure any observable of a system with as high a precision I want, and I'm not in any way restricted in the ability to choose to measure any observable of the system I like due to the state preparation.

The uncertainty relation also does not say anything about the disturbance of the system by measurement. It can't, because it's a fundamental principle derived without any reference to a specific measurement procedure, and how the system is disturbed by the interaction with the measurement apparatus of course depends on the specific device.

But in the end, this is a fight over words, or perhaps about "how to talk about that stuff". It seems mostly unrelated to the physics.

The above is of utmost significance for the correct interpretation of the formalism, and it doesn't in any way depend on the specific interpretation you prefer. It's one of the objective scientific properties of the theory.

 

[PeterDonis]

He says "system", not "ensemble". To me, it looks as confirmation of my claim that in the case of known preparation he associates the state with a single system

No, he associates the state with the preparation procedure that was used, exactly as he said in what you quoted from his p. 46. He uses "system" to refer to the thingie that comes out of the preparation procedure, precisely in order to distinguish that thingie from the preparation procedure and the abstract ensemble that results from it, which are what he says on p. 46 that the state describes.

Thus he makes clear that the ensemble interpretation of the state is not the only (in fact, not even the primary) interpretation of the state

I don't think so. See above.

 

[Morbert]

But in both cases, the "state as an ensemble" will be tested by a series of individual measurements events, that cannot be reduced nor averaged.

"equivalence class of preparations" is quite vague, as it cannot be equivalent as defined by QM itself (no cloning). Either way that ensemble is the most non-local thing there is in physics.

There's no issue with cloning. We can sketch a theory of a preparation procedure to better understand its meaning by extending our theory to include laboratory degrees of freedom. Consider a microscopic system $s$ in a lab $L$, and a desired quantum state $\rho_s(t_0)$. A preparation procedure $P$ is characterised by $$P:= (\rho_{s+L}(t_{-1}), \{C_i\})$$ such that $$\rho_s(t_0) = \frac{\mathrm{tr}_L C_i \rho_{s+L}(t_{-1})C^\dagger_i}{\mathrm{tr}_{s+L}C_i \rho_{s+L}(t_{-1})C^\dagger_i}$$where $\rho_{s+L}(t_{-1})$ is an earlier state of the system + lab and $\{C_i\}$ is an appropriate set of operators. And when we say the state represents a class of procedures, we mean there are many such $P$ that would satisfy the above desired $\rho_s(t_0)$.

Of course, if we extend our theory to include lab degrees of freedom, it can lead to some peculiar interpretations. The preparation can be associated with a POVM on the lab. Or we might consider an infinite ensemble of similarly prepared labs, of which a subensemble is associated with $\rho_s$. Some interpretations might even reject such a macroscopic application of Lueder's rule.

 

[vanhees71]

The latter idea may be expensive. Fortunately we don't need a million CERNs all doing once the same scattering experiments. It's fine to have one CERN and using its equipment to perform a million pp collisions ;-)).

 

[Morbert]

But by reading the many interpretation (of SEI flavors) explained by many people here, I am still looking for at least once factual problem that it addresses (except maybe the strong embedded denial that could help someone sleep at night), or only the vaguest way that map would relate to the territory.

Ballentine argues that it eliminates assumptions that i) play no role in the application of quantum theory, and ii) lead to conceptual difficulties with measurement processes.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

After some cleanup, the thread is reopened.

 

[gentzen]

Or maybe you are unhappy because I also used the word "system" in that sentence.

I'm unhappy about your formulation

The degrees of freedom and the Hilbert space you choose to describe them says something about what you intent to measure on the system.

because it seems to state a very common misconception about quantum mechanics. It seems as if you have an interpretation of the Heisenberg uncertainty relations in mind as if it would prevent from meausring the one or the other observable with arbitrary precision. This is, however, in no way what's implied by the uncertainty relation. I can always measure any observable of a system with as high a precision I want, and I'm not in any way restricted in the ability to choose to measure any observable of the system I like due to the state preparation.

On rereading this, I see again why I didn't know how to respond. One thing I could do is to explain the concrete example I had in mind:
When you describe a Stern-Gerlach experiment, your magnet may be described as fixed such that the magnetic field (except for its inhomogenity) points in z-direction, or you may describe it such that the magnet can be rotated around the particle beam. In the second case, you probably need to allow density matrices to describe the actual state of the incoming particles. But most introductory QM textbooks have not yet introduced density matrices at the point where they describe and analyse SG, so they typically go with the description using a fixed magnet. Independent of this, the silver atoms have more degrees of freedom in their quantum state than just the spin of the unpaired electron in the outer shell. But we won't describe those in our Hilbert-space for analyzing SG, because we don't intent to measure anything for which they would be relevant.
Another thing I could do is to explain why I participated in this unhappy discussion at all:
Morbert tried to explain some distinctions, and because we both have some "background" in the consistent histories interpretation, I thought I could help. Or at least, I didn't want to distance myself (more than I already did here and here) from that interpretation by staying silent.

The uncertainty relation also does not say anything about the disturbance of the system by measurement. It can't, because it's a fundamental principle derived without any reference to a specific measurement procedure, and how the system is disturbed by the interaction with the measurement apparatus of course depends on the specific device.

This was the actual reason why I still wanted to respond, because here you highlight a specific misconception of Heisenberg from his paper "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". I noticed that I never asked you whether you have any concrete papers (like the one above or his Solvay paper with Born) or books (like "Physics and Philosophy" or "Physics and Beyond") in mind when you say that Heisenberg is responsible for much of the confusion about QM. Here is a summary of some of his works, maybe it simplifies your "selection process":
https://www.informationphilosopher.com/solutions/scientists/heisenberg/

My own interpretation what you dislike about Heisenberg was his focus on the role of the observer (to which I can agree to the extent that it neglects the role of preparation and control), and also quite general his passion for philosophy.

 

[vanhees71]

On rereading this, I see again why I didn't know how to respond. One thing I could do is to explain the concrete example I had in mind:
When you describe a Stern-Gerlach experiment, your magnet may be described as fixed such that the magnetic field (except for its inhomogenity) points in z-direction, or you may describe it such that the magnet can be rotated around the particle beam. In the second case, you probably need to allow density matrices to describe the actual state of the incoming particles. But most introductory QM textbooks have not yet introduced density matrices at the point where they describe and analyse SG, so they typically go with the description using a fixed magnet. Independent of this, the silver atoms have more degrees of freedom in their quantum state than just the spin of the unpaired electron in the outer shell. But we won't describe those in our Hilbert-space for analyzing SG, because we don't intent to measure anything for which they would be relevant.

Once more, this is a misconception of the quantum state. The quantum state of the system you want to observe has nothing to do with the measurement device but with the preparation of this system. E.g., in the original Stern-Gerlach experiment the Ag atoms were prepared by letting Ag vapor stream out of a small hole in an oven. Indeed, this preparation procedure is described by a corresponding mixed state of Ag atoms.

You can direct your magnet independent of how you prepare the Ag atoms and thus you can measure its spin component in any direction given by the magnetic field you like. The state in this specific experiment indeed has to be described by a Ag-atom beam, prepared in a mixed state, no matter whether you fix the field in one direction or another.

Another thing I could do is to explain why I participated in this unhappy discussion at all:
Morbert tried to explain some distinctions, and because we both have some "background" in the consistent histories interpretation, I thought I could help. Or at least, I didn't want to distance myself (more than I already did here and here) from that interpretation by staying silent.

I don't know enough about the consistent-history interpretation to comment on this. I've once read about it and didn't find it in any way convincing compared to the minimal statistical interpretation, which reflects how QT is used in the physics community when discussing real-world experiments and not some philsophical isms...

This was the actual reason why I still wanted to respond, because here you highlight a specific misconception of Heisenberg from his paper "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". I noticed that I never asked you whether you have any concrete papers (like the one above or his Solvay paper with Born) or books (like "Physics and Philosophy" or "Physics and Beyond") in mind when you say that Heisenberg is responsible for much of the confusion about QM. Here is a summary of some of his works, maybe it simplifies your "selection process":
https://www.informationphilosopher.com/solutions/scientists/heisenberg/

For me Heisenberg and Bohr wrote the most incomprehensible papers compared to the other "founding fathers" of QT. Particularly matrix mechanics has been worked out by Born and Jordan in crystal-clear mathematical form, quickly followed by Pauli's derivation of the hydrogen spectrum using matrix mechanics. These authors are a much better read than Heisenberg and Bohr if you like to investigate the early development of this branch of the first discovery of modern quantum theory. This also includes the famous 2nd part, the "Dreimännerarbeit", written together by Born, Jordan, and Heisenberg. Heisenberg had ingenious ideas but always needed translators to clarify his insights for the normal physicist. This usually were Born and Jordan but also very much Pauli.

The most underrated of all these was Born, who was really the one recognizing the mathematics behind Heisenberg's weird Helgoland paper, which reflects the discovery process of the Göttingen group pretty nicely. Also see

https://arxiv.org/abs/2306.00842
https://doi.org/10.1140/epjh/s13129-023-00056-1

My own interpretation what you dislike about Heisenberg was his focus on the role of the observer (to which I can agree to the extent that it neglects the role of preparation and control), and also quite general his passion for philosophy.

What I dislike is his nebulous writing and overemphasizing philosophy over physics. He also had many misconceptions, which for some reason unfortunately still stuck in modern textbooks (although they don't play much of a role anymore in contemporary research), e.g., his first paper about the meaning of the uncertainty relation, which he first published claiming it were about the disturbance of the system by interaction with the measurement device, which leads to the misconception discussed in the first part of this posting. This was corrected immediately thereafter by Bohr, who was even more nebulous in his writing but often had the better physical instinct.

The disturbance of the system by measurement is of course also an important point following from the atomistic structure of matter. E.g., you cannot have an arbitrary small "test charge" to measure an electromagnetic field. You need at least one particle with 1e charge to probe the field, which inevitably disturbs it on these "microscopic scales". To describe this is much more complicated an after all can only be done for specific experimental setups analyzing the dynamics of the measurement process under investigation.

 

[Fra]

My own interpretation what you dislike about Heisenberg was his focus on the role of the observer (to which I can agree to the extent that it neglects the role of preparation and control)

As far as I recall reading som the historical descriptions, there was a early internal tension in the Copenhagen group, where Heisenberg focused more on the individual "observer", but while Bohr suggested that it's all of the classical reality taken together, that is the foundation for preparation, control and observeation. This doesn't mean Heisenberg was wrong and Bohr was right, as I understand it both views were unified because the internal communication within the classical domain is "trivial" as information can be copied/shared at least in principle, and that all different classical observes supposedly will AGREE on what actual records are. So Bohr and Heisenbergs views was consistent. (The dispersion of observations within classical world, is that due to special relativity, which of course wasnt what Bohr worried about, but to the extent QFT solves this, the conceptual idea of CI still holds).

So as I see it the idea is that the macroscopic measurement device is always "in tune" with the information implicit in the preparation and control of the source; as well as any prior information from process tomograpghy to determine hamiltonians etc. Changing of a dial at the detector, does not change the preparation. In this case the observer is always in principle "informed" about the preparation - via the classical communication channels. There is nothing that prevents this. Any issue between difference classical measurement devices are if nothing is wrong, supposed to be resolved by the state transformation of SR.

This is all conceptually fine, as long as we have the classical background spacetime and macrocsopic reality to back this up. I think of the "statistical interpretaion" as beeing litteraly processed and encoded in the macroscopic environment as well. So for me, the statistical interpretation are not really at face with CI. I think they get alon fine. The statistcal information, is the "knowledge of the obsever", and it is encoded in the macroscopic environment?

/Fredrik

 

[PeterDonis]

this preparation procedure is described by a corresponding mixed state of Ag atoms

Not really, since, as @gentzen correctly pointed out, the usual description only includes the outermost electron of the Ag atoms and leaves out the other degrees of freedom. So really the usual description of the state is a mixed state of qubits. (I say "qubits" instead of electrons because the description does make use of the fact that the Ag atoms are electrically neutral so there is no Lorentz force term in the Hamiltonian describing the interaction with the SG magnets, and therefore the only relevant interaction is the magnetic coupling to the spin-1/2 degree of freedom; for actual electrons that would not be the case.)

 

[lodbrok]

The system is what I'm investigating, e.g., an electron.

Is the system "an electron" or "an ensemble of similarly prepared electrons"? Whether you use a single CERN or one million CERNS, do the results of the measurements tell you about "an electron" or "an ensemble of similarly prepared electrons." (or preparation procedure).

 

[PeterDonis]

Is the system "an electron" or "an ensemble of similarly prepared electrons"?

That depends on which QM interpretation you are using. In statistical ensemble interpretations such as the OP describes, yes, the quantum state refers to an ensemble of electrons (more precisely, the spin-1/2 degree of freedom of the outermost electron in neutral silver atoms) all coming from the same preparation process.

Whether you use a single CERN or one million CERNS, do the results of the measurements tell you about "an electron" or "an ensemble of similarly prepared electrons." (or preparation procedure).

Same answer as above.

 

[lodbrok]

That depends on which QM interpretation you are using. In statistical ensemble interpretations such as the OP describes, yes, the quantum state refers to an ensemble of electrons (more precisely, the spin-1/2 degree of freedom of the outermost electron in neutral silver atoms) all coming from the same preparation process.Same answer as above.

Thank you, but the question was for vanhees in response to his statement that "the system" was "an electron", which implies a different interpretation than what I remember him using previously.

 

[PeterDonis]

the question was for vanhees

In post #67 he seems to be using an interpretation in which the quantum state describes the preparation procedure directly, as opposed to describing either an individual system produced by that procedure or an ensemble of such systems. But I agree it would be good for him to clarify.

 

[gentzen]

Once more, this is a misconception of the quantum state. The quantum state of the system you want to observe has nothing to do with the measurement device but with the preparation of this system.

There are indeed interpretations where the quantum state of the system has nothing to do with what I intent to measure on the system. But if a minimal statistical interpretation wants to keep some "operational verification" meaning, then it has to be a bit more careful at that point:

If you look at SEI from an operational verification perspective, then yes, you must associate a single system with a state before you know the results of the non-preparation measurements. This allows it to take part in some verification. Of course, no statistical verification can ever fully reject your state assignments, at most it can tell you that winning the jackpot of a lottery would have been more probable than your obtained measurement results given your previous state assignments.

The issue arises, because the statistical verification requires a limit process of accumulating measurement statistics. But this limit process requires that the equivalence classes of preparation procedures are kept fixed during the verification. And what I intent to measure on the system can impact those equivalence classes.

Now I admit that "good taste" allows to relax those "rules" significantly in practice. But the idealized description should still be the one explained above. It is simply too easy to fool oneself with statistics, not least because our human intuition is not very good in that domain.

For me Heisenberg and Bohr wrote the most incomprehensible papers compared to the other "founding fathers" of QT. ... Heisenberg had ingenious ideas but always needed translators to clarify his insights for the normal physicist. This usually were Born and Jordan but also very much Pauli.

The most underrated of all these was Born, who was really the one recognizing the mathematics behind Heisenberg's weird Helgoland paper, which reflects the discovery process of the Göttingen group pretty nicely. Also see
https://arxiv.org/abs/2306.00842

Indeed, looks like Heisenberg needed both Born and Pauli. Jordan, not so sure, the paper is consistent with what I learned from other sources. He didn't do himself a favor by being cavalier with truth. Maybe Paul Ehrenfest is even more underrated than Born:

Einstein’s enduring estimate appears in his response to a letter from Paul Ehrenfest. To Ehrenfest’s remark that, despite his relative nothingness, Einstein and Bohr had always supported his work, "whereas contact with other theorists totally discourages me," Einstein replied that like himself and Bohr, Ehrenfest was a Principienfuchser, a worrier about foundations, while most other theorists were virtuosi, polished mathematicians or devotees of detail, but not quite the real thing. He gave as examples of the polished virtuoso Born and his predecessor at Göttingen, Debye.

The award of the Nobel prize to Born in 1954 made a perfect emblem of the “great adventure." It brought out the main distinguishing feature of the new physics, its radically probabilistic basis, and implied its coherence and completeness by rewarding the co-inventor of MM for his elucidation of its rival WM. The earlier reservations about Born's contributions were ignored or forgotten. And so the Nobel establishment gave a fifth prize for wave mechanics, including those awarded in 1937 for the "experimental discovery of the diffraction of electrons." MM could boast but one.

What I dislike is his nebulous writing and overemphasizing philosophy over physics.

Well, thanks for answering me. I understand that you dislike his overemphasizing philosophy over physics. The general accusation of nebulous writing without reference to concrete papers or books suggests to me that you actually have not read much of his writings. Perhaps you tried to read his breakthrough paper, and didn't get what you hoped for.

He also had many misconceptions, which for some reason unfortunately still stuck in modern textbooks (although they don't play much of a role anymore in contemporary research), e.g., his first paper about the meaning of the uncertainty relation, which he first published claiming it were about the disturbance of the system by interaction with the measurement device, which leads to the misconception discussed in the first part of this posting.

Trying to turn one concrete misconception into a general accusation of many misconceptions again suggests to me that you are simply not very familiar with much of Heisenberg's work. Maybe you would also cite his infamous unified field theory of elementary particles as another example of his misconceptions. But both concrete misconceptions are common knowledge concerning Heisenberg, not an indication of any non-trivial familiarity with this work.

Anyway, thanks again for your answer. I always have to force myself to write an answer in cases where I am not so sure about my answer, because I know that it would be impolite if I just stayed silent. But still, sometimes staying silent is actually a good idea.

 

[PeterDonis]

the statistical verification requires a limit process of accumulating measurement statistics. But this limit process requires that the equivalence classes of preparation procedures are kept fixed during the verification.

Yes.

And what I intent to measure on the system can impact those equivalence classes.

No, it doesn't. You can do different measurements on systems that were prepared by the same preparation procedure. The preparation procedure (or equivalence class of such procedures--but in the case under discussion, the SG experiment, there was only one preparation procedure so the "equivalence class" question is moot) is modeled by the quantum state. The measurement you make is modeled by an operator. The two are distinct and independent.

 

[gentzen]

Jordan, not so sure, the paper is consistent with what I learned from other sources. He didn't do himself a favor by being cavalier with truth.

Here I found a favorable account of Jordan's contributions:

It was Jordan, more than anyone else, who developed a mathematically elegant formulation of matrix mechanics (Born and Jordan 1925; 1926). It was Jordan who went on to consolidate matrix mechanics with Dirac’s alternative operator calculus (Dirac 1925) and Erwin Schrödinger’s wave-mechanical formulation (Schrödinger 1926a; 1926b) in the comprehensive formalism known as statistical transformation theory (Jordan 1927a; 1927b, see also Duncan and Janssen 2009). It was Jordan who did more than anyone other than Dirac to inaugurate the program of quantum field theory, in ways such as developing the second quantization approach and being the first to discover the problem of divergences in quantum field theory (Jordan and Klein 1927; Jordan and Wigner 1928). And it was Jordan who, along with von Neumann and Eugene Wigner, was developing more abstract algebraic frameworks for quantum mechanics (Jordan 1933b; Jordan et.al. 1934).

But in case you believe that Jordan was less deep into philosophy than Heisenberg, see:
https://www.informationphilosopher.com/solutions/scientists/jordan/

 

[gentzen]

And what I intent to measure on the system can impact those equivalence classes.

No, it doesn't. You can do different measurements on systems that were prepared by the same preparation procedure. The preparation procedure (or equivalence class of such procedures--but in the case under discussion, the SG experiment, there was only one preparation procedure so the "equivalence class" question is moot) is modeled by the quantum state. The measurement you make is modeled by an operator. The two are distinct and independent.

To stay with the SG experiment, should a different isotope composition of the beam of silver atoms be considered to be a different preparation procedure? It doesn't matter for the correlation between spin and momentum. It has a slight impact on the correlation between momentum and position. And if it were important to us, we could simply measure the isotope composition of "our" beam. For some experiments, it might even be important for the beam to consists only of a single isotope, because of its impact on the exact trajectory.

 

[vanhees71]

In post #67 he seems to be using an interpretation in which the quantum state describes the preparation procedure directly, as opposed to describing either an individual system produced by that procedure or an ensemble of such systems. But I agree it would be good for him to clarify.

The quantum state is the formal description of a preparation procedure on a single system. In this sense the quantum state refers to the single system. The meaning of the quantum state is, of course, entirely statistical, i.e., the possible outcome of any observable is given by Born's rule, i.e., the possible outcomes are the eigenvalues of the self-adjoint operator, representing the measured observable, $\hat{O}$. Then let $|o,\alpha \rangle$ be the complete orthonormal set of eigenvectors of $\hat{O}$ with the eigenvalue $o$, then given the quantum state $\hat{\rho}$, the probablity to get $o$ when measuring $O$ (the observable is defined by a measurement procedure) then is given by
$$P(o|\hat{\rho})=\sum_{\alpha} \langle o,\alpha|\hat{\rho}|o,\alpha \rangle.$$
This can, of course, only be verified experimentally by preparing an ensemble of independently prepared systems in the state described by $\hat{\rho}$.

 

[vanhees71]

There are indeed interpretations where the quantum state of the system has nothing to do with what I intent to measure on the system. But if a minimal statistical interpretation wants to keep some "operational verification" meaning, then it has to be a bit more careful at that point:

That's the standard interpretation and practice in real-world physics laboratories. There's some preparation procedure (e.g., set up a Ag-atom beam by heating up silver in an oven with a little hole, letting it go through some slits to collimate it, etc.) and independently you can measure any observable on the so prepared system you like (e.g., the component of its spin in an arbitrary direction by putting a corresponding magnetic field in this direction and a screen registering the Ag atoms at the corresponding places behind the magnet; the magnet entangles the spin component with the position at the screen modulo some small systematical error since the inhomogenous magnetic field has at least a small component in a different direction).

The issue arises, because the statistical verification requires a limit process of accumulating measurement statistics. But this limit process requires that the equivalence classes of preparation procedures are kept fixed during the verification. And what I intent to measure on the system can impact those equivalence classes.

Of course, the preparation procedure has to be kept fixed to get an ensemble described by the intended state you want to investigate.

Now I admit that "good taste" allows to relax those "rules" significantly in practice. But the idealized description should still be the one explained above. It is simply too easy to fool oneself with statistics, not least because our human intuition is not very good in that domain.

I've no clue, what you mean by this. This has nothing to do with QT but holds for any statistics, including "classical statistics".

Indeed, looks like Heisenberg needed both Born and Pauli. Jordan, not so sure, the paper is consistent with what I learned from other sources. He didn't do himself a favor by being cavalier with truth. Maybe Paul Ehrenfest is even more underrated than Born:

Jordan was behind most of the math. E.g., he helped born to prove the famous commutation relation between position and momentum, he developed em-field quantization (in 1925/26 before Dirac!), etc. He is the most underrated of the founding fathers of QT. That's partially, because he was "too mathematical" for the taste of many physicists but also because he was foolish enough with his political involvement with the Nazis in 1933-1945.

Well, thanks for answering me. I understand that you dislike his overemphasizing philosophy over physics. The general accusation of nebulous writing without reference to concrete papers or books suggests to me that you actually have not read much of his writings. Perhaps you tried to read his breakthrough paper, and didn't get what you hoped for.

I read the Helgoland paper. Of course, you can understand it, having learnt the elaborated theory nearly 100 years later. If you, however, try to understand it without assuming this knowledge, it's hopeless. Matrix Mechanics was worked out in clear mathematical form by Born and Jordan (of course under participation of Heisenberg) shortly afterwards.

Trying to turn one concrete misconception into a general accusation of many misconceptions again suggests to me that you are simply not very familiar with much of Heisenberg's work. Maybe you would also cite his infamous unified field theory of elementary particles as another example of his misconceptions. But both concrete misconceptions are common knowledge concerning Heisenberg, not an indication of any non-trivial familiarity with this work.

Sure, it's telling. Pauli withdraw his name from the paper before publication. I think I'm quite famliar with QT in a hopefully not too trivial way ;-).

Anyway, thanks again for your answer. I always have to force myself to write an answer in cases where I am not so sure about my answer, because I know that it would be impolite if I just stayed silent. But still, sometimes staying silent is actually a good idea.

Well, one can only learn something in these matters by discussions.

 

[vanhees71]

If I prepare single-electron states, I deal with a single electron. Of course, as with any experiment, I have to prepare and entire ensemble of a single-electron state to verify the probabilistic predictions implied by this preparation. I don't need a million of CERNs to prepare an ensemble of proton or heavy-ion beams. This I can do with 1 CERN as is done in the real world ;-)).

 

[martinbn]

The quantum state is the formal description of a preparation procedure on a single system. In this sense the quantum state refers to the single system. The meaning of the quantum state is, of course, entirely statistical, i.e., the possible outcome of any observable is given by Born's rule, i.e., the possible outcomes are the eigenvalues of the self-adjoint operator, representing the measured observable, $\hat{O}$. Then let $|o,\alpha \rangle$ be the complete orthonormal set of eigenvectors of $\hat{O}$ with the eigenvalue $o$, then given the quantum state $\hat{\rho}$, the probablity to get $o$ when measuring $O$ (the observable is defined by a measurement procedure) then is given by
$$P(o|\hat{\rho})=\sum_{\alpha} \langle o,\alpha|\hat{\rho}|o,\alpha \rangle.$$
This can, of course, only be verified experimentally by preparing an ensemble of independently prepared systems in the state described by $\hat{\rho}$.

Just a side comment, that this is just one possible interpretation. I know that you are clarifying what you view is, but there are other possibilities. For example you have interpretations where the state describes the ensemble of equally prepared systems, not any single system.

 

[vanhees71]

This is not an interpretation, that's how QT is used in analyzing real experiments.

To build an ensemble you have to prepare single systems independently from each other. So there must be a meaning of "state" for a single system, and that's the preparation procedure (or rather an "equivalence class of preparation procedures") applied repeatedly to many single systems to prepare an ensemble.

Since the physical meaning described by the state, $\hat{\rho}$, is entirely probabilistic, you can say that it indeed describes an ensemble in the sense that the probabilities it predicts, can only be observed on (sufficiently large) ensembles, but to build the ensemble you have to refer to each single member of this ensemble, and that leads to the operational meaning of the state as a preparation procedure.

 

[martinbn]

This is not an interpretation, that's how QT is used in analyzing real experiments.

No, it is an interpretation. I think you are confused about it, and you don't see the difference.

To build an ensemble you have to prepare single systems independently from each other.

The ensemble in this interpretation is an abstract entity. You don't build it. You cannot build it. It is an equivalent class, it ha infinitely many members.

So there must be a meaning of "state" for a single system, and that's the preparation procedure (or rather an "equivalence class of preparation procedures") applied repeatedly to many single systems to prepare an ensemble.

Why there must be? Just because you prefer it that way?

Since the physical meaning described by the state, $\hat{\rho}$, is entirely probabilistic, you can say that it indeed describes an ensemble in the sense that the probabilities it predicts, can only be observed on (sufficiently large) ensembles, but to build the ensemble you have to refer to each single member of this ensemble, and that leads to the operational meaning of the state as a preparation procedure.

And as I said, there infinitely many members in the ensemble, what do you mean by "build it"?

 

[Fra]

The ensemble in this interpretation is an abstract entity.

I in part agree, but isnt the probabilistic embedding of a single instance as well?

But approximately information encoded in the macro environment (rhe lab and all its computers) is real, and not an astraction. Its how i would defend this.

/Fredrik

 

[Fra]

And as I said, there infinitely many members in the ensemble, what do you mean by "build it"?

How about that the environment(observer) is "learning" about what state distribution the preparation produces. Building ~ learning, tuning the procedure? Makes sens to me.

/Fredrik

 

[vanhees71]

No, it is an interpretation. I think you are confused about it, and you don't see the difference.

The ensemble in this interpretation is an abstract entity. You don't build it. You cannot build it. It is an equivalent class, it ha infinitely many members.

It's not an abstract entity, it's a real-world item to be investigated with real-world experimental setups.

Why there must be? Just because you prefer it that way?

You must be able to associate the abstract entity $\hat{\rho}$ ("statistical operator") of the formalism with the real-world system you want to investigate, and that's the preparation procedure done on this system in the real-world experiment you do with it.

And as I said, there infinitely many members in the ensemble, what do you mean by "build it"?

In a real-world experiment there's a finite ensemble, and you have the corresponding statistical errors (in addition to the usually also apparent systematical ones) when comparing the predicted probabilities to the measured averages over a finite ensemble.

 

[martinbn]

It's not an abstract entity, it's a real-world item to be investigated with real-world experimental setups.

You must be able to associate the abstract entity $\hat{\rho}$ ("statistical operator") of the formalism with the real-world system you want to investigate, and that's the preparation procedure done on this system in the real-world experiment you do with it.

In a real-world experiment there's a finite ensemble, and you have the corresponding statistical errors (in addition to the usually also apparent systematical ones) when comparing the predicted probabilities to the measured averages over a finite ensemble.

You are simply making claims based on a different interpretation instead of providing any arguments!!! You cannot reject one interpretation using another!

 

[Fra]

For example you have interpretations where the state describes the ensemble of equally prepared systems, not any single system.

How would one(an observer or a physicists) alternatically practically or operationally determine/define the probabilistic description of a single system by any level of confidence? unless history repeats itself. In a way one might say that provided that field quanta are indiguishable, can we really say wether history DID repeat itself, or if we observed the "same thing once again", without using auxiliary information? (would that be cheating btw?)

/Fredrik

 

[martinbn]

How would one(an observer or a physicists) alternatically practically or operationally determine/define the probabilistic description of a single system by any level of confidence?

What do yiu mean by that? Once the system is prepared, you can make only one measurment. After that the system is no longer in the same prepararion.

unless history repeats itself. In a way one might say that provided that field quanta are indiguishable, can we really say wether history DID repeat itself, or if we observed the "same thing once again", without using auxiliary information? (would that be cheating btw?)

/Fredrik

 

[PeterDonis]

To stay with the SG experiment, should a different isotope composition of the beam of silver atoms be considered to be a different preparation procedure?

That would depend on how much detail about the state you want to represent. In the usual treatment of the SG experiment, all the details that the isotope composition can affect are left out.

 

[PeterDonis]

It's not an abstract entity, it's a real-world item to be investigated with real-world experimental setups.

Not according to, for example, Ballentine, Section 2.1. From p. 46 in my edition:

"However, it is important to remember that [the] ensemble is the conceptual infinite set of all such systems that may potentially result from the state preparation procedure, and not a concrete set of systems that coexist in space."

In a real-world experiment there's a finite ensemble

Ballentine disagrees with this as well. In addition to the above, there is this from the same page:

"In the example of the scattering experiment, the system is a single particle, and the ensemble is the conceptual set of replicas of one particle in its surroundings. The ensemble should not be confused with a beam of particles, which is another kind of (many-particle) system."

 

[Fra]

What do yiu mean by that? Once the system is prepared, you can make only one measurment.

That's of course the problem, but I meant that to me my question to you

I understand the matheamtical concept that you have imaginary ensembles, but the the question is, how do you justify this probabilistic reasoning, in an empirical and setting (that at least in principle, is doable for the class of "obsevers" we entertain, in a real interaction), without beeing able to repeating the process.

/Fredrik

 

[PeterDonis]

how do you justify this probabilistic reasoning, in an empirical and setting (that at least in principle, is doable for the class of "obsevers" we entertain, in a real interaction), without beeing able to repeating the process

You can repeat the process. That is the whole point of having a preparation procedure that you can run the same way many times.

 

[Fra]

You can repeat the process. That is the whole point of having a preparation procedure that you can run the same way many times.

Yes agreed. But I have a feeling we have some misunderstanding here.

I read martinb to suggest that, when do to repeat, it's no longer the SAME system. Ie. it's not the SAME electron beeing fired etc. And at the same time, he wants to attribute say the the density matrix to the system, and not the "ensemble" produced by a given preparation procedure given infinite number of samples?

I did I miss something?

/Fredrik

 

[PeterDonis]

I read martinb to suggest that, when do to repeat, it's no longer the SAME system.

It's not the same system, but that doesn't mean it's not the same preparation process. The same preparation process can prepare multiple systems.

at the same time, he wants to attribute say the the density matrix to the system, and not the "ensemble" produced by a given preparation procedure given infinite number of samples?

There are interpretations that do this. However, I'm not sure @martinbn is using one. Is there something specific he said that makes you think so?

 

[Fra]

It's not the same system, but that doesn't mean it's not the same preparation process. The same preparation process can prepare multiple systems.

Agreed.

There are interpretations that do this. However, I'm not sure @martinbn is using one. Is there something specific he said that makes you think so?

I might have misinterpreted his post #81 as I read it it again.

What I meant in post #84 is that I agree that strictly speaking the ensemble as well as the p-distribution itself are fictions or abstractions. My point was based on that i thought it was questioned, without this empirical side, how do we justify the abstractions?

Does real observations or information processing approximate our theories, or is it the other way around? We interpret theories, but do we interpret information processing?

/Fredrik

 

[PeterDonis]

without this empirical side, how do we justify the abstractions?

We can't. We justify the abstractions ultimately by the fact that they make accurate predictions (if they do) about what we observe and measure empirically.

Does real observations or information processing approximate our theories, or is it the other way around? We interpret theories, but do we interpret information processing?

I'm not sure what role "information processing" plays here. We use our theories to construct models, and we compare the models with reality through their predictions about what we should observe and measure, compared with what we actually observe and measure. I'm not sure how this fits in to whatever picture you have. But that is how I would describe what is going on in as plain and simple language as I can.

 

[Fra]

We can't. We justify the abstractions ultimately by the fact that they make accurate predictions (if they do) about what we observe and measure empirically.

I don't think we disagree here, perhaps what martinb tried to say was unclear to me.

I'm not sure what role "information processing" plays here. We use our theories to construct models, and we compare the models with reality through their predictions about what we should observe and measure, compared with what we actually observe and measure.

Yes, thats the de facto information processing we do and what i meant in this case. Trying to stick to basics. In the more general picture but "information process" I would probably count every physical process in the macropscopic environment, that implicitl encodes information about subsystems. But the principle is the same.

Anyway, I can't see I disagree with you anywhere here.

/Fredrik

 

[martinbn]

Yes agreed. But I have a feeling we have some misunderstanding here.

I read martinb to suggest that, when do to repeat, it's no longer the SAME system. Ie. it's not the SAME electron beeing fired etc. And at the same time, he wants to attribute say the the density matrix to the system, and not the "ensemble" produced by a given preparation procedure given infinite number of samples?

I did I miss something?

/Fredrik

That is not what I meant. It is the same system, say the same electron, but it is not a member of the same ensemble any more. For example if it was prepared so that it is a member of the ensemble with state spin |up>, and you measure spin in a horizontal direction and you get "left", now this same electron is a member of an ensemble in the sate |left>. So any following measurement will be on the original system, but not as originally prepared.

And if you prepare another electron in the spin up, then it is not the same system, but it is still the same ensemble.

 

[PeterDonis]

perhaps what martinb tried to say was unclear to me

In post #81 he was just responding to the statement he quoted by @vanhees71. There is a difference between saying that the quantum state describes the results of a preparation procedure on a single system (which is what @vanhees71 said) and saying that the quantum state describes an ensemble of systems all prepared by the same preparation procedure (which is the type of interpretation that the title and OP of this thread refer to).