Contextuality originates with Bohr, who was the first to recognize the important contribution of the measurement arrangement in obtaining knowledge on the microscopic world. Bohr used contextuality to defend his idea of completeness against Einstein's contention that quantum mechanics is an incomplete theory. Bohr was right, but only partially so. He underestimated the influence the interaction between microscopic object and the measuring instrument has in establishing a quantum mechanical measurement result. Where contextuality means that there is an influence exerted by the measuring instrument on our knowledge about the object, even more important is the influence exerted by the object on the measuring instrument. This latter influence is nowadays treated in many textbooks employing a quantum mechanical measurement theory. However, Bohr's contextuality still plays a confusing role in EPR experiments even today.

I think the best way to get a feeling what it means to live with contextuality is to learn de Broglie-Bohm theory.  It is a realistic, causal interpretation, and avoids all this Copenhagen subdivision into classical and quantum parts and the measurement problem. And it is, of course, contextual.  In other words, living with contextuality is not worse than to live in a dBB world. 

I disagree with  Ilja Schmelzer: Hardy's thought experiment shows that the de Broglie-Bohm interpretation (it's an interpretation, not a theory) of QM, is at odds with the special relativity.

I am explaining this contradiction below, but first of all about the underlying structure of the micro-universe: if we believe that before measurement a quantum particle travels on a well-defined trajectories, as the Bohm interpretation says, e.g. after a beam-splitter the quantum particle travels either on  the transmitted path, or on the reflected path, this experiment shows that we come to a contradiction.

To put it in short, Hardy uses a pair of particles, an electron e- and a positron e+. Each particle passes through a beam-splitter, respectively BS1+ and BS1-as in the attached figure. The branches u+ and u- intersect and produce gamma rays. Thus in a single experiment we either obtain a detection in the detectors D, or we obtain gamma rays. Both cases aren't possible.

In a single experiment in which no gamma rays were obtained, the electron and positron remain entangled. At the beam-splitters BS2+ and BS2- the branches of the positron and of the electron are reorganized into new branches, c+, d+, and c-, d-. We are interested in detections in coincidence in the detectors D+ and D-.

The problem with the relativity appears when considering travelling frames. Imagine that the beam-splitter BS2+ is very far from BS2-. For an observer O+ that travels from BS2- to BS2+, the relativity implies that the detection in D+ occurs before the electron reaches BS2-, and the wave-function that this observer obtains shows that by the time of the detection of the positron in D+, the electron should be on the branch u-. But for an observer O- that travels from BS2+ to BS2-, the relativity implies that the detection in D- occurs before the positron reaches BS2+, and the wave-function that this observer obtains shows that by the time of the detection of the electron in D-, the positron should be on the branch u+. However, if before reaching the beam-splitters BS2+ and BS2- the particles were on u+ respectively u-, we would have obtained gamma rays.

dBB is an interpretation if one restricts himself to quantum equilibrium, but it is a well-defined theory even outside equilibrium.  And, then, it of course presumes a preferred frame, and one does not even have to think about descriptions based on different preferred frames being consistent with each other. 

Ilja, good evening,

Can you tell me if the ResearchGate has such an utility as "chat"? I feel that this issue may take some exchange of comments. The fact that the de Broglie-Bohm interpretation (or theory, if you prefer, but Bohm called it interpretation) requires a preferred frame, is not good. As you know, the theory of relativity allows no preferred frame. What would you say if a theory would require the violation of the 2nd principle of the thermodynamics? I am currently investigating if we can say something in favor, or against the de Broglie-Bohm, even without frames. But is too soon to say something precise.

Happy New Year

Sofia, I don't know about a chat here. But the theory of relativity does allow for a preferred frame - it is a question of which interpretation you use.  The classical Lorentz interpretation has a preferred frame, the spacetime interpretation does not allow it.  

In fact, given the violation of Bell's inequalities, every realistic and causal interpretation needs a preferred frame. Here with "realistic" I mean the quite weak sense of realism used in Bell's theorem, and with "causal" I mean some theory where Reichenbach's principle of common cause holds. This is quite a lot for giving it up, and one obtains nothing in exchange.

Ilja, I was quite sure that this talk of us will take some exchange of letters. Now, what is that classical Lorentz interpretation? I am not aware of it. What I know is the special relativity. Do you doubt it? About Bell's inequalities, they are so controversed that it is for a long time that I don't work with them anymore. Hardy's so called paradox, is much simpler. And in principle it's not good to speak of different things at once. I prefer to focus on Hardy.

Classical world i.e, individual parts. But quantum world consider universe as a whole not individual parts. This is obvious in Bohmian or deterministic or causal quantum mechanics not in standard quantum mechanics. In this way, the force exerted on a particle is dependent on all particles and all factors. Bohr mentioned this in his complementary principle but Bohm showed it in term of a quantum potential. For study, these books are very useful:

The quantum theory of motion by Holland

Undivided universe by Hiley

No, Faramarz, Bohm's interpretation is not contextual. Fr whom wants to understand contextuality I disrecommend Holland and Hiley's books. These are good for studying Bohm's interpretation, but, as I say, Bohm's interpretation is not contextual. See my explanation of Hardy's experiment, accompanied by picture.

Hi, Sofia

Probably yo think that Bohmian quantum potential in one-particle case is local. But this is not true. Unfortunately, people think that non-locality exists only in many- particle case. As far as I know, Bohmian potential is context-dependence.

What you mean by contextuality ? 

May be i do not understand it? Teach me if I'm wrong.

Thank you

Hi, Faramarz

The truth is that the Bohmian mechanics doesn't have problems with single particles' non-locality. To the contrary, it provides very convenient explanations for single particle effects, even non-local. By the way, this iis not a trivial issue, and if someone would post a question I would gladly answer. But the Bohmian mechanics has problems with relativity, e.g. two particles experiments, if we examine the predictions made by travelling observers. Such predictions clash.

About contextual experiments, the definition is as follows:

Let A, B, and C, be three operators, such that A commutes with B and with C, but B and C don't commute. In this case, the result obtained in a measurement of A which is accompanied by a measurement of B, may differ from the result that we would obtain if the measurement of A were accompanied by a measurement of C.

Now, if you read Hardy's thought-experiment, you see that this is a contextual experiment. For an observer at rest, a joint detection in D+ and D-  can be described in operatorial terms, as a measurement of the operator D+ for the positron together with the operator D-. However, an observer moving toward the beam-splitter BS2+ sees that by the time the positron is detected in D+, the electron didn't reach BS2-. If we would place the detector for electron before BS2-, we should find the electron on the path u-, i.e. the measurement of the operator D+ would be together with the operator U-. But the operators D- and U- don't commute.

So, Hardy's though experiment is an example of contextual experiment: we can't assume that the result of measuring the operator D+ were the same if instead of measuring it together with D- we would measure it together with U-.

Happy New Year!

Thank you very much.

best

Sofia, sorry but I have to disagree heavily with your claim that dBB interpretation is not contextual.  It is, and in a quite unquestionable way.  This already follows from the standard equivalence results of dBB and QT and the impossibility results for non-contextual hidden variable theories.   

And, again, dBB interpretation assumes a preferred frame, so there will be no different accounts by differently moving relativistic observers, for every physical situation there will be only one description, those made in the preferred frame.  If a moving relativistic observer thinks, errorneously, that he is at rest, his account would be simply wrong, thus, contradictions with the correct description should be expected but is not a problem for the theory itself, because dBB theory does not claim any equivalence principle.  

Charles, of course in the Lorentz interpretation the rest frame of the ether is the preferred one, and this has nothing to do with any subjective choices, but that this frame is physically preferred. In the context of dBB, it is in this frame where the hidden FTL causal influences happen.  

The impossibility theorems prove that some special hidden variable theories (Einstein-causal realistic or non-contextual) are impossible.  Of course, they cannot say that "there can be no scientific definition" of a hidden variable theory,  dBB theory is a well-defined scientific theory and contains hidden variables. 

Then, dBB theory has a full measurement theory, which includes also momentum measurements, together with an equivalence theorem which also holds for momentum measurements. So, there is no such failure. 

Charles, you define what "physically exists in a scientific way" in a way which would be better named "physically exists in positivism".  Because in the standard (Popperian) scientific method what exist is defined by complete theories.  It is the complete theory which allows to make empirical prediction, and it is the complete theory which defines what really exist if this theory is true. And it has to be derived from the theory which of the really existing things are observable. 

Your paper has been already addressed, so if you have not made major modifications I see no reason to look at it again. Any claims of "demonstratively inconsistent" are simply false. And, given your continuing complete misunderstanding of elementary scientific methodology, there would be no reason to take your claims about dBB - to evaluate it requires at least some basic understanding of the scientific method - seriously. 

Unfortunately, most scientists do not consider physical reality and think that mathematical formalism is adequate or is equivalence with physics of phenomena.

Others, such as Bohm and Hiley are realistic and their quantum theory is more logical than standard quantum mechanics. In their view there is no a sharp distinction between classical and quantum world. In fact (quantum world=classical world+ non-local quantum potential). It is closer to wisdom than standard quantum mechanics. 

Charles, support your accusations against Popper with explicit quotes. Rereading your paper, I don't even know where to start. The description of Bayesian probability is horrible, I don't understand how such a description is possible after reading Jaynes. "truth value corresponds to the expected frequency of a particular result" facepalm. The very point which makes the greatest difference between Bayesianism and frequentism is that Bayesianism can be applied to the plausibility of theories. But theories are true or false, they are clearly not true with some frequency and wrong with another one.

Then, if you use a discrete and finite lattice, then you should know that the momentum states on this lattice space will be also discrete and finite, and not continuous. (Ok, here one can at least interpret the "continuous" to mean the whole finite-dimensional vector space. Anyway, a misleading formulation.)

The reasoning below p.7 is simply laughable. States, which are in dBB quantum equilibrium states, are of course not dispersion-free. And the set of dispersion-free hidden parameters you can take from dBB theory - the positions of the system itself and the configuration of the measurement device.

Simply think about this: The result of the "measurement" (better named "interaction") depends not only on the hidden variables of the system itself, but also on the hidden variables of the "measurement device". In such a situation one obviously cannot expect a deterministic outcome of the "measurement" based on the hidden variables of the system alone.

The very idea to claim "that Schrödinger’s equation is required by the general considerations of probability theory" is laughable, given that the classical equations of motion, say in Liouville form, which are not quantum, are also in agreement with any possible general considerations of probability theory.

The reasoning on p.9 is horrible. Bell's trivial example is simply a trivial example of a hidden variable theory, used to show some simple properties of such theories, in particular that they do not fulfill the condition required by von Neumann. And what are you doing? You apply your typical positivistic argumentation to this trivial theory "...it is not meaningful to describe a parameter whose only scientific definition is that a particular measurement result was obtained at given time and then to arbitrarily apply that definition at earlier times", then you seem to modify it in some unclear and mystical way, and then claim that the resulting nonsense fulfills your conditions.

In other words, you simply reject theories which do not fulfill von Neumann's assumptions because of your own positivistic prejudices against hidden variable theories. But this is, of course, not the point the theorem. If you want to reject hidden variable theories for positivistic reasons, fine, your choice, but don't distribute fairy tales that such theories are somehow impossible.

Jauch and Piron somehow define "a and b" and "a or b" for "propositions", and claim, without any base for this, that these have the usual meaning of the common sense "and" and "or". They are obviously not, because if the corresponding measurements are incompatible, there is no natural definition of a projection operator - thus, a "proposition" - which has the normal (common sense, logical) meaning. This normal language meaning of "a or b" is "if a is measured, then the result is certainly 1, or, if b is measured, the result is certainly 1". The set of vectors which fulfills this condition is not a vector space and cannot be obtained from a projection operator, thus, a "proposition" in the language of Jauch and Piron. So, here the manipulation of language is the fault of Jauch and Piron, not of Bell who tries to find a meaningful interpretation of their proof.

About dBB: "Specifically, for single particle states, (6.4) shows that particle mass and momentum depend on summing over all positions y, and not only on x." So what? Momentum is a property of the wave function, which is also part of the reality according to dBB theory. "if x is now to be interpreted as the position where a particle would be found in measurement, then f : y → f ( y ) = 〈 y | f 〉 cannot simultaneously be interpreted as giving the probability of finding the particle at y." Of course, and nobody does such nonsense. If we consider a pure state of dBB theory, then there is a certain position x where the particle would be located. But quantum theory is dBB theory not in pure states, but in quantum equilibrium. And in this equilibrium the wave function defines the probability distribution over y. "Add parameter x to the momentum state | p 〉 to obtain the Bohmian state | x , p 〉 according to (6.10) and act with the momentum operator P on the first parameter" means you don't know how momentum measurement is defined in dBB theory and simply write complete nonsensical formulas. Learn dBB theory before refuting it.

Sorry, too tired to look at the remaining text.

Ok, let's omit all personal accusations and answer what remains. Not much: "No one spoke of the frequency of theories being true." But Bayesian probability theory allows to assign probabilities to theories, thus, there is no connection between probability and frequencies in Bayesian probability which your quote suggests.

Your considerations in no way prove that the Schrödinger equation holds in classical mechanics.  If you want to transform the Liouville equation into some sort of Schödinger-type equation, the result would be funny and quite meaningless, without any physical meaning for the phase, and would not worth to be named Schrödinger equation. 

Justify your accusations against Bohm with explicit quotes.  

If the observation that Bayesian probability can be assigned to theories is irrelevant for your quote, as you claim, why don't you explain what the quoted phrase really means?  May be I have misunderstood it - who knows, but if you don't explain, but make simply implausible claims, ...

You have not even written down the "trivial" classical "Schrödinger equation", so if there is any justification  to name it a Schrödinger equation remains completely open. 

The situation with the remaining unanswered criticism is quite simple and clear: I have presented arguments, with explicit quotes from your text.  To meet these arguments, you would have to answer these arguments.  In an adequate way, with arguments, and sometimes quotes are at least useful.  If there are some other parts of your text, containing formulas, quotes and equations, is completely irrelevant.  The only question is if you are able to anwer my objections, up to now this seems not the case.

You have no idea about my mathematical qualifications, so stop to make claims a la that I "do not have any particular mathematical qualification". It is wrong, but not your business. Where you have learned math is also not interesting at all. 

Maybe the discussion between Ilja and Charles might be concluded by giving a definition of Bayesianist probability. It seems to me that Charles is applying an interpretation that is taylored to the Copenhagen interpretation of quantum mechanics, considering quantum probability to be an objective property of a microscopic object rather than a subjectivistic one in which the probability just describes relative frequencies in an ensemble. I remember having been frustrated a long time ago by not being able to make out which of the two interpretations corresponds to Baysianism.

Willem de Muynck

Please forgive me!  In science, as in all civilized dialogues and debates, we have one argument versus another.  Each side should submit its proofs and counterproofs as objectively as possible.  There is no need for accusations or insults.  That way we hope to reach the 'truth' asymptotically (often nonlinearly). 

I have a simple view of the matte at hand.  The Universe exists with or without us.  As soon as  we interact with it through measurement, contextuality comes into play.  QM is ultimately a theory of measurement; according to it, the act (or process) of measurement influences the measured objects.  Hence its contextuality.  On the other hand, Classical Theory  considers the act (or process) of measurement not to have any impact on the measured objects.  Hence its noncontextuality.  I hope this is simple; not simplistic or oversimplified.

Charles, let's start with Bohm: I say you have used a definition of momentum measurement which is not the one used in dBB theory.  This, if correct, would make your proof nonsensical. But to show this, you would have to verify that your "momentum measurement" is that proposed by Bohm.  I have asked you to support your accusation with quotes from Bohm.  No quotes from Bohm followed, thus, it is clear that your "momentum measurement" is your invention and has nothing to do with Bohmian theory. 

Then, of course, in a deterministic theory you can also consider a corresponding stochastic theory. In classical mechanics, you can, for example, consider $\rho(p,q)$ and the Liouville equation for how it evolves. Does this possibility destroy the deterministic character of classical mechanics?  No, you simply consider, in deterministic classical mechanics, states where you have uncertainty because of insufficient information about the initial values. 

In the same way you recover quantum mechanics in quantum equlibrium, that means, in states, where you do not have exact information about the hidden variables, but know only their probability distribution $\rho(q)=|\psi(q)|^2$.  The consideration of such uncertain states does not make the Bohmian evolution equations and, therefore, Bohmian theory, indeterministic. . 

About Bayesianism: Your claim (I quote) is "Bayesian probabilities are defined from the idealisation of the notion of relative frequency, not from actual frequencies."  This is false, because, of course, whatever meaning you assign to "idealisation of the notion of relative", it cannot be defined in any way from frequencies, given that theories have no frequencies (as a consequence also no relative frequencies, notion of frequencies, or their idealizations) but have Bayesian probabilities. 

Then, I know what lattices are, and my point was that you confuse the common sense and standard logic language use of "and" and "or" with that used in these different mathematical domains. You accuse Bohm and Bub "apparently they still did not recognise that the calculus of propositions described by Jauch and Piron is a treatment of statements about measurement results".  This is nonsense. What Jauch and Piron use is the mathematical language of lattice theories, which is of course free to use a non-distributive lattice.  "Statements about measurement results" do not have this freedom, they have to follow classical logic, and their "and" and "or" follow Boolean logic. 

Then, in A.14 we have, of course, some Schroedinger equation, but what is H, what is x, what f, in the case of classical mechanics? Unclear. But, ok, let's ignore it, it is clear anyway that your theorem that everything follows a Schroedinger equation is misleading nonsense once you describe classical mechanics in this way too.  As a consequence, it means nothing: If you find a completely classical hidden variable theory, ok, you can also find then a Francis-Schroedinger equation for it, but it will probably have nothing at all to do with the original Schroedinger equation. 

About "The only argument you have is that physics might be determined by undefinable and indescribable quantities":  Of course, I have never made such an argument.  Of course, dBB theory is a well-defined theory, all of its parts are describable. I agree with you that there probably is no coherent version of positivism at all, but this nicely fits with everything I see from you, SCNR.

And of course I continue to pretend that dBB theory, which makes the same predictions as quantum theory, is not excluded by Popper's criterion of falsifiability.

To:  Dr. Charles Francis

I have just seen your answer of 6 hours ago.  Many thanks.  But, being a Cambridge scholar, you know better.  A mathematical proof can only be refuted by mathematical logic (apologies for stating the obvious).  We, physicists and mathematicians, are fortunate in that we can argue about our fields to our hearts' content without resorting to mean prejudices of the sort one observes among (most) religious and sectarian groups.  We are trained not to be fanatic (at least, in principle).  No pretensions here and no apathy!  Thank God!

Cheers.

Charles, "Ilja, I have used the definition of momentum used in quantum mechanics." 

No. You write "Add parameter x to the momentum state | p 〉 to obtain the Bohmian state | x , p 〉 according to (6.10) and act with the momentum operator P on the first parameter". This is an own invention of you and nor the definition used in quantum mechanics nor in Bohmian mechanics. 

"I have already pointed out that in qm probabilities apply to measurement results, not to theories." Fine, but in this case reformulate your claims so that it becomes clear that you make statements about QM and not statements which have something to do with a definition of Bayesian probability.  

"If "Statements about measurement results ... have to follow classical logic" as you claim, then there would be no such thing as incompatible measurement, which is patently false."

Wrong. Of course, everything related with "incompatible measurements" can be adequately described in propositions which are in agreement with standard classical logic. Of course, in this case the operations of lattice theory named there "and" and "or" should be renamed to avoid confusion with the usual logical meaning of "and" and "or".  And one would better avoid the choice of Jauch and Piron to name "propositions" only some very particular propositions about the outcome of certain experiments.  And one would also better avoid to name something "measurement" as long as it is not clear if the outcome of this "measurement" is independent of the "measurement device".  

Then, in the situation discussed your "trivial" is very unclear. There are a lot of different forms for the evolution equations of classical theories: Euler-Lagrange equations, Hamilton equations, Liouville equations, Hamilton-Jacobi, each of them well-known so that educated people could name them all "trivial"  None looks like a Schroedinger equation. 

"At least we appear to agree that Bohmian mechanics does not allow you to use the quantum mechanical notion of momentum." 

Not at all. I insist that you have to learn standard measurement theory of Bohmian mechanics.  Which includes momentum measurements.  There are sufficient introductionary texts into dBB theory where you can learn this, to copy an intro here would be too long.  Only very short:  Almost the same way as the quantum measurement, except that the Copenhagen "classical measurement pointer" is also quantum, and then the pointer position - the Bohmian configuration of a macroscopic device - is observed as the measurement result. So we have here the same measurement process as in the Copenhagen interpretation, the only difference is how the macroscopic part of it is described:  by observing a classical pointer of the classical part of Copenhagen, or by observing the configuration of the pointer in dBB theory. 

"...if you want to claim the same predictions as quantum theory, then the additional "hidden" variables of Bohmian mechanics are not falsifiable..." 

No, dBB theory as a whole is as falsifiable as quantum theory, once it makes the same predictions as quantum theory. And the configuration is in no way "additional" or "hidden", our own state as well as what is the classical part of the Copenhagen interpretation is the Bohmian trajectory of the configuration. 

But even if you could separate some part of dBB theory and name it "unobservable", it does not matter.  Popper's criterion is about theories, not about parts of theories. 

What makes a part of a theory (say, some principle) scientific is that it allows, in combination with other hypotheses and theories, derive some non-trivial predictions which otherwise would be impossible.  It is not the principle taken alone, which should be sufficient to derive a prediction.  Simple case:  From A and B follows a nontrivial prediction, from A alone as from B alone follows nothing.  Following your understanding of Popper, one would have to throw away A as well as B, because taken alone they make no predictions.  Following the real Popper, A and B together define an empirical theory which makes falsifiable predictions. 

Charles, in GDZ 1992 I find as number 11 a pair (Q,psi), which is of course not the problem, that's correct. But your way to define a momentum measurement for this is your invention. Your invention you seem to justify as a consequence of "the conjugacy relationship between position and momentum", but it nonetheless remains your invention, and the way how momentum measurement is described in BM is different. Learn BM, and then come back. My short description was, obviously, not sufficient to teach you, more is impossible here. 

"I was talking about simply bolting the "hidden" parameter x onto the quantum state f, to produce the Bohmian state (x,f)" 

But you don't understand that for macroscopic objects this x is simply the classical trajectory we see. In comparison with Copenhagen dBB it only extends x to the quantum part, and psi to the classical part, and obtains a unified picture.  The alternative - a unified picture without classical part and without x, is the many worlds nonsense, which is .... [celf-censored]. 

The minor differences between Copenhagen and von Neumann-Dirac may be of interest for those who like them, but not for me. And the purpose for mentioning Copenhagen is only explanatory. But it is, of course, meaningless to try to explain something to someone who does not want to understand. 

You seem unable to understand the difference between a theory which makes the same predictions as quantum theory and one which makes no predictions.  For theories making the same predictions, Popper's criterion gives no indication what has to be preferred, and other criteria have to be used: Simplicity, beauty, explanatory power, clarity, internal consistency, compatibility with other principles and others.  Here dBB has clear advantages, but to discuss it with somebody who does not even want to learn how momentum is measured there is hopeless. 

Charles, no, the QM momentum operator acts on psi, thus, it gives, if you like to write it as a pair (which is a quite meaningless exercise)  (Q,psi(q)) to (Q,-id_xpsi(q)).  dBB preserves all results about quantum measurements.  And also preserves the relation between the quantum "position measurement operator" defined by (Q,psi(q)) to (Q,x^ipsi(q)) and the "momentum measurement operator".  

And, I repeat myself, x is far away from being irrelevant, the x of the "measurement device" defines into which of the possible "measurement results" the effective wave function of the "measured system" collapses, and it also defines what we are and what the universe around us is - this is defined by the configuration.  At least my own state is defined by a trajectory of my configuration, your state may be described by some wave function describing a superposition of incompatible states of thinking, (observing your reasoning I would not wonder, SCNR) but not my. 

Thus, I know, of course, that the Bohmian configuration is not hidden - it describes all what we see. Sorry for trying, sometimes, to argue with people who do not recognize this without caring much about this particular wrong choice of words and not objecting to every ocassion than the whole world we see around us is named a "hidden variable".