Bell made a mistake: in proof of his "theorem" both of the factors A & B are implicitly interpreted as absolute probabilities, but one of them must be a 'conditional probability.'  When this is taken into account, the algebra of his inequality derivation does not go through. Thus, Bell's conclusion is valid only if both are absolute probabilities (i.e., both = 0), which is useless for his purpose. So there is an error, perhaps not the one you are focused on.

Thank You, Scientist! Perhaps you have details in your publications? Thank You for information. Be well!

I am sorry, for not informing my brothers in love. Namely my problems were solved by: B. C. Sanctuary, "The Statistical Interpretation of Entangled States".

http://arxiv.org/abs/quant-ph/0404011

Quantum mechanics  (QM) is not wrong because relativistic QM, quantum electrodynamics, gives us opportunity to calculate any actually measured physical quantity as energy level, transition probability or cross-section for all known atoms. Calculated values practically coincide with experimental ones. All of us must know these facts and obey them. There are talks about defects in QM, because we cannot explain QM rules in terms of habitual classical mechanics, that is impossible and, as a result, many talks are empty.

QM is, at best, incomplete.  That's part of the reason it is incompatible with GR.  Also, some things said on the authority of QM are contradictory. Such logical defects very likely also contribute to the problem with GR.  In any case, having a recipe to calculate to calculate observables, especially one with lots of adjustable parameters, is just simply unsatisfying!  The world has enough religion, let's try to do better.  One obvious target is superluminal entanglement.  Dmitri's unease is well placed in this context. 

@A.Krachlauer

Only wrong theories are complete because their boundaries are known (cf. the phlogiston theory which does not account for work). Any theory alive is incomplete by definition. The only important thing is, we are not aware of any experiment not described by QM. The rest is OUR PROBLEM, NOT THAT OF QM.

QM is not just "our" problem as its interpretation (at least) encompasses contradictions. Also, note that, per formal logic, within any would-be logical structure (e.g., a physics theory) that is based on contradictory hypothetical inputs (hypotheses) all theorems can be proved true---even wrong ones.  In this light, QM is suspicious!  BTW, it is not true that only wrong theories are complete; in fact, any theory for which an additional hypothesis is redundant, is by definition: "complete" (wrt to its set of hypotheses, example: group theory----for which boundaries are well known!).  But, Martila's point is, I judge, that, something is fishy with entanglement and singlet states (never "seen" in the lab, just in QM calculations.)     

``as its interpretation encompasses contradictions.'' 

What actual contradictions do you have in mind? I know a fair number of strange things implied by QM, but no contradictions. Further, an actual contradiction would immediately open the way to an experimental refutation of QM: it would be enough to attempt to decide experimentally which of the two terms of the ``contradiction'' actually hold: a contradiction cannot be found in an experiment!

As for ``singlet states'' not being seen in the lab: that may technically be correct, since we never see anything in the lab, except results of measurements. But surely we have seen plenty of manifestations, in the lab, of such quantum peculiarities as violation of Bell's inequalities, which are explained in terms of the underlying quantum theory.

The singlet state is the combination of mutually exclusive alternatives---a crass "contradiction."  QM is actually a grab bag full of ad hoc features.  Dispensing with any one of them could well leave the rest standing.  We agree: Nature is not contradictory.  On the other hand, the human interpretation of what it thinks it sees in nature is very often illogical, contradictory nonsense.  Arguably, v.Neumann's "Projection Hypothesis"  is in this category.   

If the singlet state were a contradiction, it would not be possible to make consistent computations with it. I will, of course, gladly admit that your interpretation of the singlet state is indeed contradictory. But that may not be altogether the singlet state's fault.

A contradiction arises whenever, in a given theory, you can prove both a given statement and its negation. This does not, to my knowledge, happen in quantum mechanics.

von Neumann's projection postulate should be employed ``whenever a measurement takes place'' and this replaces unitary evolution. Admittedly, this is not absolutely crystal clear (what is a measurement? when does it take place?) but it is very far from involving a contradiction. Further, it can be shown that it does not matter where you put the dividing line between unitary evolution and measurement, as long as the measuremet is ``macroscopic enough''. I have no doubt that improvements will some day arrive on that front, nevertheless, I must say I am more interested by the fact that the very strange predictions of quantum mechanics (entanglement, non-locality, as well as superposition of ever more macroscopic objects) have recently become experimental reality rather than theoretical imaginations.

Excuse me!  It is not "my interpretation" that the singlet state is by definition applicable to a single entity or circumstance of an electron in a particular orbit in an atom.  That is textbook material.  My interpretation is that the singlet state is an artificial, average of an ensemble comprising 50::50 of its component states.  Thus, in computations in which the average makes sense using the singlet state expression gives the right answer.  Nevertheless, the orthodox interpretation is, illogically, that, it describes an extent physical situation.  

That "measurement," and its impugned collapse, lies outside QM (i.e., is not described by Schröedinger's equ.) was taken be at least half of the founding fathers of QM as reason to doubt the total coherence of the theory.  After all, experimental apparatus and experimenters themselves are all composed of the very same sort of material (atoms) as the subjects of experiments.  That special rules obtain whenever a human is involved is a dead give-away that there are fundamental inconsistencies in the theory or at least its interpretation.

Further, NO experiments have validated entanglement or superluminal relations of any sort. In particular by measuring someTHING moving at "c".  What they have validated is just violation of a statistic derived by Bell.  The interpretation of this violation depends on the rectitude of Bell's arguments, long ago, starting evidently with Edwin Jaynes, found to be in fundamental error.

The current acceptance of weirdness as promulgated by QM-enthusiasts is a sociological phenomenon!  It is a modern secular stand-in for classical religion.  And not a bad way to procure funding!  

``That "measurement," and its impugned collapse, lies outside QM (i.e., is not described by Schröedinger's equ.) was taken be at least half of the founding fathers of QM as reason to doubt the total coherence of the theory.''

Somewhat exaggerated, but with a grain of truth to it. Nevertheless, there is no reason, in Science, to give undue weight to ``founding fathers''. They did not have access to present experiments, in which quantum effects are measured on macromolecules (C60) or on microwave fields in cavities. Nor did they know of theoretical advances such as decoherence, which go a long way towards resolving the inconsistency between von Neumann postulate and unitary evolution.

So this authoritarian argument has little value. The true issue is, do we in actuality have a problem? We well may, I agree, but none that could be called a contradiction: the theory yields unambiguous results, which are confirmed by experiment. Thus your claim that ``Also, note that, per formal logic, within any would-be logical structure (e.g., a physics theory) that is based on contradictory hypothetical inputs (hypotheses) all theorems can be proved true---even wrong ones.’’ distinctly does not apply to quantum mechanics.

As to your claim that the textbook interpretation of the singlet state is contradictory: why then can one perform experiments on single particles, the results of which are in perfect agreement with the calculations made in quantum mechanics?

Of course, no one has ever measured anything moving at velocity higher than c, nor has anyone claimed to: you seem to be making up an untenable position, the better to attack it. What has been observed are correlations which cannot, apparently, be explained in local terms. This is Bell's work, and it is, in principle, very straightforward. I am not aware of any work showing this to be in error, but am willing to learn.

A Physics theory, to be science and not religion (arbitrary dogma), in principle aspires to be a "logical structure."  Thus, it is in fact subject to the criteria of formal logic, even when the reality is that it fails to meet the implicit standard.  What is sociologically called "QM" is larded with goofy terminology which is lexicographically contradictory.  One consequence is, that the two main physics theories, namely QM and GR, are mutually incompatible.  If this problem does not interest one, so be it, that's just a personal issue.

If experiments on single systems give results in accord with QM, then there is a logical lecuna involved.  QM gives ONLY "expectations" in principle. And, it is manifestly impossible to "be" at once of two mutually exclusive natures.  Shoe-horning empirical facts into some complex of terminology so as to appear coherent, is nothing new.  Logically the circle cannot be closed. But such is not valid science.

All the correlations Mr. Leyvraz references, can be understood as engendered by a "common cause."  The QM story employing superluminial entanglement is science-fiction! Again, those happy with such an explanation are of course perfectly free to believe what they chose.  Those possibly interested in what I have had to say (or write) about these matters at a technical level can find it at:  www.nonloco-physics.ocatch.com 

I am firmly convinced that quantum mechanics is intrinsically contradictory because the fact that measurements have outcomes and we have definite perceptions cannot be accounted for by a linear theory. I believe this does not deserve any specific comment and I stress that decoherence is not at all a solution to the problem, from a fundamental point of view. It is important on the practical point of view because it states that, due to our limited capabilities we have enormous difficulties in putting into evidence superpositions of macroscopically different states. But I do not look for theories which are just useful because they give the results we get in the experiments we are (presently) able to do. I want that any proposal which aims to be "a theory" be logically consistent. Quantum mechanics is not because the standard linear evolution and wave packet reduction are contradictory. On the other hand the theory does not contains the mathematically appropriate ingredients to allow us to say: up to this stage (mesoscopicity?) the theory is OK, for larger systems it is not! So Bell was perfectly right in claiming that "nobody understands quantum mechanics because nobody knows where to put the split between quantum and classical, reversible and irreversible, microscopic and macroscopic. I think that these considerations should be clear to everybody and should push serious people to try to go beyond the standard theory.

In the large, I agree with Prof. Ghirardi above---modulo stylistic factors and emphasis.

May I take the occasion to pose a question for Prof. G., for which I have awaiting an opportunity for over 20 years:  Namely:  could the GRW notion be tested by  examining the interference pattern of a particle beam passing though a slit for deviations from the ideal?  It seems to me that if some "particle  waves" have spontaneously collapsed while in the slit, thereby at least temporarily becoming "pure" particles, they should then tend to make an undiffracted shadow of the slit superimposed on the diffraction pattern from the non-collapsed, still wave-like portion, of the beam.  Do such ideas make sense in view of your understanding of the GRW notion?

Insofar as "gold is where you find it," I suspect that the detailed study of those aspects of QM found to be troublesome (however) might lead ultimately to a breakthrough.

``Quantum mechanics is not because the standard linear evolution and wave packet reduction are contradictory.''

Why? I do agree that they do not follow from each other in any simple way. But why contradictory? In the same vein, one might argue that classical thermodynamics and classical mechanics are contradictory: the second law implies irreversibility and classical mechanics denies it.

As to the position of the ``measurement split'', I fully agree that nobody knows exactly what to do about it. But one also knows that shifting the split, as long as one does not come too close to the microscopic world, has negligible observable consequences. While I do agree that it would be wonderful to be able to say exactly what a measurement is and when, and how, it takes place, the fact that the theory itself tells us that it does not matter, may be telling us something.

I fully agree with the arguments of Professor Ghirardi, and I do fully agree with him that the decoherence theory is not a solution of the problems connected with the transition from quantum to classical physics. On the other hand, I think that the interesting observation made by professor Leyvraz about the conflict between classical thermodynamics and classical mechanics should be used for the rather different purpose of attracting the attentions of the researchers on the fact that in addition to the problem of settling the conflict between quantum  and classical mechanics, which in my opinion is satisfactorily solved by the GRW theory, there exists the closely related problem of making quantum  mechanics compatible with the second principle of thermodynamics, or, vice-versa. Is the second principle of thermodynamics  a consequence of the limitations of the observer or is it a fact independent of the human observation of nature? This depends on wether randomness is an objective property of nature or it is a convenient way to take into account the limitations of the human observer. I am personally convinced that the GWR theory can be conveniently used to handle also the second  problem, favoring the view that the second principle is a realistic representation of reality, while quantum mechanics does not fit yet that condition, if it is not supplemented  by. the GRW theory

I would like to thank Paolo for his nice words and for his appreciation of our phenomenological and surely not fundamental model. I attach to it some value because it has shown that, contrary even to Einstein's opinion in Reply to critics, one can work out a scheme which is completely free, at its nonrelativistic level, of all conceptual problems affecting the standard theory. Moreover the scheme is mathematically precisely formulated and indicates where the deviations from quantum mechanics which are implied by  it  might be found. Moreover, a large set of investigations by many scientists (Adler, Pearle, Bassi, Tumulka etc), have clarified that if one wants a quantum theory which agrees with quantum mechanics at the micro-level and accounts for reductions in measurement processes and for the classical behavior of macroscopic systems, then one unavoidably is led to consider a theory which exhibits many, if not all, features of GRW. This makes the approach as deserving some attention.

I also would like to take this opportunity to call the attention of Prof.  Kracklauer that there are many many papers suggesting tests of quantum mechanics versus GRW, and the most promising ones, which have already led to put some bounds on the parameters of the model, come precisely from diffraction experiments with large molecules, like fullerene and so on, up tho macromolecules containing 10**6 atomic masses. However, to really have a crucial test between the two theories,it seems that one should increase the mass of the particles giving rise to the diffraction pattern, of a factor of about 1000. The Vienna Group is rather confident that this will be possible in not too long times.

Since I am firmly convinced that the linear nature of the theory must break down at a certain level I believe that experiments aimed precisely to answer this question are important. However, to understand whether precisely to look at, one needs some indications. I believe that GRW, being mathematically precise can give an idea of where to try to look for violations of linearity. This is the real merit of the theory we have proposed (besides the one of being, contrary to the standard theory, internally and logically consistent and in agreement with our definite perceptions)

There are two points made above which i find to open open. 

One, is the fact that classical mechanics is said to be time reversible.  So far so good; but non relativistic mechanics is obviously an approximation. I have proposed a fully SR action-at-a-distance but on the light cone version of mechanics which is not time symmetric and therefore proposes a resolution for this issue consistent with intuition.

Two, it can be reasonably be expected that if the GRW notion is correct, then diffraction patters of particle beams should exhibit distorted features.  So far as I know such has not been observed; at least not in Zeilinger's neutron beam experiments (the last I personally examined with care). Larger particle beams surely will involve larger irrelevant secondary (size vis-a-vis the slit) effects, muddying up the waters.

That QM is "good enough for all practical purposes" is an old claim and a back-handed recognition that QM is not good enough for idealistic (i.e., fully rigorous) purposes!  Those who are happy with the current state of affairs wrt the interpretation of QM appear not to be disturbed with lexicographic contradiction---reflecting underlying conceptual contradiction.  

It seems to me that the statement "quantum mechanics is good enough fapp" is becoming inappropriate. If the alternative is something on the line of nonlinear and stochastic modifications of quantum mechanics, experimental test are in course of being performed and there is an enormous literature on the matter. I suggest to look at the papers by A. Bassi, in which there is a recent one which makes the point on the status of experimental search. So it might be that one succeeds (it has not yet happened) in proving that quantum theory is not good enough.

Quantum mechanics (QM) was intended by its fathers as a universal theory. But the cat paradox says that this theory, in the present form, can not play this role. I agree entirely with Prof. Ghirardi that “quantum mechanics is intrinsically contradictory” and that “the linear nature of the theory must break down at a certain level”. I well respect the GRW theory which is, perhaps, the only approach to the cat paradox which is based on these two ideas.  However, in my opinion, the cat paradox is not a measurement problem. This paradox appears at the micro-level, rather than at the road between the micro-level (radioactive atom) to the macro-level (cat). The cat paradox is rather the problem of constructing an internally consistent quantum-mechanical description of microcat states – such states as the linear superposition of the macroscopically distinct states, “decayed atom” and “undecayed atom” must be treated in quantum mechanics as mixed states, rather than pure states. QM with its standard approach to microcat states is indeed internally inconsistent and, as a consequence, at present it is impossible in principle to interpret it in a logically consistent manner.

 In particular, the Statistical interpretation like others is inconsistent at present. The point is that the squared modulus of the wave function to describe a microcat state can not be treated as the probability density, because it describes statistical data which do not belong to a Kolmogorovian probability space. Only the squared modulus of each state that enter into the superposition can be interpreted as the probability density.     

 The Hilbert space, associated with any one-particle quantum process in which there appear microcat-states, has a nontrivial structure. In this space there is a superselection rule according to which microcat-states must be treated as mixed states. Only with this interpretation of microcat-states quantum mechanics is really becoming a universal theory. I present this idea, by the example of a one-dimensional completed scattering, in my paper arXiv:1412.7657.

 Sincerely Yours.    

Note also that quantum processes, where such states occur are combined ones which consist of alternative subprocesses. For example, a one-dimensional completed scattering is a combination of two subprocesses – transmission and reflection. It is important that quantum evolution crosses the superselection sectors in the Hilbert spaces associated with such combined processes. As a result, the linearity and unitarity of their subprocesses are violated at the stage when they cross superselection sectors.