My STOE accepts dBB and the transaction interpretation as a closest (but not correct in the sense of describing experiments) to the STOE and the experiments that reject all other interpretation.

John Hodge

I am happy for you!

Dear Sofia,

I agree with you and I think that you have explained very well:

What I understand from Einstein's words: "too cheap" is "too much simplistic", i.e that Einstein thought that the indeterminism of QM has a more subtle explanation.

But let me to add something more. I think that the Einstein problem was not really with the determinism or not, but with locality or not in quantum mechanics. In one of these discussions with Born, he answered to him:

When a system in physics extends over the parts of space A and B, then that which exists in B should somehow exist independently of that which exists in A. That which really exists in B should therefore not depend on what kind of measurement is carried out in part of space A; it should also be independent of whether or not any measurement at all is carried out in space A.

This is in fact the main idea of the EPR paper so cited nowadays for the quantum entanglement and actually Bell what has proven is the non-locality of QM as can be the collapse of the wave function under a measurement. The hidden variables is another step of knowledge for trying to solve this difficulties of QM and I don´t think that Einstein could think that the dBB theory could solve it really. Actually there are nothing new in this theory which can be claimed to distinguish it from the usual Copenhagen school (may be it is my ignorance). Einstein always made an "ontological" interpretation of QM while Bohr and most followers did it "epistemological". That's to say, Einstein thought that QM has real problems while the school of Bohr thought in limits of our knowledge for the microscopic world. Perhaps both are right and it only depends of the real knowledge that we have of the phenomenology linked with this subtle issue.

Daniel, my wise and kind friend,

It's for long that I didn't "hear" of you. How are you?

Now, a few comments to what you said:

1. First, thanks for the citation you mentioned - can you tell me at which page it is?

2. Please note the word "determinism" in my citation from Einstein's letter. This is what he said that displeased him. As to the nonlocality, do you want to say that Einstein didn't know that Bohm's interpretation is nonlocal? Please see what Bohm wrote on pages 174-175

"We introduce the wave function, ψ = R(x1, x2, . . ., xn) exp[iS(x1, x2, . . ., xn)/ħ] and define a 3n-dimensional trajectory, where n is the number of particles, . . . . The velocity of the ith particle is vi= ∇i S(x1, x2, . . ., xn)/m. The function P(x1, x2, . . ., xn) = R2 has two interpretations. First, it defines a "quantum-mechanical" potential . . . Secondly, P(x1, x2, . . ., xn) is equal to the density of representative points (x1, x2, . . ., xn) in our 3n-dimensional ensemble."

I trust Einstein that he read Bohm's articles, which appeared just 4 months before his letter to Born.

3. You say "actually Bell what has proven is the non-locality of QM". No Daniel, this is not rigorous. In his proof Bell did two assumptions: i) locality, ii) that the algebra of complex amplitudes can be mapped one to one onto the algebra of positive numbers. So, we cannot claim that the violation of Bell's inequalities disproves the assumption i. It's rather the assumption ii that is disproved.

4. You say "The hidden variables is another step of knowledge for trying to solve this difficulties of QM and I don´t think that Einstein could think that the dBB theory could solve it really. Actually there are nothing new in this theory which can be claimed to distinguish it from the usual Copenhagen school (may be it is my ignorance)."

I quite fail to understand what are you trying to say. dBB is indeed of hidden-variable theory, but a NONlocal one. The Copenhagen view didn't admit any substructure of QM, no hidden variables. So how can dBB be nothing new in comparison with "the usual Copenhagen school"? What are you trying to say, in fact?

5. "Einstein always made an "ontological" interpretation of QM while Bohr and most followers did it "epistemological". "

Aren't you too quick to jump to conclusions? What makes you sure that Bohr had an epistemological view on quantum systems? As far as I know, Bohr thought that we can't know more on such systems than the wave-function tells us. But he spoke of what we can know, not of the nature of the wave-function. Please correct me if I am wrong.

6. Now tell me: what is Daniel's view on the wave-function? Ontological or epistemiological? You see, I am asking myself how could the wave-function describe all the influence of the fields through which a quantum system passes, without the wave-function being ontological?

With kind regards and best wishes

Dear Sofia,

These subjects belong to my curiosity and I'm not at all an expert if I have answered is mainly for giving my humble opinion in your thread. Let me try to respond you.

1. In page 164.

2. Yes I think that Einstein knew the Bohms papers.

3.Perhaps I explained me very badly, sorry. What Bell showed was that the assumption that the values of the photon polarization (or spin if it were with electrons) pre-exist their “measurement” ,combined with the perfect anti-correlation when the axes along which measurements are made are the same, and the 1/4 result for the frequencies of correlations when measurements are made along different axes, leads to a contradiction. Thus no physical theory of hidden local variables can ever reproduce all the predictions of QM. But for me what is fundamental here is the locality, because if it were made without assuming this property we could develop a theory as dBB or others in agreement with this theorem and Bell always presented his results respect to EP. What is more important, a topological quantum theory perhaps could solve these difficulties of Einstein and the Copenhagen school.

4. The dBB theory is nonlocal and I don't know where have said other thing. Perhaps you deduced it from my comment that Einstein thought that this theory could solve the problems of locality. But the locality in dBB is different than the usual QM, the guiding equation depends of boundary conditions for the system.

5. It seems to me that Bohr assumed easily the uncertainty principle and we had

thinks impossible to know. His model against the radiation of electric charge never worries him if the atomic spectra were telling us other thing.

6. I have both positions in some parts I think that QM is epistemic and only interested in an statistical knowledge, while in others the wave function is associated to states that are ontic. For instance the low spin or high spin of certain

My kind Daniel,

"3. . . . . . . What Bell showed was that the assumption that the values of the photon polarization (or spin if it were with electrons) pre-exist their “measurement”, combined with . . . . leads to a contradiction. "

Yes, my dear friend, but how did Bell show this? Using the calculus with positive probabilities. The algebra of complex numbers (amplitudes) is not isomorphic with the probabilistic calculus, as the former contains both elements x and -x.

"Thus no physical theory of hidden local variables can ever reproduce all the predictions of QM."

No, that's not true. One can build a theory in which the basic concept is amplitude (defined as complex number), and not probability. The probability has to be a secondary concept, defined as the absolute square of the amplitude. In this way one would get directly the algebra of the quantum amplitudes.

"But for me what is fundamental here is the locality, because if it were made without assuming this property we could develop a theory as dBB . . . . "

I lost you here. dBB also contradicts Bell's inequalities, it is non-local. The drawbacks of dBB are the conflict with the relativity and with the quantum contextuality.

"What is more important, a topological quantum theory perhaps could solve these difficulties of Einstein and the Copenhagen school."

How do you envisage a topological quantum theory, all the more reproducing the non-locality?

"I have both positions in some parts I think that QM is epistemic . . . while in others the wave function is associated to states that are ontic"

Hmmm! You see, it is sure that in the apparatus travels something, i.e. there exists something ontic. And it behaves according to instructions given by the wave-function. But, if the wave-function is only on the paper, i.e. the result of a calculus, how that ontic thing gets those instructions? It does not communicate with the paper. So, that ontic thing has to carry also the instructions. Bohm supposed that the wave-function is real.

Dear Sofia,

Using the calculus with positive probabilities. The algebra of complex numbers (amplitudes) is not isomorphic with the probabilistic calculus, as the former contains both elements x and -x.

The probabilities are always positive real number within the closed interval [0,1]. Thus there are never an isomorphism between probabilities (or its statistical distributions) and the complex numbers, independently of sign of x.

"Thus no physical theory of hidden local variables can ever reproduce all the predictions of QM."

No, that's not true......

Please look at the link

https://en.wikipedia.org/wiki/Local_hidden-variable_theory

Bell's 1964 paper (see Bell's theorem) showed that local hidden variables of certain types cannot reproduce the quantum measurement correlations that quantum mechanics predicts.

"But for me what is fundamental here is the locality, because if it were made without assuming this property we could develop a theory as dBB .

I think that here is a problem with my english, when I say locality I don't understand that dBB is local theory, what I want to say is that we always try to find local theories and that there is a problem of locality or how local a theory is.

How do you envisage a topological quantum theory, all the more reproducing the non-locality?

As usually the topology does, using homotopies, homologies, Chern classes or so on.

Hmmm! You see, it is sure that in the apparatus travels something, i.e. there exists something ontic

There are different forms to see the wave function, one is just by collapting due to measurement and another is to have states as the atomic ones. For instance if you have d-electrons you can see how you full eg or t2g states using Hund's rules as if they were real objects. Thus you have an epistemic and an ontic interpretation. Both are employed in Physics or at least this is how I see them.

Both the concept of amplitude and probability density are essential as seen in interference/diffraction where amplitudes are added prior to taking the square modulus.

Then you have local guage invariance because of the arbitrary phase factor, quite essential if you study the effect of a magnetic field in the quantum.(when the phase factor is related to a line integral of the vector potential, also a sort of not well defined object, indeterminate within a gradient of a scalar)

Also I see the statistical interpretation essential to

answer to be or not to be questions. The debate may be if statistics is relevant to reality, the answer of QM seems to be yes.

The fundamental problem of what is a microscopic object remains unsolved. QM is somewhat of a solipistic theory, it tells you just so far, and then you are left on your own, ideas, dreams, whatever.

Yes, Daniel is right, the form of orbitals is quite usefull in Chemistry.

Daniel,

"The probabilities are always positive real number within the closed interval [0,1]. Thus there are never an isomorphism between probabilities (or its statistical distributions) and the complex numbers, independently of sign of x."

You missed what I said. Bell proved nothing because he worked improperly. Moreover, QM is LOCAL, not NON-local. See below:

Given two particles, A and B, and a certain result UA & VB that you want to obtain with them, you proceed as follows: you multiply LOCAL amplitudes, e.g. a1,U b1,V , you add all the contributions, Q = a1,U b1,V + a2,U b2,V + . . . + an,U bn,V , The probability for UA & VB is |Q|2. Bell did not work this way. What he did was P = |a1,U|2 |b1,V|2 + |a2,U|2 |b2,V|2 + . . . + |an,U|2 |bn,V|2 . Of course that P differs from |Q|2. This is why I say that there is no isomorphism between the two types of calculus.

By the way, the sum Q may be null, and so begin questions how the particle B "knows" that particle A produced the result U, s.t. B should not produce the result V. And the answer is that B can know nothing, because that all the amplitudes ai,U and bi,V are local.

"As usually the topology does, using homotopies, homologies, Chern classes or so on."

My dear friend, here you speak a language I don't understand. I didn't learn these things, s.t. I can't follow you. But, in order to mimic the QM results one should work with complex amplitudes or an isomorphism of their algebra. It would be a very interesting and illuminating question whether one could mimic the QM predictions with an algebra poorer than that of complex numbers - but I am not a mathematician. Maybe I will decide to ask this question, if I'd have time.

How about something like

sum over k,l c(k,l) a(k))b(l))

c(k,l) amplitudes and a(k)) b(l)) states consistent with results?

The probabilities of each case are c(k,l)* c(k,l)?

Further you have to include the statistics if they are identical (Fermi or Bose)

Maybe there are conditions both together have to fullfill.

If one compares dBB with the other realist interpretations (Nelsonian stochastics, Caticha's entropic dynamics) or even with dBB after Valentini's h-theorem, it is indeed cheap. They have reached more, derived the Schrödinger equation, or derived the quantum equilibrium. In this sense, the "too cheap" is justified.

Dear Sofia,

Bell used inequalities instead of equalities which justifies the product he has done.

No, my kind @ Daniel,

This is not an answer. Bell worked with positive probabilities, the concept of complex amplitude does not exist in his proof. I wonder, is his proof fresh in your mind? In my mind it is fresh because I criticized it. So, I repeat, in Bell's proof the total probability of an event is P = P1 + P2 , if there are two ways of pbtaining that event. In QM it doesn't go this way.

It's a conceptual difference. The basic concept in QM is amplitude of probability, not probability. It is our classical tendency to speak of probability of an event. But at the microscopic level, Nature works with amplitudes. So, if there are two amplitudes of probability a1 and a2 that contribute to the event, then P = |a1 + a2|2. This result is equal with the classical one only if a1 and a2 don't produce interference, i.e. if a1* a2 = 0.

With kind regards

Dear @ Roman

It is very interesting what you say:

"However, from a biological perspective, the deBroglie-Bohm interpretation allows for consciousness to be outside of the box, while in orthodox interpretation, consciousness remains fundamental. Therefore from the point of view of mind-brain studies it would seem that Bohm's views are guided away from mysticism and more into reality. This is very important in understanding the mind."

But I have some problem with your saying that in orthodox interpretation, consciousness remains fundamental. In understand what you say, but I have a problem, because the standard QM did not mix the human brain in the measurement process. It only considered a classical apparatus of measurement, e.g. an ion or a bubble chamber. No humans are mixed in the process, and no biology.

On the other hand, our apparatuses of measurement are constructed so as we THE HUMANS be able to read them.

You also say

"In biology, there is a nonreductive vista that needs to be attained and Bohm's views are central to the full integrative picture of the molecular level beyond Watson & Crick's genes. We know that there is no gene that codes for consciousness so Bohm's work is pivotal here.

You can call it theoretical biology, integrative biology or biophysics, yet orthodox QM is encompassing to the biological need for nonreductive physicalism. Therefore, Bohm's views are essential.

Universal consciousness and other metaphysical anolomies (aspect monism, panpsychism, . . . . "

Wow!!! Many of us are not biologists and do not understand the terminology you use. Would you speak more explicitly?

With best regards

No @ Juan,

"The debate may be if statistics is relevant to reality, the answer of QM seems to be yes."

The debate is on how you calulate, theoretically, the statistics. Please look at what I answered to Daniel. (But, as I believe that you won't look, I repeat below.)

The basic concept in QM is amplitude of probability, not probability. It is our classical tendency to speak of probability of an event. Thus, in Bell's proof the total probability of an event is P = P1 + P2 , if there are two ways of obtaining that event. In QM it doesn't go this way, at the microscopic level, Nature works with amplitudes. So, there are two amplitudes of probability a1 and a2 that contribute to the event, s.t. P = |a1 + a2|2. This result is equal with the classical one only if a1 and a2 don't produce interference, i.e. if a1* a2 = 0.

This is why I say that what Bell proved, in fact, was that there is no isomorphism between the calculus with complex amplitudes and the calculus with positive probabilities.

@ Ilja,

would you be more generous with explanations? What special assumptions made Nelsons, Caticha, and Valentini to the difference of Bohm? Please give us the main ideas in a few lines. You see, people are busy, if you realize, we have an enormous amount of literature to read. So, a bit of filtration is usefull.

Dear Sofia,

Complex numbers have many different formulations, one of them is the rotation in a plane. I attach you one link where it is well calculated and presented the Bell's inequalities. They are right and this is normal because many brilliant people has been using them for a long time.

Dear Sofia,

About relation between the calculus with complex amplitudes and the calculus with positive probabilities.

I think I have a nice method to deal with positive probabilities and deduce from them the concept of the amplitude of probability that we use in Quantum mechanics.

Please see my paper:

Preprint The theory of disappearance and appearance

I mean this two sections:

4. ALIKE ACTION PRINCIPLE

5. DERIVE PATH INTEGRAL FORMULATION

With best regards.

It is not so hard, start from normalized amplitudes, take the square modulus and that should do it. There is an arbitrary phase factor which may multiply an amplitude; special attention to it should be taken in cases of interference, diffraction or cases like Bohm_Aharonov effect, all cases where amplitudes should previously be summed, and phase differences should be important.

Lovely Mazen,

I will try to read your work. I am very agglomerated, but I will try. You see, a complex number can be represented as a vector of two real numbers, one positive (the modulus) and the other one being the phase. The trouble is that the phase may be negative. This is why we cannot map the algebra of complex numbers on the calculus with positive numbers.

Anyway, I will try to read your material.

With kind regards

My dear Daniel,

I am busy, I won't have time for links. Let's be practical. Tell me how do you represent the imaginary number i, and the real number -1, in positive numbers. Please represent them within the frame of the same algebra, not in two different algebras, s.t. I could operate with them both. (In QM the complex numbers belong to a single algebra.)

Juan,

The issue is how to map complex numbers algebra onto positive numbers? Look what you answer,

"There is an arbitrary phase factor which may multiply an amplitude . . . "

You map the complex number onto positive numbers and then multiply the result by a phase factor? And what you do if the phase factor should be exp(-iπ) = -1 ? How do you map -1 on positive numbers?

And then you say "special attention to it should be taken in cases of interference, diffraction or cases like Bohm_Aharonov effect, all cases where amplitudes should previously be summed, and phase differences should be important."

What are these negotiations? How do you map the "special attention" on positive numbers? And it's the same "special attention for all the cases, or each case with its particular "special attention"? Please note that the complex calculus obeys one single algebra, and needs no "special attention".

With kindest regards ;-D

@ Roman,

I am not competent in matters of non-equilibrium states. I would just want to tell you a thought: I am not sure that the evolution of every quantum system is according to the Schrodinger equation. Neither am I sure that every quantum system evolves unitary. When saying this I don't refer to the famous "collapse" of the wave-function, but to evolution before being disturbed by a classical apparatus. As an example, the decay of an excited atom, I don't know whether is a unitary transformation.

Well, maybe what I said here has nothing to do with the non-equilibrium states of which you speak. In that case I appologize for the digression.

Best regards

There is nothing wrong with my statement, since you mark yourself that the -1 is a special case of phase factor,

when you take the complex number and multiply it by its complex conjugate this phase factor goes away, as will a -1

also. And the result is real.

Are you just trying to vex me?

The result of the calculation is a real number, and if the amplitude is normalized the result is a proper probability. Yes the phase differences are important in interference, again you know this. You also know that in this case amplitudes are summed first, before finding intensity or probability.

In general mass is conserved by proof using Schrodinger.Maybe we should know exactly what is meant by equilibrium.

You cannot bi uniquely map, but you obviously can map. and this guage freedom is very well known and discussed galore.

Dear Sofia D. Wechsler

How Nature works, is a question of interpretation. In Caticha's entropic dynamics, all we have is a random process influenced by the entropy of the rest of the world and guided by energy conservation.

And it is not "our classical tendency" but the simple fact that what QM predicts via the Born rule and what we can measure are probabilities.

Ilya

Well, you seem to be ceding to the more reasonable statistical viewpoint.(or I did not understand sooner) Personally I dont see that big a difference statistically speeking if the entities are a big mixup of lines (ddB), or particles with uncertain location(normal QM). In any case both make use of the same Schrodinger equation. There were some efforts to overrule specific kinds of classical behaviour , but Bell has some success showing such supposed proofs have loopholes.

I cannot see how a real wave would collapse, so far.

regards, juan

About ceding: I support, first of all, realist causal interpretations. Because of Bell's theorem they need a preferred frame. So, even if I prefer Caticha's entropic dynamics, which is a stochastic one, attacks against dBB are also, implicitly, attacks against Caticha's interpretation too, so I care about them. Note also that all the realist interpretations prefer configuration over momentum, they all use the continuity equation which is one part of the Schroedinger equation as a base for having a continuous trajectory in configuration space. QM in the minimal interpretation does not have such a trajectory.

This trajectory allows to use the same way as dBB to define, out of the wave function for the whole thing (quantum system + measurement device) and the trajectory of the measurement device (which we can see) an effective wave function of the quantum system, and to see how it behaves. This effective wave function of the measured system collapses.

Juan,

In QM you always multiply a complex number by its complex conjugate? When you add two complex amplitudes a1 and a2, you first multiply each one by the respective complex conjugate? I don't understand what you are trying to say.

Now, I was a bit joking and if I irritated you please accept my appology. No need to tell you that I have no bad intention. I only tried to say that you propose a complication. In fact, neither the complication is the problem but the lack of isomorphism between a linear algebra under addition and scalar multiplication, and the calculus with positive numbers. See below:

The space of complex numbers satisfy a linear algebra of an abelian group under the addition and scalar multiplication operations. The properties of this group which are relevant for our talk, are the null element, 0, and the so-called "inverse element" under addition, i.e. for each element x in the space of complex numbers there exists an element -x, such that x + (-x) = the null element 0, and -x also belongs to the space. The positive numbers P are not at all a group under addition, because the "inverse element" under addition, -P, does not belong to the positive numbers.

Ilja,

Thanks for explanations on Caticha.

Ahhh! As I told Daniel, in the past it was possible to browse among the pages of a thread. Now, they changed the system, s.t. I can only require one previous post, then another previous post, and so on. And I give up. So, I can't see my previous relevant posts.

I am therefore trying to answer you as I can. Of course, it's not a pleasure to talk under restrictions.

Well, you say "How Nature works, is a question of interpretation" No doubt that in trying to understand the microscopic world, the macroscopic human construction is an impediment. But the Nature is ONE. If there are two different interpretations, it means that we didn't understand enough, and new phenomena are expected to be revealed, that would remove this degeneracy.

Lovely Sofia,

If god willing I think you will love this approach,

I think it is not natural to have a complex number in the main equations of physics, I understand the benefits of using the complex number in physics but only as useful tools, While it is not acceptable to use the complex number as a primary physics concept (as we have in Schrodinger equation, Path integral formulation, Matrix mechanics),

so in my approach, I avoid the using of the complex number as a startup concept of the derivation of main quantum formulations, so I only deduce it as a mathematical consequence, as you can see in my paper.

With best regards

Sofia

No, I said first add up the wave functions or amplitudes and only then multiply the result with its complex conjugate.

That is the way even in introductory QM. Like in optics you add up all the electric fields and only then take the scalar product with itself. (This is in interference, diffraction)

Of course you map into a different type of algebraic structure, a many to one map.

regards, juan

My dear Mazen,

We need to use complex numbers in wave phenomena, because waves have intensity and have PHASE. We represent them as A0exp(iφ) where A0 is the square root of the intensity, or, the maximal amplitude.

Juan,

Do you pay atention to what I say, or you just throw an answer? I thought that you pay some attention to my words. What kind of answer is "Of course you map into a different type of algebraic structure, a many to one map." Do you pay any attention at what is the issue?

What I said that is wrong in Bell's SO-CALLED PROOF against locality, was that he tried to mimic the complex numbers algebra of QM, with the calculus with positive numbers. There is no isomorphism between the two, and THIS IS whatever Bell proved, not non-locality.

Dear Roman,

I am an ignorant in your domain, s.t.my words may be not worth much. But the brain components are macroscopic systems. QM describes microscopic systems. Because of this, I remain "at the door" of what you say, I can't get in.

With kind regards

Dear Sofia D. Wechsler

what Bell has written was (different from your so-called proof) a serious and correct proof, not a "so-called" proof. The assumption have been well-defined and clear, the conclusion follows from the assumptions.

He did not try to mimic anything, quantum theory is not even mentioned in the theorem, neither in the assumptions nor the conclusions. Once QT violates the conclusion of the theorem, one of the assumptions has to be wrong in QT.

It is your decision to think which of the assumptions can be simply rejected. Your choice is either Einstein-locality (locality with a velocity much higher than c would be a possibility too), or realism (EPR criterion of reality) together with causality (Reichenbach's common cause principle). That you have to give up causality too follows from another variant of the proof. Alternatively, if you follow my variant of the theorem ( see Article EPR-Bell realism as a part of logic

My dear Sofia,

Sure we use complex numbers in wave phenomena, but this as usefull tool only, we can use only the real sinus functions, but in quantum formulation it is better to start without using complex number (specially the Phase concept) to know how the probability concept is related to particle via space and time.

With best regards

Mazen

Most of the time you can get away with using only real eigenfunctions. In certain situations, ie. if there is a magnetic field,

you must use complex functions. In any case probability density is what counts physically. An eigenfunction you can multiply by any real or complex constant and it is still an eigenfunction.

Regards, juan

Sofia

You do not necesarily have to multily by anything before you add the amplitudes. The relative phases of different paths is important when you add. Usually in interference the phase difference between two beams comes from a distance effect. The only thing I say is that in the calculation of probability density an arbitrary phase factor makes no difference, it does not matter if an eigenstate is a real or complex function, just before you calculate probability density. So psi or -psi are the same physically.

With respect to your other remarks, I would say that complex numbers are a field, not a group. I still fail to see why you are

repeating these well known algebraic properties. Ok I get it, it is wrong in Bell calculation.

If you have an eigenvector or eigenstate, you can also multiply by some real or complex scalar an it is still an eigenvector or eigenfunction.

For a simple superposition, where all states are orthonormalized, both proceedures are the same.(add probabilities or calculate complex time complex conjugate of the whole)

But perhaps before using it in an addition, you must normalize it first, or see what phase it must carry in relation to other paths. In each physical situation, you must use comon sense, to get the right physical effect.

regards, juan

Sofia, any complex number is representable by real numbers using the matrix

x -y

y x

with real numbers x, y, in liu of x+iy

The algebra is the same in both systems.(numbers or matrices)

Juan,

I again remind you that the problem is that one cannot mimic the complex algebra with the calculus of positive numbers.

So, are both x and -x positive numbers?

Also, I repeat that in Bell's calculus are added positive probabilities P = P1+P2, which is always greater than zero if P1 > 0 and P2 > 0. In QM calculus P = |a1 + a2|2 where a1 and a2 are complex number, and even if |a1| > 0 and |a2| > 0, P may be zero.

Ilja,

"what Bell has written was (different from your so-called proof) a serious and correct proof, not a "so-called" proof. The assumption have been well-defined and clear, the conclusion follows from the assumptions."

I am endlessly happy.

"He did not try to mimic anything, quantum theory is not even mentioned in the theorem, neither in the assumptions nor the conclusions."

Look what he wrote in the article "On the Einstein Podolski Rosen paradox"

"The paradox of Einstein, Podolski and Rosen was advanced as an argument that quantum mechanics could neo be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. In this note the idea wil be formulated mathematically and shown to be incompatible with the statistical predictions of the quantum mechanics."

But if you say that Bell's inequalities were not written for comparing with QM predictions, so it is without doubt - obviously, Bell didn't understand what he wrote.

"That you have to give up causality too follows from another variant of the proof. Alternatively, if you follow my variant of the theorem . . . . "

I speak of Bell's inequalities not of yours. You are unable to understand that my proof is one thing, Hardy's proof is another thing, and in the same way, Bell's proof is something else than your proof.

"Your choice is either Einstein-locality (locality with a velocity much higher than c would be a possibility too) "

You have no right to falsify Einstein's theory, which says that the light velocity cannot be surpassed, with your inventions.

You disply all the time inability to distinguish between works of different people. You have to solve your problem if you want attention to be paid to your posts.

I just dont get you, even though your initial statement is correct.The complex uses two real numbers, the real numbers only one.

Where do you imagine such a contradiction.? Why do you imagine it?

Where do you get such a contradiction from?

I repeat that if a1 and a2 are both orthogonal , the two methods of calculation are the same.

Remember that states can also be complex functions, and not just complex numbers.

A probability may also be zero as well as positive.

Take a1) +a2) and multiply on the left with the bra form

where (a1, a2)=0

Then (a1+a2, a1+a2) = a1* a1+a2*a2 = P1+P2

So if this is the case that Bell is adding orthogonal states, his calculation holds.

If you wanted to trick me at this pont, maybe you would say that a1+a2=0, with both being nonzero.

However this would contradict the fact that a1 and a2 are taken orthogonal, so such assumption is impossible.

Dear Juan,

Yes, but I mean it is not natural to see a complex number in a fundamental equation like Schrodinger equation!

I understand the benefits of using the complex number in physics but only as useful tools, so we can start with a purely physical concept (I mean with a real number only) then deduce the formalism with a complex number as a useful tool only.

with best regards.

My Juan,

You mix everything. What's the matter, what is so complicated in what I said?

I will repeat, but I already said it many times. The issue under discussion (and which in fact has nothing to do with the question here, s.t. I want to cut it short) was that in proving his inequality, Bell used the common calculus of probabilities. This is a problem in his proof.

This usual calculus of probabilities operates with positive numbers. If an event that can be obtained on two ways, the total probability P is calculated from the two contributions i.e. P = P1 + P2.

In the QM formalism, probabilities are not a primary concept. Complex amplitudes are the primary concept. For the above event, one calculates first the total amplitude of probability, a = a1 + a2. Only after that one calculates the probability P = |a|2 = |a1 + a2|2.

Thus, while P1 + P2 is never zero, except if P1=P2 = 0, the value |a1 + a2|2 may be zero if a1 = -a2, despite the fact that both a1 and a2 are nonzero. Probabilities are always positive (or null) they cannot cancell one another, while complex amplitudes can be opposite to one another, in particular a1 can be positive and a2 negative.

What I claimed is that one cannot map ISOMORPHICALLY the calculus of complex numbers onto the calculus with positive numbers because of the above difference in properties.

So, it is not clear what Bell proved: did he prove non-locality, or did he prove that the calculus with positive probabilities is unsuitable for obtaining the QM probabilities?

This is the issue, but I don't want to continue with it, because it is not in line with the question of this thread.

sofia

The two ways of obtaining P are only equivalent if the states are taken orthogonal.

You obtain the result using orthogonality (a1,a2)=0

After that you cannot assume a1 and a2 to be linearly dependent as in a1+a2=0

If you do the whole calculation only using a1+a2=0, you will get 0=0, the cross terms then compensate. Do it!

This is also my last answer, about this topic.

Do not know what Bell was trying to do, but I consider the non isomorfic aspect devoid of any real problem, it proves nothing.

regards, juan

Dear Sofia D. Wechsler

"But if you say that Bell's inequalities were not written for comparing with QM predictions"

If ... I did not say such a thing. I said "quantum theory is not even mentioned in the theorem".

Then, please stop personal insults about what I'm "unable to understand".

"You have no right to falsify Einstein's theory, which says that the light velocity cannot be surpassed, with your inventions."

What's this? Sofia D. Wechsler being now in a position to decide which scientists are allowed to do which things?

Sofia, we cn only guess at what Einstein was thinking when he said Bohm's theory was too cheap. Whatever it was not complex numbers or Bell's inequalities, both of which as discussed above I disagree with some of the comments, but they should be separate discussions. As for Bohm's theory, uncertainty remains there, and probabilities remain because while ψ is deterministic (in any QM theory) you don't know the phase so uncertainty follows. My guess at what he meant as being too cheap is Bohm's quantum potential. The problem with the expression is there is no way to determine a value for it because the same variable occurs in the denominator and the numerator, and hence it has the same value when multiplied by any integer. I suspect that Einstein thought that just adding an indeterminate term to solve the problem was cheap. But I do not know.

Thanks @ Ian

Indeed the issue with complex numbers should be a separate thread.

About the quantum potential I don't understand why you say that it is undeterminate. But I also incline to think that the quantum potential may not have been an appealing idea to Einstein.

With kind regards

The reason I say it is indeterminate is the quantum potential reduces to [h^2 ∇^2A]/[2mA]. A is in both the numerator and the denominator.