Assumptions of the Bell theorem

[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.

 

[PeterDonis]

To try and outline the reasoning a bit more clearly

You're not doing that in the rest of your post. You're just restating your personal opinion. You are still not giving any argument for why anyone else should care about your personal opinion or adopt the personal definition of "elements of reality" or "complete description of reality" that you have adopted.

 

[WernerQH]

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.

I only read the abstract and was puzzled. Theorists are unable even to define the trajectories, but smart experimentalists can observe them anyway. ;-)

 

[PeterDonis]

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.

You are shifting your ground. Before, you wanted the theory to tell you whether the system was capable of interacting with a given measurement device. Now, you want the theory to tell you which specific measurement device, out of multiple measurement devices that the system is capable of interacting with, the system will actually interact with. In other words, now you want to require the theory to be deterministic; just having the necessary "elements of reality" isn't enough.

 

[Lynch101]

You're not doing that in the rest of your post. You're just restating your personal opinion. You are still not giving any argument for why anyone else should care about your personal opinion or adopt the personal definition of "elements of reality" or "complete description of reality" that you have adopted.

It's not my personal definition.

From the EPR paper

Whatever the meaning assigned to the term complete, the following requirement for a complete theory appears to be a necessary one: every element of the physical reality must have a counterpart in the physical theory. We shall call this the condition of completeness.

Some interpretations of QM simply define themselves as incomplete according to this criterion. Namely instrumentalist/'anti-realist'/minimal statistical interpretations*.

Am I correct in saying that psi-epistemic is the umbrella term?

What I have described is anti-realism as it is used in literature: The claim that the properties in QM do not refer to real properties of the system.

When EPR talk about a complete description of physical reality, they are talking about describing every element of the system under consideration. What they refer to as 'elements of [physical] reality'.

As I have been saying, any interpretation that only gives us the probability of measurement outcomes e.g. the aforementioned interpretations is, by the EPR definitions, an incomplete description of physical reality.

We can further establish this by asking the simple question, do those interpretations describe the system prior to measurement. IF they only predict the outcome of interactions with measurement devices then, again by definition, they do not describe the system prior to measurement.

The question then becomes, is there a system prior to measurement? If the answer is yes, then we ask do the above interpretations fully describe it? The answer to that is no, because they don't describe it prior to measurement. Therefore, they are incomplete according to the criterion set out by EPR.I think, perhaps, the baby is getting thrown out with the bathwater. Violations of Bell inequalities demonstrate that quantum systems do not have a single, pre-defined value for location prior to measurement. This, however, does not mean that the system has no location whatsoever.

I am contending that location is 'an element of reality' of the system. Not that it has a single, pre-defined value, simply that location is a property of the system that must be described - whatever form that takes. My contention is that any interpretation that fails to describe the location of the system prior to measurement cannot be considered complete, according to the EPR criterion.

It has been suggested that the probability distribution does describe the location of the system prior to measurement. We can explore what exactly this description does and does not tell us and make certain inferences about the 'complete description of physical reality'.

The alternative is to claim that there is no location to be described whatsoever i.e. it is not located in the universe. This too has consequences which we can explore.

 

[EPR]

It has been suggested that the probability distribution does describe the location of the system prior to measurement. We can explore what exactly this description does and does not tell us and make certain inferences about the 'complete description of physical reality'.

The alternative is to claim that there is no location to be described whatsoever i.e. it is not located in the universe. This too has consequences which we can explore.

In the MWI, which is a very sensible interpretation btw, every particle has a location before being measured. The MWI is pure QM as it is.
QM without the need for an interpretation and with no human baggage.

 

[PeterDonis]

It's not my personal definition.

From the EPR paper

This paper gives a general definition of "completeness" that involves elements of reality, but it does not claim that "location" is an element of reality. Only you are claiming that.

 

[Lynch101]

This paper gives a general definition of "completeness" that involves elements of reality, but it does not claim that "location" is an element of reality. Only you are claiming that.

And those interpretations that 'claim the properties in QM do not refer to real properties of the system' i.e. instrumentalist/'anti-realist'/minimal statistical are, according to that general definition, incomplete. Would you agree with that?

 

[facenian]

This paper gives a general definition of "completeness" that involves elements of reality, but it does not claim that "location" is an element of reality. Only you are claiming that.

Pardon me for stepping in. I think that according to the paper "position" is an element of physical reality(at least in special cases). However, I also think that it is an unnecessary metaphysical concept that Einstein himself immediately rejected. As it is well known, he preferred to argue against completeness using his separation principle. In my opinion, the infamous metaphysical concept of "elements of physical reality" has produced much confusion in the literature, and experts in quantum foundations do not even mention it. They only use Einstein's "separation principle".

 

[PeterDonis]

I think that according to the paper "position" is an element of physical reality(at least in special cases).

There was one particular thought experiment proposed (not, IIRC, in the paper, but in the general discussion around that time that the paper was part of--I think this particular proposal was during one of the Solvay conferences) in which it would have been, yes: basically, you prepare an entangled pair of particles in an eigenstate of total momentum and separation (i.e., difference in position), such that measuring the position of one enables you to know the position of the other without measuring it. However, IIRC it was never clear that the states assumed in this thought experiment were actually valid states (because of possible lack of normalizability, which would make them not physically realizable).

In any case, the qualifier you give, "at least in special cases", is already sufficient, IMO, to cast doubt on the whole concept of "elements of reality" (which you appear to be dubious about as well). Intuitively, one would not expect what things count as "elements of reality" to depend on what states we prepare or what particular experiment we are doing.

 

[PeterDonis]

And those interpretations that 'claim the properties in QM do not refer to real properties of the system' i.e. instrumentalist/'anti-realist'/minimal statistical are, according to that general definition, incomplete. Would you agree with that?

With that very weak claim, yes, I agree. But you have been making much stronger claims than just that.

 

[Demystifier]

I thought a photon does not have a position operator.

It does, but it does not have a Lorentz covariant position operator. A photon detector detects a photon at a certain position, so photon position certainly has an operational meaning.

How do you compute the trajectory of a photon in Bohmian mechanics?

In this case, by postulating a preferred Lorentz frame with respect to which the position operator is defined. As you might know, a preferred Lorentz frame is needed in Bohmian mechanics for other reasons as well.

 

[Demystifier]

Of course, because the Born rule is about probabilities for the outcome of measurements.

Please answer with yes or no to the following questions:
1. Can measurement be defined in a closed (not open) system?
2. If not, does it mean that the Born rule only makes sense for open systems?
3. If so, does it mean that the Universe as a whole is deterministic?
4. Or would you prefer to say that the whole Universe cannot be measured, so it's a meaningless concept?

I think that your answers are this:
1. no
2. yes
3. no
4. yes
but please confirm or correct!

 

[Lynch101]

With that very weak claim, yes, I agree. But you have been making much stronger claims than just that.

I'm not sure how you view that as the weaker claim since it is effectively establishes the overarching claim of EPR, that the QM description of physical reality is incomplete. Those interpretations essentially define themselves as incomplete descriptions of physical reality.

Weaker Claim
The subsequent claim I am making is much weaker than you might think. It simply amounts to the question, 'where is the system prior to measurement?'. Usually, answers to 'where' questions will take the form of a 'location'. We might, however, prefer to talk about 'position' instead.

So, I'm saying that the system must be 'somewhere' prior to measurement - a very weak claim. Therefore, to provide us with a complete description of the system we need a description of where it is 'located' prior to measurement. Or, if we prefer, we need a description of its position prior to measurement.

This is where the baby appears to get thrown out with the bathwater, since the description of the position of the system [prior to measurement] does not require a single, pre-defined value. There are myriad possible descriptions. To give two (of possibly very many) examples:
- it might have multiple pre-defined values for position
- it might be spread out across a broader spatial region such that a single pre-defined value doesn't fully describe its 'location' or 'position'.

Whatever form the description of the system's location/position takes is immaterial. What is required for the purpose of completeness, however, is some description of location/position prior to measurement; any description. By their own definition, interpretations which only give us probabilistic predictions about measurement outcomes do not give us any description of the systems location/position prior to measurement whatsoever. Therefore, by definition, those cannot be said to be complete descriptions of the system or, in the words of EPR, physical reality.

Alternatives
Of course, we are free to reject this reasoning. However, doing so leaves us with a very clear alternative. That alternative* is simply that the system is 'nowhere' prior to measurement. If this is the position we wish to adopt, then we can explore the implications of it.What might appear as the middle-ground, could be the position that the probability distribution does tells us something about the location/position of the system prior to measurement. However, it is less of a middle ground since it either tells us that the system is 'somewhere' prior to measurement or that it is 'nowhere'.

What we can do is probe what the probability distribution tells us about the location/position of the system prior to measurement. This might not give us the complete description of reality we are looking for, but it can certainly narrow the domain of possible descriptions. It might point, inexorably, to certain unavoidable features.*A further alternative might be that notions such as 'somewhere' and 'nowhere' do not apply at the quantum level. This too would have potential consequences for observations at the classical level. We could probe whether or not this idea could lead to observational results which appear to be superluminal at the classical level.

 

[Lynch101]

In the MWI, which is a very sensible interpretation btw, every particle has a location before being measured. The MWI is pure QM as it is.
QM without the need for an interpretation and with no human baggage.

The MWI, as far as I can tell, would fall under the category of 'potentially complete descriptions of physical reality' - as long as it describes all possible worlds. It would, at least, fall under the category of 'potentially complete descriptions of this world'.

The point I am making is that those 'interpretations' which remain agnostic would not, by their own definition, fall under the category 'potentially complete descriptions of physical reality'. They rule themselves out by declaring that the mathematical formalism does not correspond to 'physical reality'.

 

[Lynch101]

It does, but it does not have a Lorentz covariant position operator. A photon detector detects a photon at a certain position, so photon position certainly has an operational meaning.In this case, by postulating a preferred Lorentz frame with respect to which the position operator is defined. As you might know, a preferred Lorentz frame is needed in Bohmian mechanics for other reasons as well.

Is a preferred Lorentz frame necessary, or is it more that absolute simultaneity is required?