A Assumptions of the Bell theorem

[Fra]

While it might be a classical bias to expect a definite value for position, prior to measurement, the system in question must be somewhere as it cannot be nowhere. If the mathematics doesn't describe this 'somewhere' it cannot give a complete description of physical reality.

Some interpretations ascribe an ontology to the wave function which attempts to describe this 'somewhere', which make them [an attempt at] a complete description of physical reality. Any interpretation that only gives probabilistic predictions for the outcomes of measurements cannot be said to describe the location of the system prior to measurement. Since the system must be somewhere - even if that doesn't imply a definite location - such an interpretation/theory cannot be said to be a complete description of physical reality.

This is of course interpretation dependent, but there is also the possibility that the nature of inference - what an observer can state about the system - is limted in a much deeper way than thinking that it's an "incomplete description" in the sense of the observers ignorance. The apparent incompleteness may come from the nature of asking many non-commutative questions, and trying to bring the answers together into one "state space". And it may very well be that the physical interaction between two subsystems are determined, merely by their mutual expectations.

This is why I think that the best way to understand QM interactions is not in terms of physical objects bouncing around, but as two "expectations" interacting via information exchange. That seems the natural way to make sense of things, at least for me. Though I know some will frown upon introducin the notion of "information" as a new ontology.

/Fredrik

 

[Lynch101]

No, the burden of proof lies on you to support your positive claim that there are no other possibilities, since your argument rests on that claim. Just saying that you can't think of any other possibilities is not enough.

I am not making a positive claim so there is nothing for me to have to prove. I am simply pointing out that you have not given sufficient support for your claim.

I am not saying that no other possibilities exist, I am saying the emboldened part.

If other possibilities exist, then the burden of proof is on anyone (not necessarily you) to demonstrate these possibilities. It is possible to reject the possibilities I have suggested and not proffer any alternatives but to not do so leaves us with an incomplete description of physical reality.

If "location" means "the spatial region in which the system is capable of interacting with a measurement device", then yes, as I have already explained.

OK, so then you are not contending that the wave function only gives probabilistic predictions for measurement outcomes, but also describes the location of the system prior to measurement.

Does the wave function tell us the probability of measuring the system in "the spatial region in which the system is capable of interacting with a measurement device"?

No, the contention by others in this thread has been that the wave function does not describe the system being in a definite location, by which they meant "a single point at which the system is known to be located".

Vanhees has taken the aforementioned position

My conclusion from this simply is that we just have to take the quantum state as what it tells us: The probabilities for the outcomes of measurements.

and you have stated as much yourself (emphasis added by me)

The wave function always tells you the probabilities of measurement outcomes; but depending on which interpretation of QM you adopt, it can also be a description of the system prior to interacting with the measurement device.

From that we can have the position that wave function tells you the probabilities of measurement outcomes or we can have the position that wave function tells you the probabilities of measurement outcomes and be a description of the system prior to interacting with the measurement device.

It is the former that is the incomplete description of physical reality.

You attempted to gerrymander the definition by replacing "point" with "finite region", but that still doesn't work, because you are conflating two different claims. The claim that the system can interact with a measuring device in any spatial region in which its wave function is nonzero--which is the only claim that your arguments about measurements being "elements of reality" actually justify--is a much weaker claim than the claim you are trying to argue about, which is the claim that the system must be "definitely located" in the spatial region occupied by the measurement device. That claim, at least according to QM, is false: QM in no way requires that a system's wave function be nonzero only in some particular finite spatial region, presumably the one occupied by the measurement device (which is what "definitely located" means), in order to interact with that device.

I'm simply stating that there are two options, as per your statement above:
1) The wave function tells us the probabilities of measurement outcomes only
2) The wave function tells us the probabilities of measurement outcomes and be a description of the system prior to interacting with the measurement device.

I am saying #1 cannot be a complete description of reality because it does not describe the system prior to measurment.

If we adopt position #2, we can ask what the wave function tells us about the location of the system prior to measurement.

This is where we have a range of possible explanations. I have listed some and, in general, this is where 'interpretations' of QM come into play. I am by no means saying that the options I listed is an exhaustive list, it is possible that there are more. At this juncture there are [at least] 3 options:

A) Suggest that the list is exhaustive
B) Suggest and addition to the list
C) Reject all options on the list and do not proffer any additional options.

To adopt position #C effectively puts us back to position #1 above, which leaves us with an incomplete description of reality.

Where? Who has contended this?

Vanhees has been attempting to reconcile positions #1 & #2 (emphasis added by me)

As I repeatedly said, the description of the system prior to measurement is given by the quantum state (statistical operator) at the initial time (after the "preparation" is finished). In classical mechanics it's given by the point in phase space at the initial time.

No, we do not know this. We know that we have to apply the collapse postulate in order to make accurate predictions about future measurements, but what, if anything, that says about the "actual" location of the system depends on which QM interpretation you adopt.

In summary: you are reasoning from premises that no one else but you accepts, and you have given no reason why anyone else should accept them. That is why no one else is accepting your arguments.

I'm not talking about future predictions, I'm talking about observations. When we measure the system we get a definite value for its location. The probability of the system being (or having been) where it registered on the measurement device is 1.

 

[PeterDonis]

It is this position which cannot be considered a complete description of physical reality because the system it does not describe the system prior to measurement.

To support this argument, you need to show that "the system prior to measurement" is an element of physical reality. You have not done so.

 

[PeterDonis]

If other possibilities exist, then the burden of proof is on anyone (not necessarily you) to demonstrate these possibilities.

Only if they are making a positive claim that depends on the existence of those other possibilities.

If you are making a positive claim that depends on the possibilities you listed being the only ones, then you need to demonstrate that that must be the case.

 

[Lynch101]

To support this argument, you need to show that "the system prior to measurement" is an element of physical reality. You have not done so.

The alternative is that it is not 'an element of reality'. If it is not an element of reality, then it could not possibly interact with the measurement device. Unless we contend that things which are not part of the universe can interact with things that are.

 

[PeterDonis]

then you are not contending that the wave function only gives probabilistic predictions for measurement outcomes, but also describes the location of the system prior to measurement.

I am saying that's a possible interpretation. That claim seems unproblematic since there are existing interpretations that say that.

Does the wave function tell us the probability of measuring the system in "the spatial region in which the system is capable of interacting with a measurement device"?

Since the wave function in the position representation tells you the probability of measuring the system in any spatial region, that would include the spatial region you describe.

 

[PeterDonis]

The alternative is that it is not 'an element of reality'. If it is not an element of reality, then it could not possibly interact with the measurement device.

Why? You are using a particular (implicit) definition of "element of reality", but you have given no argument for why I should care about this definition. You certainly have not argued, except by definition, that something must be an "element of reality" in order to interact with a measurement device. But if that is only true by definition, then I have already explained how QM meets this requirement (because it tells you whether or not the wave function is nonzero in the spatial region occupied by the measurement device)--in other words, on this interpretation, it is an element of reality, because it meets the definition (since it tells you whether or not the system can interact with a measurement device).

 

[Demystifier]

Then no physics is realistic or complete, because all of physics describes or predicts observations and thus, in quantified form, the outcome of measurements. How do you want to describe anything if not by referring to the phenomena we can observe about it?

Classical mechanics can consistently be formulated without referring to measurements. There are also such formulations of quantum mechanics (Bohmian mechanics is an example), but standard formulation of QM is not such a formulation.

 

[Demystifier]

If I never look, I can't check any predicted probabilities.

Sure, but if you take a look at a classical/analytical mechanics textbook, you will not see mentioning of measurements in the formulation of the theory. In quantum mechanics it's different. Why is that?

 

[Demystifier]

Can you specify more clearly, what you mean by the Born rule were not local? I have no clue, how the Born rule may be considered as local or non-local at all.

Read again what I said in the bracket! The Born rule by itself is neither local nor nonlocal. But when it is combined with other axioms, its consequences can be local or nonlocal, depending on definition of locality.

 

[vanhees71]

Classical mechanics can consistently be formulated without referring to measurements. There are also such formulations of quantum mechanics (Bohmian mechanics is an example), but standard formulation of QM is not such a formulation.

Classical mechanics as a physical theory first of all bases on a proper definition of (inertial) reference frames, i.e., it's based on how to quantify events in space and time. For that you use some geometry as a mathematical language. To make it a physical theory you have to define how to measure distances and times.

 

[vanhees71]

Read again what I said in the bracket! The Born rule by itself is neither local nor nonlocal. But when it is combined with other axioms, its consequences can be local or nonlocal, depending on definition of locality.

That's the problem. The word "locality" is as burnt as "realism". Nobody knows what you are talking about if you don't specify what you mean by "local". For me "locality" is synonymous with "microcausality" in relativistic QFT. In Newtonian physics there's no necessity for any kind of locality. Actions at a distance are kind of a paradigm there.

 

[Demystifier]

To make it a physical theory you have to define how to measure distances and times.

Maybe, but QM cannot even be formulated as a mathematical theory without a notion of measurement. For instance, Takhtajan in the book "Quantum Mechanics for Mathematicians" writes 4 axioms of QM, the last of which is this:

 

[vanhees71]

Yes, sure. So what? It's just the standard QT written in a more formal mathematical way.

 

[Demystifier]

So what?

So measurement is more fundamental in standard quantum mechanics than in standard classical mechanics. In particular, the Born rule in standard QM cannot even be stated without a notion of measurement.

 

[vanhees71]

I don't know any physics that can be stated without measurement. Physics is about quantified observations of nature. To quantify observations you need to operationally assign (usually real) numbers to phenomena, and this is done using appropriate measurement devices. Why is this different for you in classical physics in comparison to quantum physics?

 

[Demystifier]

I don't know any physics that can be stated without measurement. Physics is about quantified observations of nature. To quantify observations you need to operationally assign (usually real) numbers to phenomena, and this is done using appropriate measurement devices. Why is this different for you in classical physics in comparison to quantum physics?

Then choose some book on classical mechanics for mathematicians and find a quote where measurement is mentioned explicitly!

 

[Demystifier]

I don't know any physics that can be stated without measurement.

I know a physicist who claims that Noether theorem implies conservation of energy without measurement. If you wonder who this physicist is, look at the mirror.

 

[Lynch101]

Only if they are making a positive claim that depends on the existence of those other possibilities.

If you are making a positive claim that depends on the possibilities you listed being the only ones, then you need to demonstrate that that must be the case.

I am not making a claim that depends on the possibilities I listed being the only ones. I am saying it is not an exhaustive list and that there could be more. However, rejection of all those options and failing to provide an alternative explanation leaves us with an incomplete description of physical reality.

To try and outline the reasoning a bit more clearly, because there are different tracks the debate can go down, and it seems as though we are jumping between them.

1) Giving only the probability of measurement outcomes i.e. interaction of the system with the measurement device does not describe the system prior to measurement. This is a simple matter of definition. To do this leaves us with an incomplete description of physical reality.

From here, we have the reasonable request to justify this claim. The justification for this is that the system is part of the universe prior to measurement and so, it requires a description. Your argument here appears to be that the system doesn't have a single, pre-defined value prior to measurement i.e. a single, pre-defined value for location is not an element of reality. I think you might, inherently, be assuming that this is what I am claiming, but I am not.

While the system might not have a definite pre-defined value for location, it does have location. This location requires a description for the purpose of completeness.

The alternative is that the system has no location. However, this would mean that the system is not part of the universe and, therefore, could not interact with the measurement device in the first place. Again, this location does not have to be a single pre-defined value, but it does require a description. An interpretation which only gives the probability of measuring a single, well defined value upon measurement necessarily lacks this description.

2) Alternatively, we might say that the probability distribution does tell us something about the location of the system prior to measurement. In doing this, we are dropping the above claim that the mathematics only predicts the interaction with the measurement device.

That still doesn't give us a complete description of physical reality, however, because we need to investigate what the probability distribution tells us about the location of the system prior to measurement. We can probe this by asking questions and by applying 'the rules of the game' that we have already established.

Does the probability distribution tell us that the system has a single, pre-defined value for position but we are missing some information about the system, which means we can only predict with probability where we will measure it?

If the answer to this is no, then what does the probability distribution tell us about the location of the system prior to measurement?

An answer that has been proposed to this is that it tells us that the system does not have a single, pre-defined value for location.

OK, well what does it mean for a system to not have a single, pre-defined value for location?

Does it mean:
a) the system has more than one pre-defined value for location?
b) the system pops in and out of existence?
c) the particle is being guided by a pilot wave?
d) [insert another explanation/description]

The above list is not exhaustive, there could be many more. However, to reject all of the above and not propose an alternative leaves us with an incomplete description of physical reality.We can probe the question further to see what shape an explanation might take. I'll do this in response to your point below.

I am saying that's a possible interpretation. That claim seems unproblematic since there are existing interpretations that say that.

Indeed, and such interpretations are potentially complete descriptions of physical reality. The minimal statistical interpretation is not if only gives predictions for measurement outcomes.

If it tells us something about the location of the system prior to measurement, then we can investigate what it tells us.

Since the wave function in the position representation tells you the probability of measuring the system in any spatial region, that would include the spatial region you describe.

So, what does this tell us about the location of the system prior to measurement? If I put detectors in multiple spatial regions will all of the detectors register an interaction? If not, why not? How does the system 'choose' to interact with only one measurement device at a time?

Why? You are using a particular (implicit) definition of "element of reality", but you have given no argument for why I should care about this definition. You certainly have not argued, except by definition, that something must be an "element of reality" in order to interact with a measurement device.

The definition of 'element of reality' I have used is 'in or part of the universe'. This is just to clarify the statements I am making.

I presume that you agree that there is 'a universe'. In my reading of EPR they are calling for a complete description of physical reality i.e. a complete description of the elements of reality i.e. a complete description of the [parts of] the universe.

According to this definition, if something is not 'an element of reality' then it is not a part of the universe. Something which is not part of the universe cannot interact with things that are part of the universe. I think that is fairly uncontroversial. I am open to correction, of course.

But if that is only true by definition, then I have already explained how QM meets this requirement (because it tells you whether or not the wave function is nonzero in the spatial region occupied by the measurement device)--in other words, on this interpretation, it is an element of reality, because it meets the definition (since it tells you whether or not the system can interact with a measurement device).

Having a non-zero value for the wave function in a given spatial region is not sufficient for interaction with a measurement device, since we can put measurement devices in all of those regions with a non-zero probability and not observe interactions with all of the measurement devices.

What we can do, however, is probe what it means for the wave function to be nonzero in the spatial region occupied by the measurement device. As we said, we can put measurement devices in all regions with a non-zero probability yet not observe an interaction with the measurement device. Why is that? Is it because the system wasn't actually in the given region or did it spontaneously collapse into a single, well-defined value? This might not be the only option, but failure to describe what happens leaves us with an incomplete description of physical reality.We can ask further questions like, can a system interact with a spatially separated measurement device? If not, then this forces restrictions on us with regard to the possible location of the system.

 

[vanhees71]

Well, yes. Mathematicians don't care about the physical interpretation of theories. Their task is to make the physical theories mathematically consistent and thus don't need to bother about the physical meaning. They can also discuss QT without any reference to the physics it should describe. Then you have some purely mathematical theory about probabilities without reference to the physical observations. The same holds for classical mechanics or field theory. You just have a theory about certain types of ordinary and partial differential equations.

What Noether's theorem has to do with all this I don't know. It's of course also a mathematical theorem for a given theory.

 

[Lynch101]

I don't know any physics that can be stated without measurement. Physics is about quantified observations of nature. To quantify observations you need to operationally assign (usually real) numbers to phenomena, and this is done using appropriate measurement devices. Why is this different for you in classical physics in comparison to quantum physics?

There might be a slight disconnect here again between the two questions.

The question at hand is the completeness of the description of physical reality. Some of the the theorems of QM seem to point to the limitations of human inquiry in that regard. It might simply be the case that measurement alone cannot give us a complete description of physical reality.

In Bohmian Mechanics, the pilot wave is posited to exist regardless of whether it is measured. This is potentially a complete description of physical reality.

 

[vanhees71]

Well, whether or not Bohmian trajectories are "real" or not, is not clear to me either. For me again it all hinges on the question, whether Bohmian trajectories are observable or not. I'm not aware of any measurement of such a Bohmian trajectory for, e.g., a particle in a double-slit experiment.

 

[Demystifier]

Mathematicians don't care about the physical interpretation of theories. Their task is to make the physical theories mathematically consistent and thus don't need to bother about the physical meaning. They can also discuss QT without any reference to the physics it should describe. Then you have some purely mathematical theory about probabilities without reference to the physical observations.

And yet even a mathematician is not able to state the Born rule without measurement, as the post #703 demonstrates.

 

[Lynch101]

My apologies, I missed this in all the back and forth. I saw PeterDonis had referred to it, but I got caught up in responding to him.

I think the tautology that a "particle is located somewhere in the universe" is the very weak assumption that, given that there is a particle of a certain kind and that it has a position observable, then
$$\int_{\mathbb{R}^3} \mathrm{d}^3 x \rho(t,\vec{x},\vec{x})=1.$$
This is indeed already in the very foundations of quantum theory, because it merely says that a quantum state is described by a statistical operator (self-adjoint positive semidefinite operator of trace 1).

It is a tautology, indeed, or at least it should be. If it the particle is located somewhere in the universe, prior to measurement, then that location needs a description for the description to be considered complete. An interpretation which says that the probability distribution only gives predictions for the outcomes of experiments, by definition, lacks that part of the description.

If we say that the probability distribution does tells us something about the location prior to measurement, then the above claim is dropped and we can explore what the probability distribution tells us about location prior to measurment.

Are there regions of space/the universe with a zero probability value for finding the particle/system?

 

[Lynch101]

Well, whether or not Bohmian trajectories are "real" or not, is not clear to me either. For me again it all hinges on the question, whether Bohmian trajectories are observable or not. I'm not aware of any measurement of such a Bohmian trajectory for, e.g., a particle in a double-slit experiment.

But the Bohmian interpretation posits that they are real, doesn't it?

 

[Demystifier]

What Noether's theorem has to do with all this I don't know.

It's your own counterexample to your own general statement (that physics cannot be formulated without measurement).

 

[Demystifier]

I'm not aware of any measurement of such a Bohmian trajectory for, e.g., a particle in a double-slit experiment.

A quote from
https://www.researchgate.net/publication/51187205_Observing_the_Average_Trajectories_of_Single_Photons_in_a_Two-Slit_Interferometer
"In the case of single-particle quantum mechanics, the trajectories measured in this fashion reproduce those predicted in the Bohm–de Broglie interpretation of quantum mechanics (9,10)"

 

[WernerQH]

A quote from
https://www.researchgate.net/publication/51187205_Observing_the_Average_Trajectories_of_Single_Photons_in_a_Two-Slit_Interferometer
"In the case of single-particle quantum mechanics, the trajectories measured in this fashion reproduce those predicted in the Bohm–de Broglie interpretation of quantum mechanics (9,10)"

I thought a photon does not have a position operator. How do you compute the trajectory of a photon in Bohmian mechanics?

 

[vanhees71]

And yet even a mathematician is not able to state the Born rule without measurement, as the post #703 demonstrates

Of course, because the Born rule is about probabilities for the outcome of measurements. If you want to do physics within a theory you have to say what the mathematical symbols operationally mean. That's not different in classical physics too. To be able to define what the basic equation $\vec{F}=m \vec{a}$ means you have to operationally define what their symbols mean. In theoretical-physics books that's usually done in a view lines on the first pages, when discussing Newton's postulates. It's not very surprising, because the meaning of the mathematical objects is qualitatively pretty well known from everyday experience. It's no surprise that this is not so in realms, where we don't have too much experience like when dealing with single electrons, atoms, or even most puzzling photons ;-).

 

[vanhees71]

I thought a photon does not have a position operator. How do you compute the trajectory of a photon in Bohmian mechanics?

AFAIK there is no satisfactory Bohmian interpretation of relativistic QFT, and indeed photons don't have a position observable to begin with nor a consistent first-quantization formulation and thus also no wave function in the literal sense. At least QED is conceptually simpler than Dirac's "hole-theoretical formulation".

What's measured in the quoted article are approximate photon momenta. I've to read the details to understand, how they can claim that what they calculate out of these measurements were Bohmian trajectories of photons though photons don't even have a position observable to begin with.