I Where is the quantum system prior to measurement?

[Lynch101]

TL;DR Summary

To explore the idea that the system has location/position* prior to measurement which requires description for a theory/interpretation to be considered complete.

*Not necessarily a single, pre-defined value.

Continuing the discussion in the 'Assumptions of Bell's Theorem' thread, I'm hoping to explore the question of the location/position of the QM system prior to measurement.

I may have some bias or underlying assumption that is affecting the conclusion that I am drawing and, by exploring this question, my bias might become clear - or the alternative .

There are a probably a number of different ways to explore this question, but I'm hopeful that by making some general statements and asking some general questions (and having those probed, questioned, and challenged) the discussion will develop organically.I think the following are the only necessary assumptions but, as I say, there might be some hidden assumptions I am making:

Assumptions
1. The universe is spatially extended.
2. The system [we prepare and consider] is a subset of the universe i.e. it is not the entire universe*.Is it the case that one of the two following propositions must be true:
1) Location/position is an 'element of reality' of the system, prior to measurement.
2) Location/position is not an 'element of reality' of the system, prior to measurement.

Do the following make sense:
A) If someone tells you that they have hidden something 'somewhere in the field over there'. Would you know where to look for that 'something'?

B) Does the idea of being somewhere in the universe make sense?

B) If we have two laboratories, one in Paris the other in Rome. Does it make sense to say that, as part of our experiment to test quantum theory, a system was prepared in the lab in Paris?

C) If the system is prepared in the lab in Paris, does it make sense to say that the system is located somewhere in the lab in Paris and not in the lab in Rome?

 

[gentzen]

C) If the system is prepared in the lab in Paris, does it make sense to say that the system is located somewhere in the lab in Paris and not in the lab in Rome?

Yes, it makes sense to say that the system is located somewhere in the lab in Paris. I would be more careful with the negative assertion "and not in the lab in Rome". There might be nonlocal correlations that make negative assertion dangerous, at least sometimes. Positive assertions on the other hand are normally unproblematic.

 

[Lynch101]

Yes, it makes sense to say that the system is located somewhere in the lab in Paris. I would be more careful with the negative assertion "and not in the lab in Rome". There might be nonlocal correlations that make negative assertion dangerous, at least sometimes. Positive assertions on the other hand are normally unproblematic.

Would it be safe to say that the system is not everywhere in the universe?

 

[gentzen]

Would it be safe to say that the system is not everywhere in the universe?

I don't want to be drawn into your discussion with PeterDonis. Your assertion is so weak that I believe the only reason why you wanted to say it was your discussion with PeterDonis. It is simply not a useful assertion, because it has no chance to imply any observable consequences.

 

[Lynch101]

I don't want to be drawn into your discussion with PeterDonis. Your assertion is so weak that I believe the only reason why you wanted to say it was your discussion with PeterDonis. It is simply not a useful assertion, because it has no chance to imply any observable consequences.

The discussion isn't exclusively between PeterDonis and I. It's open to anyone.

The assertion is indeed quite weak. I don't believe it needs to be any stronger to draw the inferences I believe can be drawn.

The consequences need not necessarily be observable in order to draw the conclusion of incompleteness.

 

[PeterDonis]

I don't want to be drawn into your discussion with PeterDonis.

The discussion isn't exclusively between PeterDonis and I. It's open to anyone.

More than that, I am intentionally not posting in this separate thread, so that other people can have a discussion with @Lynch101 without being influenced by my particular opinions. I have already said all that I can usefully say in the previous thread. So everyone please feel free to post here about the OP question; that is the purpose of having this separate thread.

 

[msumm21]

Summary:: To explore the idea that the system has location/position* prior to measurement which requires description for a theory/interpretation to be considered complete.

*Not necessarily a single, pre-defined value.

A) If someone tells you that they have hidden something 'somewhere in the field over there'. Would you know where to look for that 'something'?

B) Does the idea of being somewhere in the universe make sense?

B) If we have two laboratories, one in Paris the other in Rome. Does it make sense to say that, as part of our experiment to test quantum theory, a system was prepared in the lab in Paris?

C) If the system is prepared in the lab in Paris, does it make sense to say that the system is located somewhere in the lab in Paris and not in the lab in Rome?

I’m not an expert and I haven’t read the other forum where this originated, but it seems like your questions are ones that nobody knows the answer to. Does something “have a location” when it’s not “measured”? Is that a core question here? Some interpretations of QT (e.g. pilot-wave) say yes, others say no, right? I realize I’m probably missing a deeper question here, maybe you could clarify?

Regarding C, I assume you are aware of Bell experiments? You could measure/prepare something in Paris from a lab in Rome, but only if very carefully setup that way.

 

[Lynch101]

I’m not an expert and I haven’t read the other forum where this originated, but it seems like your questions are ones that nobody knows the answer to. Does something “have a location” when it’s not “measured”? Is that a core question here? Some interpretations of QT (e.g. pilot-wave) say yes, others say no, right? I realize I’m probably missing a deeper question here, maybe you could clarify?

Yep. The original discussion stemmed from a claim I made, that those interpretations which only provide statistical predictions for the outcomes of measurements i.e. only tell us the probability of what we will observe on measurement devices are, by definition, incomplete descriptions of physical reality because they do not describe the quantum system prior to measurement. To my mind this seems pretty obvious but I might be making an invalid assumption somewhere.

There are a couple of different arguments, all of which are pretty simple and straightforward:

The argument from Somewhere
1) The quantum system must be located somewhere in the universe prior to measurement.
2) A complete 'description of physical reality' must provide a description of the location of the system prior
to measurement.
3) The description of the location/position of the system does not need to be a single, pre-defined value.
4) Given 3) we don't need a single, pre-defined value for position but we do need some description of
location/position.

The argument from Nowhere
5) If the system is not located anywhere in the universe then it is not in/part of the universe.
6) If the system is not in/part of the universe then it cannot interact with measurement devices which are in/
part of the universe.

The argument from Near here
7) The QM probability distribution tells us the probability of measuring the system in a given spatial location.
8) One of the following propositions must be true:
- The QM system is located in (or adjacent to) the given spatial region.
- The QM system is not located in (or adjacent to) in the given spatial region.
9) If the QM system is not located (or adjacent to) in the given spatial region then it cannot interact with the
measurement device.

The argument from Everywhere
10) To have the possibility of interacting with a measurement device, the QM system must be located in (or
adjacent to) the given spatial region, otherwise see 9) above.
11) If the probability distribution is not the result of a lack of information and there is a genuine possibility
of measuring the system in a spatial region with a zero-probability, then the system must be located in
all of those locations with a non-zero probability.
12) If the system is located in all of those spatial regions, then some form of FTL 'collapse' must occur to
give rise to a single measurement outcome.

Regarding C, I assume you are aware of Bell experiments? You could measure/prepare something in Paris from a lab in Rome, but only if very carefully setup that way.

I am aware of Bell experiments msumm. The point I am trying to get at is, is it possible to say that there is somewhere that the system is not?

If we prepare a system in Paris from a lab in Rome, is it possible to say that the system is not on the moon?

The point I am trying to make is:
1) If the system is not located everywhere, and we can state where it is not located then, by the process of
elimination we could [theoretically] have a description of where the system
is located, prior to measurement.

 

[EPR]

The quantum system is mostly around the peak of the probability amplitude.

 

[Lynch101]

The quantum system is mostly around the peak of the probability amplitude.

But also partially around the spatial regions with non-zero probabilities of measurement?

 

[EPR]

But also partially around the spatial regions with non-zero probabilities of measurement?

Of course.

You should disregard the other worlds as they don't affect yours.

 

[vanhees71]

I’m not an expert and I haven’t read the other forum where this originated, but it seems like your questions are ones that nobody knows the answer to. Does something “have a location” when it’s not “measured”? Is that a core question here? Some interpretations of QT (e.g. pilot-wave) say yes, others say no, right? I realize I’m probably missing a deeper question here, maybe you could clarify?

Regarding C, I assume you are aware of Bell experiments? You could measure/prepare something in Paris from a lab in Rome, but only if very carefully setup that way.

If you have a quantum system for which a position obsevable exists (e.g., all massive particles as well as massless particles with spin 0 or spin 1/2), then you can measure the position of this system/particle. Position, as a then necessarily continuous observable is never exactly determined but a particle can be principally localized in a small region (but usually not smaller than its Compton wavelength, because if you try to localize it even better you'll create rather more particles than localizing the single particle better). The only meaningful statement you can make is that when measuring the position of the system, you'll find it witin a volume $\mathrm{d} V$ at a given place with a probability $\rho(t,x,x) \mathrm{d} V$, where $\rho(t,x,x')$ are the position matrix elements of the statistical operator describing the state of the particle.

Concerning C: That's right. You can prepare, say, an entangled photon pair by parametric downconversion in the polarization-singlet state and direct one of them to Rome. If you make sure that this photon is not interacting with anything on its way then by measuring the other photon in Paris and let the other photon only go to Rome if your photon is horizontally polarized, then you know with certainty that the experimentalist measuring the polarization in Rome in precisely the same direction, he or she will find the photon to be vertically polarized.

 

[Lynch101]

Concerning C: That's right. You can prepare, say, an entangled photon pair by parametric downconversion in the polarization-singlet state and direct one of them to Rome. If you make sure that this photon is not interacting with anything on its way then by measuring the other photon in Paris and let the other photon only go to Rome if your photon is horizontally polarized, then you know with certainty that the experimentalist measuring the polarization in Rome in precisely the same direction, he or she will find the photon to be vertically polarized.

Does this not imply that there is a time when the system is not in Rome then?

 

[vanhees71]

Since photons don't have a position observable it doesn't even make sense to ask where they are. You can only give a probability distribution for detecting them at a place defined by the location of the detector.

 

[Lynch101]

Since photons don't have a position observable it doesn't even make sense to ask where they are. You can only give a probability distribution for detecting them at a place defined by the location of the detector.

I think this is where we are in disagreement because I think it does make sense to ask where they are. The answer may not be a single, pre-defined value but, to my mind, it makes sense to say that the system must be located (or have position) somewhere in the universe.

I think you said that this is a trivial point and that the probability of it being somewhere in the universe is 1.

 

[gentzen]

I think it does make sense to ask where they are. The answer may not be a single, pre-defined value

Your terminology is slightly dangerous here, because asking and getting answers is often associated with measurements, but that is probably not what you want.
It seemed to me that what you want is more to make certain assertions about a system that may be true independent of whether you make a specific measurement to check them.

 

[Lynch101]

Your terminology is slightly dangerous here, because asking and getting answers is often associated with measurements, but that is probably not what you want.
It seemed to me that what you want is more to make certain assertions about a system that may be true independent of whether you make a specific measurement to check them.

I understand that it usually is associated with measurements but I don't think it necessarily needs to be. As PeterDonis mentioned in the 'Assumptions' thread.

The hidden variables do not even have to be observables, and they certainly do not have to contain all possible observables. They just have to contain enough information to determine the results of the measurements being conducted.

It seemed to me that what you want is more to make certain assertions about a system that may be true independent of whether you make a specific measurement to check them.

I think it is, or at least should be, possible to make certain deductions about nature that are not limited to what we can observe.

It might actually be more accurate to say that we can make certain deductions about different models of nature based on a given set of principles, as per the arguments above.

For example, if our model says the universe is spatially extended and the quantum system is a subsystem of the universe i.e. it is not everywhere in the universe then, we should be able to make the deduction that the system has a location in the universe which a full and complete description would describe.

Similarly, if we can say that the model is not located in a given spatial region e.g. the moon, then we should be able to conclude, by the process of elimination, where the system is located, in principle at least.What form this description takes is completely open, and it seems as though some interpretations do indeed have such a description. Not providing such a description, to my mind, renders a model/interpretation incomplete.

 

[vanhees71]

I think this is where we are in disagreement because I think it does make sense to ask where they are. The answer may not be a single, pre-defined value but, to my mind, it makes sense to say that the system must be located (or have position) somewhere in the universe.

I think you said that this is a trivial point and that the probability of it being somewhere in the universe is 1.

If something doesn't have a position variable, how can it then be located? The important point is that you cannot locate a photon. All there is are probabilities to register a photon with a detector, which is located somewhere, and you can locate the detector, because it has a well-defined (macroscopic) position observable. Of course everything is somwhere in the universe.

 

[Lynch101]

If something doesn't have a position variable, how can it then be located?

That is what I think we can deduce. It doesn't necessarily have to have a single, pre-defined value for location, but that doesn't mean that it isn't located anywhere nor have any position whatsoever.

The important point is that you cannot locate a photon. All there is are probabilities to register a photon with a detector, which is located somewhere, and you can locate the detector, because it has a well-defined (macroscopic) position observable.

And there are a number of ways we can interpret what the probabilities tell us. One such way is that the system has a definite position, we just don't know what it is. Under this scenario, only one possible outcome for the measurement is ever possible. There is only the seeming possibility that spatial regions with a non-zero probability can register the photon, but in truth there is not a genuine possibility for each region with a non-zero probability.

If we reject this idea and we say that there is a genuine possibility that the measurement device can register the photon in any spatial region with a non-zero probability, then we can ask what this tells us about the system. How is it a genuine possibility, as opposed to a seeming possibility?

And, if there is a genuine possibility that the photon will register in spatial regions with a non-zero probability, why is it that we only ever observe the photon in one spatial region? What happens to 'make nature choose' one over the others?

Of course everything is somwhere in the universe.

Is the system everywhere in the universe?

 

[PeterDonis]

That is what I think we can deduce.

Moderator's comment: Please bear in mind that "I think" is not an acceptable argument here. You should be able to either make an argument yourself showing how this can be deduced from the basic math and principles of QM, or give a reference that makes such an argument.

 

[StevieTNZ]

You might find this paragraph helpful:

If an electron is expelled from a nucleus, Schrodinger's equation predicts that the ψ wave spreads out evenly through space. But when the electron is revealed, by a detector for instance, or by a TV screen, its arrives at one point only, not spatially spread out.

-- pg 25 of 'Helgoland: Making Sense of the Quantum Revolution' by Carlo Rovelli.

 

[Lynch101]

You might find this paragraph helpful:

-- pg 25 of 'Helgoland: Making Sense of the Quantum Revolution' by Carlo Rovelli.

thank you Stevie, I must give Helgoland a read.

There appears to be different ways of interpreting the idea that 'the ψ wave spreads out evenly through space'. Some physicists ascribe an ontology to the wave function, while others treat it instrumentally.

We can explore the implications of both these possibilities and see what inferences/deductions we can make.

 

[vanhees71]

BTW: Does Rovelli explain what Heisenberg wanted to tell us with this paper? I never could understand its logic. Only the follow-up papers by Born and Jordan and by Born, Jordan, and Heisenberg ("Dreimännerarbeit") make an understandable theory for me, though the Helgoland paper is of course the result of Heisenberg's original idea.

 

[Lynch101]

In the other thread you mentioned:

An experiment consists of a preparation procedure followed by a measurement. E.g. in the SG experiment you put silver vapor into an oven at a given temperature and let out the silver atoms through a little hole.
...
Then you let this stream of Ag atoms go through a nicely tailored inhomogeneous magnetic field.

Presumably the 'little hole' that the atoms go through is a finite region of space. Can we not then conclude that the system, at some time, had a position within this little hole? Otherwise, in what sense do the atoms 'go through the little hole'?

Similarly, the 'nicely tailored inhomogeneous magnetic field', presumably, represents a finite region of space. Can we not, therefore, narrow down the position of the system (at some time during the experiement) to 'somewhere in that field'?

Can we set up a similar magnetic field with a detection plate in another location? Either, in the same room in the laboratory; a different room in the laboratory; a different building; a different town; etc. etc. and say that the quantum system doesn't go through those other magnetic fields? In this way can we rule out possible locations/positions of the quantum system? If so, this too would help us narrow down the location/position of the quantum system.If not, then in what sense does the quantum system 'go through the magnetic field'?

 

[vanhees71]

Well, all our observations are localized by the equipment we use. I don't understand your problem with that.

 

[Lynch101]

Well, all our observations are localized by the equipment we use. I don't understand your problem with that.

I'm just saying, if we can say the system passes through a 'little hole' or passes through one magnetic field but not the other, then we are necessarily saying the system is located in some finite region of space.

 

[vanhees71]

Sure. So what?

 

[Lynch101]

Sure. So what?

Just was we have narrowed down the location of the system to some finite region of space, we can try to narrow it's location down further within that finite region of space. We can ask, is the quantum system everywhere in that finite region of space?

 

[Lynch101]

Reasoning might lead us to the following conclusions. This is by no means an exhaustive list but it allows us to explore the implications:
1) The system is spatially extended to occupy the entire spatial region.
2) The system is spatially extended but doesn't occupy the entire spatial region.
3) The system is localised and occupies a single pre-defined position.
4) The system is localised but occupies multiple positions.

We can rule out the possibility that the system is not located anywhere in the given spatial region, since we have agreed that the system is located somewhere in that finite region of space.

All but #3 above would necessitate some form of spontaneous and/or FTL 'collapse' of the system to a single well-defined position.

If there are other possibilities, other that 1-4 above, we can explore the implications of those also.

 

[gentzen]

BTW: Does Rovelli explain what Heisenberg wanted to tell us with this paper? I never could understand its logic.

Yes, Rovelli tries to explain it. But if you already read the paper yourself, you will probably be disappointed by explanations like:

It is very simple: the forces are the same as in classical physics; the equations are the same as those of classical physics. But the variables are replaced by tables of numbers, or "matrices."

I guess what confuses you is that Heisenberg's tables of numbers or "matrices" seem to be just as unobservable as the position of an electron or its time of circulation. Yet the abstract says: "In der Arbeit soll versucht werden, Grundlagen zu gewinnen für eine quantentheoretische Mechanik, die ausschließlich auf Beziehungen zwischen prinzipiell beobachtbaren Größen basiert ist." Not sure whether an explanation like the following would help you:

We cannot find new laws of motion to account for Bohr’s orbits and his “leaps”? Fine, let’s stick with the laws of motion that we’re familiar with, without altering them.

Let’s change, instead, our way of thinking about the electron. Let’s give up describing its movement.

After trying to read the paper, my guess is that the logic is to give up describing the time dependence. Because that is what Heisenberg does: He looks at the Fourier expansion in time of the classical variable, replaces that infinite series of Fourier coefficients by an infinite matrix, and then explains how to replace the expressions for the Fourier series of $x^2$, $x^3$ and $f(x)$ for general analytic functions $f$ by corresponding expressions for his infinite matrix. Then he notices that the expression for $xy$ is not commutative and therefore more problematic, suggests that the classical expression $v\dot{v}$ should probably be replaced by $(v\dot{v}+\dot{v}v)/2$ because it is the derivative of $v^2/2$, but that it is less clear what to do with more general terms.

So it looks like Heisenberg is trying to find out how to keep the equations of motion, but reinterpret their "mathematical" meaning for his matrices (which he obtained by giving up attempts to descibe a time dependence). This guess matches well with the title "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen."

 

[PeterDonis]

Just was we have narrowed down the location of the system to some finite region of space, we can try to narrow it's location down further within that finite region of space.

In order to narrow down the location further, you would have to use a device that is smaller. And at some point you will run into the fact that, according to quantum field theory, to confine a quantum object into a smaller space requires more energy, and at some point you have pumped enough energy into the system to create more particles, so you're no longer measuring just the original system.

This illustrates the more general point that you can't just wave your hands and say "we can try" to do something. You have to actually specify how you are going to try, and then you have to ask what QM tells you about what will happen when you do that.

 

[Lynch101]

In order to narrow down the location further, you would have to use a device that is smaller. And at some point you will run into the fact that, according to quantum field theory, to confine a quantum object into a smaller space requires more energy, and at some point you have pumped enough energy into the system to create more particles, so you're no longer measuring just the original system.

This illustrates the more general point that you can't just wave your hands and say "we can try" to do something. You have to actually specify how you are going to try, and then you have to ask what QM tells you about what will happen when you do that.

But there we've gone from not being able to say anything about the location/position of the system prior to measurement to narrowing it down to a very specific, finite region of space. We don't need to physically narrow down the location any further to draw the conclusions from propositions 1-4 above.

But we could narrow it down as much as possible, to that limit you've specified, and conclude that the system is located in that finite region of space. Here, again, we are saying that the system has location/position prior to measurement. Again ask if it is located everywhere in that smaller, finite region of space. Again, we will arrive at the same conclusions.

If the description does not specify that the system is:
A) everywhere in the finite region of space
B) not-everywhere in the finite region of space

then it does not give a complete description of the system. If it specifies B above, but does not then specify where in the region of space it is (or where it is not), it cannot be said to give a complete description of the system.

 

[PeterDonis]

we could narrow it down as much as possible, to that limit you've specified, and conclude that the system is located in that finite region of space at the instant the measurement is made.

See my qualifier in bold above. It makes a big difference.

 

[PeterDonis]

If the description does not specify that the system is:
A) everywhere in the finite region of space
B) not-everywhere in the finite region of space

then it does not give a complete description of the system.

This is not physics. It's your personal opinion. That will remain true no matter how many times you repeat it.

 

[Lynch101]

See my qualifier in bold above. It makes a big difference.

Vanhees said that, when we prepare the system, it passes through a 'little hole' and it passes through a 'nicely tailored inhomogeneous magnetic field'. If it passes through these finite regions of space then we can conclude that, at some time during the experiment the system was located somewhere in those finite regions of space.

Having narrowed down the possible location(s)/position(s) of the system to some finite region of space we can then ask if the system was located everywhere in that finite region of space. From there, we can draw the conclusions outlined above.

This is not physics. It's your personal opinion. That will remain true no matter how many times you repeat it.

This is interpreting what the physics tells us, which you seem to categorise as 'not physics'. Which is fair enough. But, that doesn't invalidate the conclusions drawn.

 

[PeterDonis]

If it passes through these finite regions of space then we can conclude that, at some time during the experiment the system was located somewhere in those finite regions of space.

Not with the meaning you are giving to the term "located", no. What @vanhees71 means by "passes through" is that the wave function is localized to that finite region of space. But you are trying to use a definition of "located" that will say more than that. So you are going beyond what @vanhees71 is saying.

This is interpreting what the physics tells us, which you seem to categorise as 'not physics'.

That is an acceptable paraphrase of what I said, given what you are using "interpreting" to mean, yes.

that doesn't invalidate the conclusions drawn.

It's not my job to "invalidate" the claims you are making. It is your job to either make physics claims, or not have this discussion in a physics forum. Granted, this particular forum, the QM interpretations forum, has somewhat looser rules; but along with those looser rules you must accept the fact that many discussions in this forum cannot be resolved because they come down to one person's opinion or preference vs. another's. That's basically where we are with your preferred concept of "location".

 

[PeterDonis]

that doesn't invalidate the conclusions drawn.

Conclusions are not valid until proven otherwise. They are invalid until evidence otherwise is presented; and they can never be "proven" valid because it is always possible that more evidence could either falsify them or limit their domain of validity in a way not previously considered. That's how science works.

 

[StevieTNZ]

Given the quote I provided, I'd say the quantum system is everywhere in the universe before measurement. This is further emphasised by Stephen Hawking in his book 'The Grand Design'.

 

[hutchphd]

Vanhees said that, when we prepare the system, it passes through a 'little hole' and it passes through a 'nicely tailored inhomogeneous magnetic field'. If it passes through these finite regions of space then we can conclude that, at some time during the experiment the system was located somewhere in those finite regions of space

This is a bad way to look at it. You have simply enlarged your detector. Before you had just a particle detector. Now you have a particle detector with a magnet in front of it. That doesn't change the behavior of the electron. Same for the slit and the photon.

 

[vanhees71]

Yes, Rovelli tries to explain it. But if you already read the paper yourself, you will probably be disappointed by explanations like:

I guess what confuses you is that Heisenberg's tables of numbers or "matrices" seem to be just as unobservable as the position of an electron or its time of circulation. Yet the abstract says: "In der Arbeit soll versucht werden, Grundlagen zu gewinnen für eine quantentheoretische Mechanik, die ausschließlich auf Beziehungen zwischen prinzipiell beobachtbaren Größen basiert ist." Not sure whether an explanation like the following would help you:After trying to read the paper, my guess is that the logic is to give up describing the time dependence. Because that is what Heisenberg does: He looks at the Fourier expansion in time of the classical variable, replaces that infinite series of Fourier coefficients by an infinite matrix, and then explains how to replace the expressions for the Fourier series of $x^2$, $x^3$ and $f(x)$ for general analytic functions $f$ by corresponding expressions for his infinite matrix. Then he notices that the expression for $xy$ is not commutative and therefore more problematic, suggests that the classical expression $v\dot{v}$ should probably be replaced by $(v\dot{v}+\dot{v}v)/2$ because it is the derivative of $v^2/2$, but that it is less clear what to do with more general terms.

So it looks like Heisenberg is trying to find out how to keep the equations of motion, but reinterpret their "mathematical" meaning for his matrices (which he obtained by giving up attempts to descibe a time dependence). This guess matches well with the title "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen."

The problem is not to understand matrix mechanics, which is just one representation of QT, and it's well understandable reading the papers by Born&Jordan and Born&Jordan&Heisenberg written shortly after the Helgoland paper, where of course also the Heisenberg picture of time evolution was established, which indeed is the most natural description: the state (statistical operator) describes the initial preparation and the time-evolution of the operators representing observables determines the time evolution of the corresponding eigenvectors and then together with the statistical operator you get the probabilities/probability distributions as a function of time, containing all the physical probabilistic content of the quantum formalism.

The interesting point of the Helgoland paper is more to learn, how Heisenberg got to his profound new idea and the logic used to derive the matrix representation. It's obviously very heuristic and based on intuition. This was Heisenberg's strength. His weakness was to formulate these things in a deductive way. This was done by his collaborators, particularly Pauli and Born and Jordan.

 

[Lynch101]

Conclusions are not valid until proven otherwise. They are invalid until evidence otherwise is presented; and they can never be "proven" valid because it is always possible that more evidence could either falsify them or limit their domain of validity in a way not previously considered. That's how science works.

A conclusion is either valid or it is invalid i.e. it is either true or it is false*. On this basis we can explore the consequences of it being true and the consequences of it being false.

So, if it is true that the system passes through one 'nicely tailored inhomogeneous magnetic field' but does not pass through a separate magnetic field (which we have set-up either in the same room in the lab or another room) then we can narrow down the location of the system at some time to having been located in the finite region of space associated with the magnetic field through which it did pass.

This is either true or it is false.

If it is false, then we have to look at our premises.Evidence is the means by which we confirm the truth or falsity of a conclusion for us. Our finding evidence for a proposition does not make the proposition true, since it must have been true in the first place for us to be able to find evidence for it - unless this particular claim is false, in which case we can explore the consequences of that.*The conclusion 'X is valid in a limited domain' is a different conclusion to 'X is valid'.

 

[Lynch101]

Given the quote I provided, I'd say the quantum system is everywhere in the universe before measurement. This is further emphasised by Stephen Hawking in his book 'The Grand Design'.

OK, then if this is true we must have some form of spontaneous FTL collapse to explain how the system then localises to a single position.

There may, of course, be alternative explanations. Again, if there are, we can explore their consequences. It might be the case that the idea of the system as being a separate 'thing' from the universe is challenged.

 

[Lynch101]

This is a bad way to look at it. You have simply enlarged your detector. Before you had just a particle detector. Now you have a particle detector with a magnet in front of it. That doesn't change the behavior of the electron. Same for the slit and the photon.

It doesn't matter if we make the set-up more complicated, particularly if it reveals more information to us. If we do have such a set up then:

If it is true that the system passes through one 'nicely tailored inhomogeneous magnetic field' but does not pass through a separate magnetic field (which we have set-up either in the same room in the lab or another room) then we can narrow down the location of the system at some time to having been located in the finite region of space associated with the magnetic field through which it did pass.

 

[gentzen]

The interesting point of the Helgoland paper is more to learn, how Heisenberg got to his profound new idea and the logic used to derive the matrix representation. It's obviously very heuristic and based on intuition. This was Heisenberg's strength. His weakness was to formulate these things in a deductive way. This was done by his collaborators, particularly Pauli and Born and Jordan.

Born and Jordan definitively were true collaborators. Pauli is different, but not in the way you would expect. Pauli was a troubled soul, more than anyone can imagine who has never read more about the details of his life. Here is an example from Arthur I. Miller's book 137: Ralf Kronig first had the idea that each electron also has an angular momentum of its own. But Bohr and Pauli dismissed his proposal, and Kronig dropped the idea. Then, nine months later, two Dutch physicists, George Uhlenbeck and Samuel Goudsmith, rediscovered spin and stacked their claim in print, warning that one should not visualize the electron as a spinning top. Pauli was deeply embarrased at having discouraged Kronig from publishing his idea and thereafter always spoke highly of him. And similarly, Pauli had also dismissed Heisenberg's idea, but both Heisenberg and Born were "sufficiently experienced" with how to handle Pauli's reactions and how to even "help" him.

Anyway, let me have another try at providing more information about "how Heisenberg got to his profound new idea". The most relevant words from
3. The Development of Quantum Mechanics (1925 – 1927)
3. Die Entstehung der Quantenmechanik (1925 – 1927)
are probably:

Nevertheless, his latest calculations in Copenhagen on dispersion theory and on complex spectra, especially the principle of “sharpened” correspondence applied in these works, seemed to point toward a future satisfactory theory.

With characteristic optimism the Göttingen Privatdozent took on a new and difficult problem at the beginning of May 1925, the calculation of the line intensities in the hydrogen spectrum. Heisenberg began with a Fourier analysis of the classical hydrogen orbits, intending to translate them into a quantum theoretical scheme – just as he had done with Kramers for the dispersion of light by atoms. But the hydrogen problem proved much too difficult, and he replaced it with the simpler one of an anharmonic oscillator. With the help of a new multiplication rule for a quantum-theoretical Fourier series he succeeded in writing down a solution for the equations of motion for this system. On 7 June 1925 he went to the island of Helgoland to recover from a severe attack of hay fever. There he completed the calculation of the anharmonic oscillator, determining all the constants of the motion. He made use, in particular, of a modified quantum condition that was later called by Born, Pascual Jordan and himself a “commutation relation”, and he proved that the new theory yielded stationary states (conservation of energy).

(Of course, I can also imagine that your real question is something else, but that you don't want to state it explicitly, because on the one hand you could just work it out yourself if that question were really sufficiently important to you, and on the other hand you don't expect that anybody could really answer that question for you, at least not without you also investing quite some work into trying to understand that answer.)

 

[hutchphd]

If it is true that the system passes through one 'nicely tailored inhomogeneous magnetic field' but does not pass through a separate magnetic field (which we have set-up either in the same room in the lab or another room) then we can narrow down the location of the system at some time to having been located in the finite region of space associated with the magnetic field through which it did pass.

Sorry I don't even know what hypothetical experiment you are talking about now. It is equivalent to two slits I believe, so perhaps we could simplify to that and you can restate your thesis.
It is clear from Feynman's formulation of the path integral that a particle samples all paths the most likely being where the action is stationary. At some level of approximation we concentrate only on likely paths.
Any other formulation is similar and one always tacitly works with a truncated hamiltonian $H_{local}+H_{rest~of~ universe}$. Whether a door is ajar in the lab down the hall seldom makes a difference to the experimental outcome but there is a tiny finite probability to detect the particle there.

 

[Lynch101]

Not with the meaning you are giving to the term "located", no. What @vanhees71 means by "passes through" is that the wave function is localized to that finite region of space. But you are trying to use a definition of "located" that will say more than that. So you are going beyond what @vanhees71 is saying.

It sounds to me like you are confusing the 'map' with the 'territory' here. The finite region of space which the magnetic field occupies is not the same as the mathematical description of the magnetic field.

So, if by the wave function is localized to that finite region of space you mean the mathematical artefact that is the 'wave function' is localized in the mathematical description of the finite region of space that is one thing. We can examine how this corresponds to 'physical reality'.

The 'wave function' [as the mathematical description of the system] cannot be localized in a finite region of space [in physical reality] because it is a mathematical artefact. If you mean the physical system to which the wave function corresponds is localized in the finite region of space, then that is precisely the point I am making.

It's not my job to "invalidate" the claims you are making. It is your job to either make physics claims, or not have this discussion in a physics forum. Granted, this particular forum, the QM interpretations forum, has somewhat looser rules; but along with those looser rules you must accept the fact that many discussions in this forum cannot be resolved because they come down to one person's opinion or preference vs. another's. That's basically where we are with your preferred concept of "location".

It might very well be the case that these matters cannot be resolved by appeal to experiment and must remain in the domain of interpretation. That doesn't mean, however, that we cannot deduce or infer the necessary consequences of a given position or determine what implications they imply.

It's also not simply a matter of opinion, they are matters of reason based on premises and conclusions. A conclusion is either true or it is false. We can explore the consequences of it being true and the consequences of it being false.

If we deny the conclusion presented then we must be denying the premises from which it follows. You have attempted this above by questioning what we mean by 'passes through'. Doing this has implications for any given position. We can explore these implications by presenting reasoned arguments which follow from the physics or which interpret the physics.

 

[Lynch101]

Sorry I don't even know what hypothetical experiment you are talking about now. It is equivalent to two slits I believe, so perhaps we could simplify to that and you can restate your thesis.
It is clear from Feynman's formulation of the path integral that a particle samples all paths the most likely being where the action is stationary. At some level of approximation we concentrate only on likely paths.
Any other formulation is similar and one always tacitly works with a truncated hamiltonian $H_{local}+H_{rest~of~ universe}$. Whether a door is ajar in the lab down the hall seldom makes a difference to the experimental outcome but there is a tiny finite probability to detect the particle there.

I believe an experimental set-up like the following:

#3 is what creates the inhomogenous magnetic field.

If, the magnetic field occupies a finite region of space and the system 'passes through' this region of space, then we have narrowed down the possible location(s)/position(s) of the system.

Now imagine, in the same room we also have an identical set-up but instead of sending particles through the second apparatus we simply turn on the magnetic field. So, now we've got two magnetic fields occupying two separate finite regions of space. If we can say that the system 'passes through' one magnetic field but not the other, then we can narrow down the locations(s)/position(s) of the system [at some time during the experiment] to the finite region of space through which it did pass.

 

[vanhees71]

Born and Jordan definitively were true collaborators. Pauli is different, but not in the way you would expect. Pauli was a troubled soul, more than anyone can imagine who has never read more about the details of his life. Here is an example from Arthur I. Miller's book 137: Ralf Kronig first had the idea that each electron also has an angular momentum of its own. But Bohr and Pauli dismissed his proposal, and Kronig dropped the idea. Then, nine months later, two Dutch physicists, George Uhlenbeck and Samuel Goudsmith, rediscovered spin and stacked their claim in print, warning that one should not visualize the electron as a spinning top. Pauli was deeply embarrased at having discouraged Kronig from publishing his idea and thereafter always spoke highly of him. And similarly, Pauli had also dismissed Heisenberg's idea, but both Heisenberg and Born were "sufficiently experienced" with how to handle Pauli's reactions and how to even "help" him.

Anyway, let me have another try at providing more information about "how Heisenberg got to his profound new idea". The most relevant words from
3. The Development of Quantum Mechanics (1925 – 1927)
3. Die Entstehung der Quantenmechanik (1925 – 1927)
are probably:

(Of course, I can also imagine that your real question is something else, but that you don't want to state it explicitly, because on the one hand you could just work it out yourself if that question were really sufficiently important to you, and on the other hand you don't expect that anybody could really answer that question for you, at least not without you also investing quite some work into trying to understand that answer.)

Well, I don't think it's so important concerning physics, because we have the fully developed theory today. It's just interesting historically to get the idea, how the original ideas developed. This is not so easy with the Helgoland paper.

It's even more difficult for Newton's Principia or Maxwell's Treatise. The content we learn of course today in a much more digestible form, going back to Euler and Heaviside, respectively.

 

[hutchphd]

If, the magnetic field occupies a finite region of space and the system 'passes through' this region of space, then we have narrowed down the possible location(s)/position(s) of the system.

This is just a slightly complicated two slit experiment. What is the surprise here?
Clearly particles can be localized more or less I do not understand the point you are trying to make

 

[Lynch101]

This is just a slightly complicated two slit experiment. What is the surprise here?
Clearly particles can be localized more or less I do not understand the point you are trying to make

The general point is that those interpretations which only predict the outcomes of an ensemble of experiments do not give a complete description of the system because they do not describe, in any way, where the system is prior to measurement.