QED Lagrangian lead to self-interaction?

[akhmeteli]

Well, here's the thing. If you decide to take unitary evolution seriously and reject the projection postulate, then you are no longer talking about standard QM. Then you are talking about a truly unitary QM theory such as deBB or MWI.

There are three statements which are mutually inconsistent:

A. The wavefunction of a system is complete, i.e. the wavefunction specifies (directly or indirectly) all of the physical properties of a system.

B. The wavefunction always evolves in accord with a linear dynamical equation (e.q. Schroedinger equation).

C. Measurements of e.g. the spin fo an electron always (or at least usually) have determinate outcomes.

If you take A + B, then that is incompatible with C. If you take B + C, then that is incompatible with A. If you take A + C, then that is incompatible with B. The resolution to A + B, is something like MWI. The resolution to B + C is linear hidden variables. The resolution to A + C is a stochastic or nonlinear QM theory like GRW stochastic collapse or nonlinear hidden variable theories.

Hope this helps.

I won't discuss A and B now, as it may take a long time, but I certainly don't need C, as it seems to contradict the Poincare recurrence theorem.

 

[Maaneli]

I certainly don't need C, as it seems to contradict the Poincare recurrence theorem.

Poincare's recurrence theorem doesn't really apply to QM measurements in this way.

What is your response to my previous post about VBI in SFED?

 

[akhmeteli]

Poincare's recurrence theorem doesn't really apply to QM measurements in this way./QUOTE]
I think the quantum recurrence theorem (Phys. Rev. V.107 #2, pp.337-338, 1957), maybe somewhat modified, does apply to QM measurements. Why not? No measurement is ever final, if you accept the theorem.

What is your response to my previous post about VBI in SFED?

I do remember that post, but I do need time to sort it out.

 

[Maaneli]

Poincare's recurrence theorem doesn't really apply to QM measurements in this way./QUOTE]
I think the quantum recurrence theorem (Phys. Rev. V.107 #2, pp.337-338, 1957), maybe somewhat modified, does apply to QM measurements. Why not? No measurement is ever final, if you accept the theorem.


I do remember that post, but I do need time to sort it out.

<< I think the quantum recurrence theorem (Phys. Rev. V.107 #2, pp.337-338, 1957), maybe somewhat modified, does apply to QM measurements. Why not? No measurement is ever final, if you accept the theorem. >>

Perhaps the recurrence theorem is applicable in some way - but I don't think it is relevant to the treatment of measurement processes in short time intervals.

 

[akhmeteli]

<< I think the quantum recurrence theorem (Phys. Rev. V.107 #2, pp.337-338, 1957), maybe somewhat modified, does apply to QM measurements. Why not? No measurement is ever final, if you accept the theorem. >>

Perhaps the recurrence theorem is applicable in some way - but I don't think it is relevant to the treatment of measurement processes in short time intervals.

In this case it does not matter whether it is relevant to the treatment of measurement processes in short time intervals. The way you formulated C, it does seem relevant (no time scales defined there). And it is certainly relevant to the projection postulate (actually, it seems incompatible with the latter). I also tend to believe that it is relevant to the Bell theorem.

 

[Maaneli]

In this case it does not matter whether it is relevant to the treatment of measurement processes in short time intervals. The way you formulated C, it does seem relevant (no time scales defined there). And it is certainly relevant to the projection postulate (actually, it seems incompatible with the latter). I also tend to believe that it is relevant to the Bell theorem.

Why does it seem so relevant to Bell's theorem or incompatible with the projection postulate? The recurrence theorem would just say something to the effect that given a long enough time, a quantum system will eventually return back to its original coherent state. That's just because a system can in principle still be put back into an approximately coherent state by human experimenters or by some complex and improbable series of natural events in the world. But it would take an extremely long time for this to happen. I don't see how this is very relevant to Bell's theorem (unless you want to take seriously something like the common past hypothesis), or how it is incompatible with the projection postulate, anymore than it is incompatible with Boltzmann's typicality argument in statistical mechanics (which it isn't).

 

[akhmeteli]

Why does it seem so relevant to Bell's theorem or incompatible with the projection postulate? The recurrence theorem would just say something to the effect that given a long enough time, a quantum system will eventually return back to its original coherent state. That's just because a system can in principle still be put back into an approximately coherent state by human experimenters or by some complex and improbable series of natural events in the world.

No, if the system has discrete energy eigenvalues, it will happen in the absence of any external interference, and with certainty. Actually, external interference can prevent recurrence.

But it would take an extremely long time for this to happen. I don't see how this is very relevant to Bell's theorem (unless you want to take seriously something like the common past hypothesis), or how it is incompatible with the projection postulate, anymore than it is incompatible with Boltzmann's typicality argument in statistical mechanics (which it isn't).

It is incompatible with the projection postulate because a particle, strictly speaking, does not stay in the eigenstate after measurement (so the projection postulate may be a good approximation or a bad approximation, but it is just an approximation). It is relevant to the Bell's theorem because the projection postulate is an essential assumption of the theorem.

 

[Maaneli]

No, if the system has discrete energy eigenvalues, it will happen in the absence of any external interference, and with certainty.

What will happen with certainty in the absence of any external interference?

It is incompatible with the projection postulate because a particle, strictly speaking, does not stay in the eigenstate after measurement (so the projection postulate may be a good approximation or a bad approximation, but it is just an approximation).

Well it depends. Perhaps you could infer that in the standard QM, it is only an approximation; but certainly not so in GRW theories.

By the way, this is why I think it is not very productive to stick with thinking about the projection postulate in standard QM, since it is already obvious that it cannot be a fundamental description of measurement processes.

 

[akhmeteli]

What will happen with certainty in the absence of any external interference?

The state vector will get arbitrarily close to the initial one (Phys. Rev. V.107 #2, pp.337-338, 1957).

Well it depends. Perhaps you could infer that in the standard QM, it is only an approximation; but certainly not so in GRW theories.

I was discussing the standard QM.

By the way, this is why I think it is not very productive to stick with thinking about the projection postulate in standard QM, since it is already obvious that it cannot be a fundamental description of measurement processes.

If we reject the projection postulate (PP), we can reject the Bell theorem, as it cannot be proven without PP.

 

[Maaneli]

The state vector will get arbitrarily close to the initial one (Phys. Rev. V.107 #2, pp.337-338, 1957).

I'll have a look but I'm skeptical of how relevant it is.

If we reject the projection postulate (PP), we can reject the Bell theorem, as it cannot be proven without PP.

No you can't just reject Bell's theorem by rejecting the projection postulate - the derivation of the Bell inequality has nothing to do with the projection postulate, and the Bell theorem (which says if an LCHV model is compatible with QM, then here is an inequality that QM correlations must satisfy) does not specifically require the projection postulate, just a means by which the QM formalism can make empirical predictions. That is why deBB theory and stochastic mechanics, which have no need for a projection postulate in their measurement theories, can also violate the Bell inequality and thus be applied to Bell's theorem. If you reject the projection postulate, you have to replace it with some other mechanism for your quantum theory to make empirical predictions (some measurement theory in other words). Otherwise, your theory can't make any predictions and is therefore totally useless.

 

[akhmeteli]

No you can't just reject Bell's theorem by rejecting the projection postulate - the derivation of the Bell inequality has nothing to do with the projection postulate,

I agree, the derivation of the Bell inequality does not require PP.

and the Bell theorem (which says if an LCHV model is compatible with QM, then here is an inequality that QM correlations must satisfy)

The Bell theorem also states that the inequality can be violated in QM, and you need PP to prove that. Maybe you use some form of the Bell theorem that does not include this statement, but that does not really matter. I think you appreciate that if this statement is wrong, the Bell inequality, although correct, loses most of its thrust, as genuine VBI will never be demonstrated (assuming QM is correct).

does not specifically require the projection postulate, just a means by which the QM formalism can make empirical predictions. That is why deBB theory and stochastic mechanics, which have no need for a projection postulate in their measurement theories, can also violate the Bell inequality and thus be applied to Bell's theorem. If you reject the projection postulate, you have to replace it with some other mechanism for your quantum theory to make empirical predictions (some measurement theory in other words). Otherwise, your theory can't make any predictions and is therefore totally useless.

I could agree that we may need some other mechanism for quantum theory to make empirical predictions. But how can we be sure that no matter how we choose that mechanism, we'll have VBI in QM? I tend to believe that PP is the real source of nonlocality, as it states that immediately after we measure a spin projection of one particle of a singlet, the system will be in an eigenstate of that spin projection. That means that the spin projection of the second particle immediately becomes definite (assuming angular momentum conservation), no matter how far the second particle is.

 

[Maaneli]

The Bell theorem also states that the inequality can be violated in QM, and you need PP to prove that. I think you appreciate that if this statement is wrong, the Bell inequality, although correct, loses most of its thrust, as genuine VBI will never be demonstrated (assuming QM is correct).

I think you're still missing the point. Bell's theorem doesn't say that the inequality can be violated in QM. Again, Bell's theorem is just the statement that if QM can be embedded into a locally causal theory, it must satisfy a certain inequality derived from that locally causal theory. QM (and this applies not just to standard QM) just violates that inequality.

I could agree that we may need some other mechanism for quantum theory to make empirical predictions. But how can we be sure that no matter how we choose that mechanism, we'll have VBI in QM?

If that other mechanism for QM measurements still involves wavefunctions (with their nonlocal configuration spaces) and Bell's causality assumption (like deBB or GRW theory), then it will always violate Bell inequalities. If you give up causality, or modify Kolmogorov's probability axioms, you can keep locality and still violate the Bell inequality. All this has been exhaustively demonstrated.

I tend to believe that PP is the real source of nonlocality, as it states that immediately after we measure a spin projection of one particle of a singlet, the system will be in an eigenstate of that spin projection.

Sorry but there are counterexamples to your belief. deBB theory and stochastic mechanics, neither of which require the PP, make explicit the fact that the entanglement of wavefunctions in configuration space is the real source of nonlocality. The PP is only an approximation. And don't forget that those alternative formulations of QM are empirically equivalent to standard QM and QED. So you have no reason to regard standard QM as primary over those alternative formulations.

 

[akhmeteli]

I think you're still missing the point. Bell's theorem doesn't say that the inequality can be violated in QM.

Some forms of BT do include that, at least implicitely. For example, in http://plato.stanford.edu/entries/bell-theorem/#2 :

"In the present section the pattern of Bell's 1964 paper will be followed: formulation of a framework, derivation of an Inequality, demonstration of a discrepancy between certain quantum mechanical expectation values and this Inequality." A discrepancy arises when VBI occur in QM.

Or in http://en.wikipedia.org/wiki/Bell's_theorem :
"No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics." Again, to prove it, you have to prove that there are VBI in QM.

Again, Bell's theorem is just the statement that if QM can be embedded into a locally causal theory, it must satisfy a certain inequality derived from that locally causal theory.

Again, it does not matter whether you formally include the statement that there are VBI in QM in the Bell theorem. If this statement is wrong, the Bell Theorem (BT) loses most of its importance.

QM (and this applies not just to standard QM) just violates that inequality.

Again, you need PP or something similar to prove this "just violates" for the standard QM. Unitary evolution is not enough.

If that other mechanism for QM measurements still involves wavefunctions (with their nonlocal configuration spaces) and Bell's causality assumption (like deBB or GRW theory), then it will always violate Bell inequalities.

I don't know that other mechanism, so I don't know if these "ifs" are true for it. All I know is the current mechanism contradicts unitary evolution.

Sorry but there are counterexamples to your belief. deBB theory and stochastic mechanics, neither of which require the PP, make explicit the fact that the entanglement of wavefunctions in configuration space is the real source of nonlocality. The PP is only an approximation. That don't forget that those alternative formulations of QM are empirically equivalent to standard QM and QED.

If entanglement of wavefunctions in configuration space is the real source of nonlocality in deBB theory and stochastic mechanics (I don't know it for sure and unfortunately don't have time and motivation to check), it does not mean this is also true for QM. These theories may be empirically equivalent to QM only after you include PP in the latter. I don't know how one can get VBI in standard QM without PP or something similar.

 

[Maaneli]

Some forms of BT do include that, at least implicitely. For example, in http://plato.stanford.edu/entries/bell-theorem/#2 :

Fair enough. Maybe I was parsing a little too much.

Again, it does not matter whether you formally include the statement that there are VBI in QM in the Bell theorem. If this statement is wrong, the Bell Theorem (BT) loses most of its importance.

If Bell's theorem is wrong, that's just as important as if it's right. That means QM can be embedded into a locally causal theory.

I don't know that other mechanism, so I don't know if these "ifs" are true for it. All I know is the current mechanism contradicts unitary evolution.

As I said, those scenarios have been exhaustively considered. They are true, as you would find out if you make the effort to study them for yourself. Of course, there are phenomenological models like stochastic optics that prevent a VBI from happening in the first place (for example with light). You are welcome to try and formulate a locally causal theory of the electron or other massive particles for which VBI appears to occur only when the locality condition has not been met, which is the current status of such experiments with massive particles. Oh and you should also try to reproduce all the other QM predictions with that locally causal theory of the electron. But ultimately, you'll have to deal with the time when a Bell or GHZ test is done with massive particles and which also implements the locality condition. If that turns out to produce VBI (which is likely), then I really see no more wiggle room. Anyway, hopefully you can see how serious of an uphill battle it is, and all the odds against it.

If entanglement of wavefunctions in configuration space is the real source of nonlocality in deBB theory and stochastic mechanics (I don't know it for sure and unfortunately don't have time and motivation to check), it does not mean this is also true for QM. These theories may be empirically equivalent to QM only after you include PP in the latter.

Actually it is. It is not the projection postulate per se that produces the correlations that violate the VBI, as indicated by the fact that the projection postulate applies even to separable wavefunctions in standard QM. If two wavefunctions were not entangled in configuration space, then the correlations between two "particles" would not VBI, even with PP, since they would be separable quantum systems.

I don't know how one can get VBI in standard QM without PP or something similar

Well you're conflating standard QM with another theory that doesn't have PP. As I said before, if you remove PP from textbook QM, then you have to replace it with a measurement theory to make predictions. The moment you do that, you have a different formulation of QM.

I see no point in dwelling on the PP from standard QM without considering the alternative formulations. It really isn't getting us anywhere. And besides, you already agreed that the PP is only an approximation, and that there must be some more fundamental measurement. So let's talk about those superior quantum measurement theories. Or at least can we get back to the argument in my previous post about VBI in SFED?

 

[Maaneli]

And besides, you already agreed that the PP is only an approximation, and that there must be some more fundamental measurement.

Some more fundamental measurement theories*.

 

[akhmeteli]

As I said, those scenarios have been exhaustively considered. They are true, as you would find out if you make the effort to study them for yourself.

Maybe you misunderstood me or I was not clear enough. I meant that this hypothetical mechanism does not have to "involve wavefunctions (with their nonlocal configuration spaces)".

Anyway, hopefully you can see how serious of an uphill battle it is, and all the odds against it.

I am not making any commitments to battle anything. All I'm trying to say can be formulated as follows. 1) The theoretical proof of VBI in the standard QM is based on assumptions that contradict each other. 2) Genuine VBI have not been demostrated experimentally.

Actually it is. It is not the projection postulate per se that produces the correlations that violate the VBI, as indicated by the fact that the projection postulate applies even to separable wavefunctions in standard QM. If two wavefunctions were not entangled in configuration space, then the correlations between two "particles" would not VBI, even with PP, since they would be separable quantum systems.

This argument does not seem conclusive. If PP does not always result in VBI, it does not mean PP is not the culprit.

Well you're conflating standard QM with another theory that doesn't have PP. As I said before, if you remove PP from textbook QM, then you have to replace it with a measurement theory to make predictions. The moment you do that, you have a different formulation of QM.

I see no point in dwelling on the PP from standard QM without considering the alternative formulations. It really isn't getting us anywhere. And besides, you already agreed that the PP is only an approximation, and that there must be some more fundamental measurement. So let's talk about those superior quantum measurement theories. Or at least can we get back to the argument in my previous post about VBI in SFED?

I have no alternative mechanism to offer. Again, what I'm trying to tell is limited to the two points above. As for VBI in SFED, I am not sure I'll be able to offer a final conclusion within a reasonable time frame, so I'll just try to share a preliminary opinion today or tomorrow.

 

[Maaneli]

Andy,

Maybe you misunderstood me or I was not clear enough. I meant that this hypothetical mechanism does not have to "involve wavefunctions (with their nonlocal configuration spaces)".

I didn't misunderstand you. I said before that IF your hypothetical alternative measurement mechanism involves wavefunctions plus the causality assumption, then it will necessarily VBI. If your alternative mechanism involves the causality assumption but does not involves wavefunctions, then it will not VBI.


I am not making any commitments to battle anything. All I'm trying to say can be formulated as follows. 1) The theoretical proof of VBI in the standard QM is based on assumptions that contradict each other. 2) Genuine VBI have not been demostrated experimentally.

But as I pointed out, 1) is trivial because a) the PP is not the direct cause of VBI, and b) the PP is not a uniquely valid measurement theory (in fact it is not even a measurement theory), as there are other superior one's that do not require PP and still VBI. So I don't think 1) has any substance with all do respect. And of course I agree with 2), but the situation is still in favor of nonlocal or causally symmetric formulations of QM, as I explained before.


This argument does not seem conclusive. If PP does not always result in VBI, it does not mean PP is not the culprit.


I disagree. Think about the example I provided earlier, if a wavefunction psi(x1, x2, t) is not factorizable (it is entanglement in configuration space), then add in the PP and you get VBI. However, if a wavefunction psi(x1, x2, t) is factorizable (there is no entanglement in configuration space), then add in the PP (because you still have reduction of the state vector) and you do not get VBI. So it is obvious that in the first case, entanglement of wavefunctions plus PP necessarily implies VBI, and in the second case, there are factorizable wavefunctions plus PP, and no VBI is possible. The same is also true of GRW collapse QM. So which do you think is more directly relevant to the cause of VBI in standard QM? Entangled wavefunctions in configuration space or the PP? Also throw in the fact that you can eliminate PP with deBB, keep unitary evolution, and still get VBI. There is no doubt that PP is not the culprit of VBI. The PP is actually a deceptive, red herring.


I have no alternative mechanism to offer. Again, what I'm trying to tell is limited to the two points above. As for VBI in SFED, I am not sure I'll be able to offer a final conclusion within a reasonable time frame, so I'll just try to share a preliminary opinion today or tomorrow.

OK.

 

[akhmeteli]

Andy,

I just uncovered that the configuration space formalism is indeed used in the Barut SFED for two entangled electrons. And there is no contradiction between this and the use of the nonlinear Hartree-Fock equations. So I am now quite certain there are indeed nonlocal entanglement correlations in SFED, and therefore VBI. I also read nightlight's posts on this subject, and I'm not sure he properly understood the Barut ansatz or the variational principle argument Barut used, in relation to the issue of entanglement nonlocality (he also seemed to improperly mix-up SED and SO with SFED). For a clear explication of this, see Barut's section 3.5 "QED of the relativistic two-body system" in The Electron:

The Electron: New Theory and Experiment
http://books.google.com/books?id=7w...&oi=book_result&resnum=10&ct=result#PPT143,M1

Also see "Relativistic two body QED" in section 4 of Barut's paper "QED based on self energy":

http://streaming.ictp.trieste.it/preprints/P/87/248.pdf

On page 10 of that section, you will see that Barut says,

<< We must now specify a variational principle. We could vary the action W with
respect to individual fields psi_1 and psi_2 separately. This results in non-linear
coupled Hartree-type equations for thse fields. Instead, we propose a
relativistic configuration space formalism to take into account the long-range
quantum correlations. >>

He then goes on to discuss how one can equivalently rewrite the SFED Lagrangian in terms of a 16-component composite wavefunction field.
~Maaneli

Sorry for the late reply. As I said, the situation does not seem clear to me. I don't know what was Barut's take on his own ansatz, whether this is just a computational trick or a new theory. In fact, what he does is he "underoptimizes" action and radically expands the set of functions under consideration. It is not clear at all whether this is necessary. nightlight's take is that this is just an artifact of linearization, and I tend to agree. If you believe that this is part and parcel of the Barut's theory, then it's another place where he "smuggles in" 2nd quantization, at least as far as contents, rather than form, is concerned.

Actually, my problem is I don't know why we need entangled wavefunctions in the Barut's theory, in the first place.

 

[Maaneli]

Sorry for the late reply. As I said, the situation does not seem clear to me. I don't know what was Barut's take on his own ansatz, whether this is just a computational trick or a new theory. In fact, what he does is he "underoptimizes" action and radically expands the set of functions under consideration. It is not clear at all whether this is necessary. nightlight's take is that this is just an artifact of linearization, and I tend to agree. If you believe that this is part and parcel of the Barut's theory, then it's another place where he "smuggles in" 2nd quantization, at least as far as contents, rather than form, is concerned.

Actually, my problem is I don't know why we need entangled wavefunctions in the Barut's theory, in the first place.

Andy,

Did you read any of those links? Barut does say what he thinks about the configuration space formulation. That's why I posted them for you. He also says it is necessary for accounting for nonlocal correlations. Mathematically, this is no different from what is done in standard QM for entangled wavefunctions. So nothing really changes there except now you have the self-field interaction which makes the wave equation nonlinear. In fact, he is not at all linearizing the Dirac equation by constructing that 16-component nonlocal wavefunction. The Dirac equation is still a nonlinear integro-differential equation, but now with a nonlocal self-field. In fact, what Barut does here is exactly what I proposed one would do in constructing the SFED version of the singlet state in an earlier post. Nightlight's comments seem misguided to me. He was incorrectly mixing up the Hartree-Fock approximation for separable wavefunctions, with nonlocality in Barut's SFED.


<< my problem is I don't know why we need entangled wavefunctions in the Barut's theory, in the first place. >>

How can you still not see it? We've just been over the fact that entangled wavefunctions are not only possible in SFED, but that it would be necessary to consistently predict the current quantum entanglement correlations for electrons, even though none of them VBI quite yet (because the seperability condition hasn't been implemented yet). Otherwise, the theory wouldn't even be able to match these nonideal correlations. You would have to construct some additional ad-hoc mechanism to do this. And you couldn't use stochastic optics to do it, since there are no ZPF fields in SFED. So I don't know what else would be availabe. And, honestly, what seems more natural, using the entanglement of the wavefunctions in configuration space which the theory permits or something entirely new and ad-hoc?

 

[akhmeteli]

Andy,

Did you read any of those links? Barut does say what he thinks about the configuration space formulation. That's why I posted them for you.

I did read the links, and what Barut says, does not make clear what he thinks, sorry.

He also says it is necessary for accounting for nonlocal correlations.

I have not found the word "nonlocal". He does mention "long-range quantum correlations". I'm not sure this is the same thing.

Mathematically, this is no different from what is done in standard QM for entangled wavefunctions. So nothing really changes there except now you have the self-field interaction which makes the wave equation nonlinear. In fact, he is not at all linearizing the Dirac equation by constructing that 16-component nonlocal wavefunction. The Dirac equation is still a nonlinear integro-differential equation, but now with a nonlocal self-field. In fact, what Barut does here is exactly what I proposed one would do in constructing the SFED version of the singlet state in an earlier post. Nightlight's comments seem misguided to me. He was incorrectly mixing up the Hartree-Fock approximation for separable wavefunctions, with nonlocality in Barut's SFED.

When I was talking about linearization, I did not mean that specific equation. I meant that "underoptimization" and the configuration space in general may appear as a result of linearization. nightlight cites the results from K. Kowalski and W.-H. Steeb, Nonlinear Dynamical Systems and Carleman Linearization (World Scientific, Singapore, 1991) (brief outline in http://arxiv.org/abs/hep-th/9212031 ): if we have a nonlinear differential equation in an (s+1)-dimensional space-time $${\partial_t}u(x,t) = F(u,{D^\alpha}u)$$ , where $${D^\beta}={\partial^{|\beta|}}/\partial x_1^{\beta_1}\ldots \partial x_s^{\beta_s}$$, $$|\beta|=\sum\limits_{i=1}^{s}\beta_i$$ , we can introduce a normalized functional
coherent state $$|u\rangle =\exp\left(-\frac{1}{2}\int d^sx|u|^2\right)\exp\left(\int d^sxu(x)a^\dagger(x)\right)|0\rangle$$ (so $$a(x)|u\rangle =u(x)|u\rangle$$, where $${a^\dagger}(x)$$ and $$a(x)$$ are the standard Bose operators of creation and annihilation) and a boson operator $$M = \int {d^s}x{a^\dagger}(x)F(a(x),{D^\alpha}a(x))$$ , and then we have a linear evolution equation in Hilbert
space $$\frac{d}{dt}|u,t\rangle = M|u,t\rangle$$, where $$|u,t\rangle = \exp\left[\frac{1}{2}\left(\int {d^s}xu^2 -\int {d^s}xu_0^2\right)\right]|u\rangle$$ and $$\qquad |u,0\rangle=|u_0\rangle$$ (I did cut some corners; you can find the details in the Kowalski's work). Thus, a less than exciting nonlinear differential equation can be linearized using 2nd quantization. If we did not know the equation for what it was, we could start talking about the configuration space, entangled wavefunctions, and so on. Are you sure we should do just that? I am not. I am not trying to convince you that entangled wavefunctions are just an artifact of linearization, but I believe this is a possibility.

How can you still not see it? We've just been over the fact that entangled wavefunctions are not only possible in SFED, but that it would be necessary to consistently predict the current quantum entanglement correlations for electrons, even though none of them VBI quite yet (because the seperability condition hasn't been implemented yet). Otherwise, the theory wouldn't even be able to match these nonideal correlations. You would have to construct some additional ad-hoc mechanism to do this. And you couldn't use stochastic optics to do it, since there are no ZPF fields in SFED. So I don't know what else would be availabe. And, honestly, what seems more natural, using the entanglement of the wavefunctions in configuration space which the theory permits or something entirely new and ad-hoc?

Frankly, a straight-forward nonlinear differential equation seems much more natural. Whether such a minimalist theory is possible, I don't know, but so far I don't see any fundamental difficulties, theoretical or experimental.

Just a few more words. I can live without locality. I can live without reality. I can live without causality. But not until I am absolutely sure I do have to go to such extremes.

 

[Maaneli]

I did read the links, and what Barut says, does not make clear what he thinks, sorry.



I have not found the word "nonlocal". He does mention "long-range quantum correlations". I'm not sure this is the same thing.



When I was talking about linearization, I did not mean that specific equation. I meant that "underoptimization" and the configuration space in general may appear as a result of linearization. nightlight cites the results from K. Kowalski and W.-H. Steeb, Nonlinear Dynamical Systems and Carleman Linearization (World Scientific, Singapore, 1991) (brief outline in http://arxiv.org/abs/hep-th/9212031 ): if we have a nonlinear differential equation in an (s+1)-dimensional space-time $${\partial_t}u(x,t) = F(u,{D^\alpha}u)$$ , where $${D^\beta}={\partial^{|\beta|}}/\partial x_1^{\beta_1}\ldots \partial x_s^{\beta_s}$$, $$|\beta|=\sum\limits_{i=1}^{s}\beta_i$$ , we can introduce a normalized functional
coherent state $$|u\rangle =\exp\left(-\frac{1}{2}\int d^sx|u|^2\right)\exp\left(\int d^sxu(x)a^\dagger(x)\right)|0\rangle$$ (so $$a(x)|u\rangle =u(x)|u\rangle$$, where $${a^\dagger}(x)$$ and $$a(x)$$ are the standard Bose operators of creation and annihilation) and a boson operator $$M = \int {d^s}x{a^\dagger}(x)F(a(x),{D^\alpha}a(x))$$ , and then we have a linear evolution equation in Hilbert
space $$\frac{d}{dt}|u,t\rangle = M|u,t\rangle$$, where $$|u,t\rangle = \exp\left[\frac{1}{2}\left(\int {d^s}xu^2 -\int {d^s}xu_0^2\right)\right]|u\rangle$$ and $$\qquad |u,0\rangle=|u_0\rangle$$ (I did cut some corners; you can find the details in the Kowalski's work). Thus, a less than exciting nonlinear differential equation can be linearized using 2nd quantization. If we did not know the equation for what it was, we could start talking about the configuration space, entangled wavefunctions, and so on. Are you sure we should do just that? I am not. I am not trying to convince you that entangled wavefunctions are just an artifact of linearization, but I believe this is a possibility.



Frankly, a straight-forward nonlinear differential equation seems much more natural. Whether such a minimalist theory is possible, I don't know, but so far I don't see any fundamental difficulties, theoretical or experimental.

Just a few more words. I can live without locality. I can live without reality. I can live without causality. But not until I am absolutely sure I do have to go to such extremes.

Andy,


I have not found the word "nonlocal". He does mention "long-range quantum correlations". I'm not sure this is the same thing.

Then you just did not read closely enough, because he does in fact use the word "nonlocal" in both references I gave you. In the paper "QED based on self-energy", he says on page 9:

The total Hamiltonian also contains the self-energy terms which we have written separately. These latters involve however non-local and non-linear potentials given in [equation] (13).

And have a look at the nonlocal self-fields in equation (13). He also says on page 7:

Whence the action (7) can be written in an "action-at-a-distance" form

then look at equation (10). The term "action-at-a-distance" literally means nonlocal. So he's writing the action (10) in a nonlocal form.

Also, in that google books link to Barut's section, he says on page 128, after showing that the linear configuration space Schroedinger equation for N particles can be rewritten as N-coupled nonlinear Hartree-Fock equations,

These two formulations are closely related but not identical. We shall see that they correspond to two different types of variational principles and actually describe two different types of physical situations. Quantum theory has a separate new postulate for two or more particles, namely that the state space is the tensor product of one particle state spaces. This leads immediately to the first formulation in configuration space. Such combined systems are called in the axiomatic of quantum theory "nonseparated" systems with all the nonlocal properties of quantum theory . But this postulate does not apply universally. There are other systems, namely the "separated" systems which are described by the second type of equations, which are described by the second type of equations...We shall now see how all this comes about from two different basic variational principles in the relativistic case (the nonrelativistic case is similar).

He then goes on to construct the 16-component configuration space wavefunction for two nonseparable particles, after showing the nonlinear Hartree-Fock formalism for two separable particles. Notice also that the Dirac equation that he constructs for this nonlocal two-particle case (equation 39) is still a nonlinear integro-differential equation, given that it still is a function of these nonlinear, nonlocal self-fields.

I meant that "underoptimization" and the configuration space in general may appear as a result of linearization. nightlight cites the results from K. Kowalski and W.-H. Steeb, Nonlinear Dynamical Systems and Carleman Linearization

I have seen nightlight do the Carleman linearization before.

Thus, a less than exciting nonlinear differential equation can be linearized using 2nd quantization. If we did not know the equation for what it was, we could start talking about the configuration space, entangled wavefunctions, and so on. Are you sure we should do just that? I am not.

There are many obvious problems with the connection you're trying to make here. First, you're saying that a nonlinear differential equation can be linearized by using 2nd quantization; problem is, the linear Schroedinger equation is a 1st quantized equation and therefore has nothing to do with any second quantization procedure. So, what you and nightlight showed is a neat mathematical procedure, but ultimately not applicable to QM in the same way. Also, as Barut has already shown, what is applicable to QM is the two different types of variational principles (one with respect to psi(x) and the other with respect to phi(x1, x2)), and they apply to different physical situations. This is the crucial point that nightlight either overlooked or did not understand. Moreover, the equations of motion are both still nonlinear. By the way, there have been attempts to describe nonseparable systems in terms of the nonlinear Hartree-Fock equations - but they don't get rid of the nonlocality in any way that you seem to hope to. Also, you will see from those references to Hartree-Fock equations, that they only constitute an approximation and have specific limitations.

Frankly, a straight-forward nonlinear differential equation seems much more natural. Whether such a minimalist theory is possible, I don't know, but so far I don't see any fundamental difficulties, theoretical or experimental.

So based on all the above, I think these sentiments are totally misguided. Sorry.


Just a few more words. I can live without locality. I can live without reality. I can live without causality. But not until I am absolutely sure I do have to go to such extremes.

I don't understand anything you said here. In particular, I don't know what you think locality, reality, and causality are, and what you mean by the term "live with". In any case, these are non sequiturs in relation to the physics definitions of locality, reality, and causality. Our discussion is about SFED and locality, and has nothing to do with reality or causality. Indeed, confirmation that SFED is nonlocal does not have any bearing on the validity of realism or causality.

 

[akhmeteli]

Then you just did not read closely enough, because he does in fact use the word "nonlocal" in both references I gave you. In the paper "QED based on self-energy", he says on page 9:


These two formulations are closely related but not identical. We shall see that they correspond to two different types of variational principles and actually describe two different types of physical situations. Quantum theory has a separate new postulate for two or more particles, namely that the state space is the tensor product of one particle state spaces. This leads immediately to the first formulation in configuration space. Such combined systems are called in the axiomatic of quantum theory "nonseparated" systems with all the nonlocal properties of quantum theory . But this postulate does not apply universally. There are other systems, namely the "separated" systems which are described by the second type of equations, which are described by the second type of equations...We shall now see how all this comes about from two different basic variational principles in the relativistic case (the nonrelativistic case is similar).

I meant I had not found the word "nonlocal" applied to correlations. Sorry for not being clear enough. My reading of what you quoted is he is discussing the standard quantum mechanics. Then he demonstrates how his theory could yield similar results. Again, it is not clear to me if he replaces his theory by a new one or demonstrates a computational trick to obtain the familiar results. Just think about it, he says: "For example, for the system hydrogen molecule the two protons are separated, whereas the two electrons are nonseparated." But the difference between the protons and electrons is quantitative only, not qualitative.

He then goes on to construct the 16-component configuration space wavefunction for two nonseparable particles, after showing the nonlinear Hartree-Fock formalism for two separable particles. Notice also that the Dirac equation that he constructs for this nonlocal two-particle case (equation 39) is still a nonlinear integro-differential equation, given that it still is a function of these nonlinear, nonlocal self-fields.

Again, is it a new theory or a computational trick?

There are many obvious problems with the connection you're trying to make here. First, you're saying that a nonlinear differential equation can be linearized by using 2nd quantization; problem is, the linear Schroedinger equation is a 1st quantized equation and therefore has nothing to do with any second quantization procedure. So, what you and nightlight showed is a neat mathematical procedure, but ultimately not applicable to QM in the same way.

What does the linear Schroedinger equation has to do with that? Barut shows how the results of the quantum theory can be obtained from his nonlinear theory. So the Kowalski-Steeb's procedure can be applied to SFED or something similar.

Also, as Barut has already shown, what is applicable to QM is the two different types of variational principles (one with respect to psi(x) and the other with respect to phi(x1, x2)), and they apply to different physical situations. This is the crucial point that nightlight either overlooked or did not understand.

Barut just showed that the standard QM results can be obtained as a result of underoptimization. The status of this underoptimization is not clear at all, if only because there is only quantitative difference between the two physical situations. He could apply his trick to a banal nonlinear differential equation and write off the differences as higher orders of the perturbation theory.

Moreover, the equations of motion are both still nonlinear.

So what? Barut emphasizes elsewhere that SFED does not coincide with QED in each order, but the sums of a greater number of orders are close.

By the way, there have been attempts to describe nonseparable systems in terms of the nonlinear Hartree-Fock equations - but they don't get rid of the nonlocality in any way that you seem to hope to. Also, you will see from those references to Hartree-Fock equations, that they only constitute an approximation and have specific limitations.

So somebody did something and did not achieve something. So what? Until there is a no-go theorem (and the Bell theorem has limitations I mentioned earlier), the issue is not resolved.

So based on all the above, I think these sentiments are totally misguided. Sorry.

Maybe they are. But not for the above reasons.

I don't understand anything you said here. In particular, I don't know what you think locality, reality, and causality are, and what you mean by the term "live with". In any case, these are non sequiturs in relation to the physics definitions of locality, reality, and causality. Our discussion is about SFED and locality, and has nothing to do with reality or causality. Indeed, confirmation that SFED is nonlocal does not have any bearing on the validity of realism or causality.

Do I really have to define every word I say? :-) And did not we discuss the Bell theorem in detail? And the Bell theorem is often regarded as a proof of impossibility of local realistic theories (whether you personally accept such interpretation of the theorem or not, is a different question). Didn't you mention causality several times? So our discussion has a lot to do with locality, causality, and reality. I just wanted to say that the burden of proof is much higher for radical conclusions.

 

[Maaneli]

I meant I had not found the word "nonlocal" applied to correlations. Sorry for not being clear enough. My reading of what you quoted is he is discussing the standard quantum mechanics. Then he demonstrates how his theory could yield similar results. Again, it is not clear to me if he replaces his theory by a new one or demonstrates a computational trick to obtain the familiar results. Just think about it, he says: "For example, for the system hydrogen molecule the two protons are separated, whereas the two electrons are nonseparated." But the difference between the protons and electrons is quantitative only, not qualitative.



Again, is it a new theory or a computational trick?




What does the linear Schroedinger equation has to do with that? Barut shows how the results of the quantum theory can be obtained from his nonlinear theory. So the Kowalski-Steeb's procedure can be applied to SFED or something similar.



Barut just showed that the standard QM results can be obtained as a result of underoptimization. The status of this underoptimization is not clear at all, if only because there is only quantitative difference between the two physical situations. He could apply his trick to a banal nonlinear differential equation and write off the differences as higher orders of the perturbation theory.



So what? Barut emphasizes elsewhere that SFED does not coincide with QED in each order, but the sums of a greater number of orders are close.




So somebody did something and did not achieve something. So what? Until there is a no-go theorem (and the Bell theorem has limitations I mentioned earlier), the issue is not resolved.



Maybe they are. But not for the above reasons.




Do I really have to define every word I say? :-) And did not we discuss the Bell theorem in detail? And the Bell theorem is often regarded as a proof of impossibility of local realistic theories (whether you personally accept such interpretation of the theorem or not, is a different question). Didn't you mention causality several times? So our discussion has a lot to do with locality, causality, and reality. I just wanted to say that the burden of proof is much higher for radical conclusions.

Andy,


My reading of what you quoted is he is discussing the standard quantum mechanics. Then he demonstrates how his theory could yield similar results. Again, it is not clear to me if he replaces his theory by a new one or demonstrates a computational trick to obtain the familiar results.


He clearly wasn't just talking about SQM, but also his own theory, when he says that the Hartree-Fock formulation applies to separable systems, and the configuration space formalsm applies to entangled systems. He also demonstrates that this is a natural and necessary aspect of his own relativistic theory, when dealing with two or more entangled or separable particles, when he takes the two different variations. This is clearly not a new theory, but just a natural property of his own original theory, no different than how entanglement is a natural property of standard QM. I think you're pushing it by trying to suggest that this a new theory or demonstrates a "computational trick", whatever that means.


Just think about it, he says: "For example, for the system hydrogen molecule the two protons are separated, whereas the two electrons are nonseparated." But the difference between the protons and electrons is quantitative only, not qualitative.


Sorry but that's totally irrelevant. Besides, Barut is correct because the wavefunctions of electrons in two H-atoms can overlap considerably, while the nuclei wavefunctions are much smaller in wavelength and are shielded from overlapping by the surrounding electron coulomb repulsion.


Again, is it a new theory or a computational trick?


No, this is just the logical and necessary extension of his relativistic two-body SFED theory to the case of entangled particles.


What does the linear Schroedinger equation has to do with that? Barut shows how the results of the quantum theory can be obtained from his nonlinear theory. So the Kowalski-Steeb's procedure can be applied to SFED or something similar.


The linear Schroedinger equation is a first-quantized wave equation, whereas the procedure you showed involves second quantization. Barut shows how the results of nonlinear SFED can be applied to separable and entangled relativistic (and nonrelativistic) two-particle systems. This has absolutely nothing to do with the Kowalski-Steeb procedure.


Barut just showed that the standard QM results can be obtained as a result of underoptimization. The status of this underoptimization is not clear at all, if only because there is only quantitative difference between the two physical situations.


This isn't underoptimization because this is the only way to describe the entangled two-particle case, as he also made clear. And the status of this is quite clear to me, since there is not just a quantitative difference between entangled and separable situations.


He could apply his trick to a banal nonlinear differential equation and write off the differences as higher orders of the perturbation theory.


I don't understand what you mean by this.


Originally Posted by Maaneli
Moreover, the equations of motion are both still nonlinear.

So what?


So what? Come on. The obvious point is that he isn't doing any Carleman linearization or underoptimization procedure if he's still getting a nonlinear integro-differential equation.


So somebody did something and did not achieve something. So what? Until there is a no-go theorem (and the Bell theorem has limitations I mentioned earlier), the issue is not resolved.


Well based on the various research that has been done by applied QM theorists on this exact topic of many-body QM systems and entanglement, I am more confident in accepting and better understanding their judgement, than in hanging my hopes on some remote chance that they might be wrong.


Maybe they are. But not for the above reasons.


Sorry I still think they are misguided for the reasons above.


Do I really have to define every word I say? :-) And did not we discuss the Bell theorem in detail?


Sorry but I was not convinced that you understood what the terms nonlocality, causality, and locality meant. In fact, you never actually said what you thought those terms meant. And I will not assume what you think.


And the Bell theorem is often regarded as a proof of impossibility of local realistic theories (whether you personally accept such interpretation of the theorem or not, is a different question).


Bell's theorem is also often regarded as a proof of impossibility of all hidden variable theories, but that's blatantly false. So is the "interpretation" that it is a proof or statement of the impossibility of local realistic theories, because it is causality, NOT realism, that is being tested.


Didn't you mention causality several times?


Yes.


So our discussion has a lot to do with locality, causality, and reality.


Yes but I still have no idea what these terms mean to YOU.


I just wanted to say that the burden of proof is much higher for radical conclusions.


The evidence for such conclusions is quite strong, and I believe I did the most accurate assessment of the current situation that is possible. If anything, the much greater burden of proof that one can retain locality, causality, and classical Kolmogorov probability theory in one complete theory of QM is on you or people like you who think it is still possible and plausible.

 

[akhmeteli]

He clearly wasn't just talking about SQM, but also his own theory, when he says that the Hartree-Fock formulation applies to separable systems, and the configuration space formalsm applies to entangled systems. He also demonstrates that this is a natural and necessary aspect of his own relativistic theory, when dealing with two or more entangled or separable particles, when he takes the two different variations. This is clearly not a new theory, but just a natural property of his own original theory, no different than how entanglement is a natural property of standard QM. I think you're pushing it by trying to suggest that this a new theory or demonstrates a "computational trick", whatever that means.

Again, there are no qualitative criteria of when the particles are deemed separable and when they are not. Therefore, I cannot understand from Barut's text if the configuration-space-based version is a fundamental theory or some approximation resulting from underoptimization.

Sorry but that's totally irrelevant. Besides, Barut is correct because the wavefunctions of electrons in two H-atoms can overlap considerably, while the nuclei wavefunctions are much smaller in wavelength and are shielded from overlapping by the surrounding electron coulomb repulsion.

I think it is relevant for the reason above.

The linear Schroedinger equation is a first-quantized wave equation, whereas the procedure you showed involves second quantization. Barut shows how the results of nonlinear SFED can be applied to separable and entangled relativistic (and nonrelativistic) two-particle systems. This has absolutely nothing to do with the Kowalski-Steeb procedure.

Again, the linear Schroedinger equation is irrelevant. And the procedure I mentioned uses second quantization, but does not radically change the meaning of the first quantized nonlinear equation that this procedure is applied to (the solutions of the 1st-quantized equation are "embedded" into the set of solutions of the 2nd-quantized eq.) Yes, Barut shows something, and this something (the arisal of the configuration space) has everything to do with the Kowalski-Steeb procedure (KSP), because the configuration space arises as a result of KSP as well, even for banal nonlinear dif. equations. What I meant is you can apply KSP to SFED before introducing the configuration space and without a radical change of the meaning of the equation, and still obtain a configuration space.

This isn't underoptimization because this is the only way to describe the entangled two-particle case, as he also made clear. And the status of this is quite clear to me, since there is not just a quantitative difference between entangled and separable situations.

I am not sure there is any evidence that we need to describe the entangled two-particle case, in the first place.
What qualitative criteria are there to decide whether the particles are separable or not?

He could apply his trick to a banal nonlinear differential equation and write off the differences as higher orders of the perturbation theory.


I don't understand what you mean by this.

I mean you can apply KSP to a nonlinear differential equation (without radically changing the meaning of the equation) and then trim the resulting Fock space to the two-particle configuration space, thus radicaly changing the meaning of the equation and obtaining entangled particles, which will be an artifact of such trimming.

So what? Come on. The obvious point is that he isn't doing any Carleman linearization or underoptimization procedure if he's still getting a nonlinear integro-differential equation.

So why is he saying that the second action optimization condition is weaker? If this is not an underoptimization, what is?

Well based on the various research that has been done by applied QM theorists on this exact topic of many-body QM systems and entanglement, I am more confident in accepting and better understanding their judgement, than in hanging my hopes on some remote chance that they might be wrong.

I fully appreciate and respect your position.

Sorry but I was not convinced that you understood what the terms nonlocality, causality, and locality meant. In fact, you never actually said what you thought those terms meant. And I will not assume what you think.

So our discussion has a lot to do with locality, causality, and reality.


Yes but I still have no idea what these terms mean to YOU.

The problem was you denied that our discussion had anything to do with reality and causality. As for how I understand those notions, I guess this is a long story, and not very relevant, so I am afraid I cannot give definitions right now without unnecessarily delaying the reply.

I just wanted to say that the burden of proof is much higher for radical conclusions.


The evidence for such conclusions is quite strong, and I believe I did the most accurate assessment of the current situation that is possible. If anything, the much greater burden of proof that one can retain locality, causality, and classical Kolmogorov probability theory in one complete theory of QM is on you or people like you who think it is still possible and plausible.

So I guess the evidence is strong enough for you, but not strong enough for me. I guess this is perfectly normal.

 

[Maaneli]

Andy,


Again, there are no qualitative criteria of when the particles are deemed separable and when they are not. Therefore, I cannot understand from Barut's text if the configuration-space-based version is a fundamental theory or some approximation resulting from underoptimization.

Why do you keep saying there are no "qualitative criteria"? In the first place, I explained qualitatively how Barut's example with the H-atom relates to nonseparable and separable quantum systems. Moreover, Barut already showed a quantitative criterion for nonseparable vs separable systems in his two different variational principles. What more do you want? By the way, Barut never implies that the configuration-space formulation is an approximation, but he does clearly say that it describes a DIFFERENT physical situation from the Hartree-Fock case. I just don't see what else you're clinging to.


Again, the linear Schroedinger equation is irrelevant. And the procedure I mentioned uses second quantization, but does not radically change the meaning of the first quantized nonlinear equation that this procedure is applied to (the solutions of the 1st-quantized equation are "embedded" into the set of solutions of the 2nd-quantized eq.)


First you were claiming the linear Schroedinger equation could be derived by a Carleman linearization method, but then I pointed out that this can't work because the linear S.E. is a FIRST quantized equation, and the Carleman method uses second quanization to get a linear SECOND quantized equation, and they are obviously not the same thing. Now you want to claim that the Carleman procedure can be applied to Barut's nonlinear 1st quantized equation, and that the 2nd quanization procedure wouldn't significantly change the physical meaning of Barut's equation. And you also seem to be suggesting that Barut is already doing this type of linearization procedure in those references. But neither can be the case, because Barut's theory is fundamentally and always a 1st quantized nonlinear equation either in Hartree-Fock form or in the configuration-space form. So, again, the method you suggest here looks to me like a non-sequitur.


Yes, Barut shows something, and this something (the arisal of the configuration space) has everything to do with the Kowalski-Steeb procedure (KSP), because the configuration space arises as a result of KSP as well, even for banal nonlinear dif. equations.


Sorry, I disagree. And you haven't even given me any evidence that what Barut does in those references is in fact the KSP.


What I meant is you can apply KSP to SFED before introducing the configuration space and without a radical change of the meaning of the equation, and still obtain a configuration space.


Has this ever actually been done (Barut certainly isn't doing this)? And if it has been done, does it really allow one to eradicate nonlocal correlations at the level of the nonlinear SFED equation? I doubt it.


I am not sure there is any evidence that we need to describe the entangled two-particle case, in the first place.


Well if you want to ignore all the current Bell inequality experiments (even if they are nonideal, it doesn't matter) and their logical implications, as I have explained numerous times already, then you are free to do so; but then I think you are leading yourself down a potentially dead end.


I mean you can apply KSP to a nonlinear differential equation (without radically changing the meaning of the equation) and then trim the resulting Fock space to the two-particle configuration space, thus radicaly changing the meaning of the equation and obtaining entangled particles, which will be an artifact of such trimming.


There are actually some ambiguities here in your argument, I noticed now. First off, the configuration space formalism can describe both entangled and non-entangled two particle systems. So just because you might be able to get a two-particle system in configuration space, it doesn't mean these two particles are necessarily entangled. Secondly, if you apply KSP to, say, Barut's nonlinear SFED equation for two particles, it sounds like you are then saying that the resultant "entangled particles" in configuration space, is just a mathematical artifact of KSP linearization, and does not correspond to a physically real situation that the nonlinear SFED equation also describes as distinct from the non-entangled two particle system. Is that correct?


So why is he saying that the second action optimization condition is weaker? If this is not an underoptimization, what is?


Where does he say that again?


I fully appreciate and respect your position.


Thanks.


The problem was you denied that our discussion had anything to do with reality and causality. As for how I understand those notions, I guess this is a long story, and not very relevant, so I am afraid I cannot give definitions right now without unnecessarily delaying the reply.


Reality is a red herring issue here. The implications of our discussion has to do with locality, and much less to do with Bell's causality assumption. Nevertheless, I still don't know if you really understand what these three terms mean, and I suspect you have a different understanding of the term locality, which is why you are so reluctant to see why Barut's SFED needs to include entanglement nonlocality.


So I guess the evidence is strong enough for you, but not strong enough for me. I guess this is perfectly normal.


I don't think your view is perfectly normal at all. You still have never given any specific physics-based reasons for why you think locally causal theories can plausibly account for all of QM phenomena. And this requires much more than just saying "well there has not yet been any true VBI yet". And, to be honest, with due respect, I am not even convinced yet that you understand exactly why in the current experimental situation VBI's have not been demonstrated. I, on the other hand, have explained my views quite thoroughly, especially in relation to Barut's SFED.

Peace,
Maaneli

 

[akhmeteli]

Why do you keep saying there are no "qualitative criteria"? In the first place, I explained qualitatively how Barut's example with the H-atom relates to nonseparable and separable quantum systems. Moreover, Barut already showed a quantitative criterion for nonseparable vs separable systems in his two different variational principles. What more do you want? By the way, Barut never implies that the configuration-space formulation is an approximation, but he does clearly say that it describes a DIFFERENT physical situation from the Hartree-Fock case. I just don't see what else you're clinging to.

I don’t agree that your explanation was qualitative. Of course, the wavefunctions of the protons overlap to much smaller extent than those of the electrons, but this is a quantitative difference. For the same reason the difference between the two physical situations is quantitative only and therefore it only exists at the approximation level.

First you were claiming the linear Schroedinger equation could be derived by a Carleman linearization method

I am afraid I don’t quite understand which of my statements you are referring to. Could you remind me the exact words? I searched in this thread, but failed to find something like this. Maybe what you had in mind was Kowalski’s words that a nonlinear dif. equation (NDE) is equivalent in some respect to a Shroedinger-like linear evolution equation? But THAT Schroedinger-like equation is 2nd-quantized.

, but then I pointed out that this can't work because the linear S.E. is a FIRST quantized equation, and the Carleman method uses second quanization to get a linear SECOND quantized equation, and they are obviously not the same thing. Now you want to claim that the Carleman procedure can be applied to Barut's nonlinear 1st quantized equation, and that the 2nd quanization procedure wouldn't significantly change the physical meaning of Barut's equation. And you also seem to be suggesting that Barut is already doing this type of linearization procedure in those references. But neither can be the case, because Barut's theory is fundamentally and always a 1st quantized nonlinear equation either in Hartree-Fock form or in the configuration-space form. So, again, the method you suggest here looks to me like a non-sequitur.

But the 1st-quantized equation in the configuration space is practically a 2nd-quantized equation, just in a different form (if you use the symmetry condition).

Yes, Barut shows something, and this something (the arisal of the configuration space) has everything to do with the Kowalski-Steeb procedure (KSP), because the configuration space arises as a result of KSP as well, even for banal nonlinear dif. equations.

Sorry, I disagree. And you haven't even given me any evidence that what Barut does in those references is in fact the KSP.

I presume that you disagree with my statement that what Barut does has something to do with KSP. If I am wrong, please advise. I did not say that what Barut does is in fact KSP. I said that it “has to do” with KSP. Let me explain. Let us assume that we start from the Barut’s nonlinear equation in 3+1 dimensions. Then we can apply KSP to it and obtain a linear equation in the Fock space. Then we can “project” this equation onto the configuration space for two particles. I agree that this reasoning is not straight-forward, because Barut has a nonlinear equation in the configuration space, and the “projection” should be linear. However, that may have to do with the fact that the Barut’s equation is integro-differential, not just differential (maybe this was a source of some of your previous remarks). Maybe KSP (or something similar) should be applied to the 1st-quantized Dirac-Maxwell equations (i.e. differential equations) before elimination of the electromagnetic field by Barut. But what is important is that the configuration space, which Barut introduces “manually” (as a new postulate or as an approximation, I just cannot understand from his texts), can naturally arise from KSP. In that sense what Barut does can have a lot to do with KSP.

What I meant is you can apply KSP to SFED before introducing the configuration space and without a radical change of the meaning of the equation, and still obtain a configuration space.

Has this ever actually been done (Barut certainly isn't doing this)? And if it has been done, does it really allow one to eradicate nonlocal correlations at the level of the nonlinear SFED equation? I doubt it.

I agree, Barut does not seem to do it, at least not more than partially. As far as I know, this has not been done (it is difficult to understand if nightlight himself or somebody else connected to him developed this idea - as you know, he ties peculiarities of quantum theory to linearization along the KSP lines; furthermore, he is much interested in SFED). Furthermore, maybe it should not be done literally (and I appreciate that this phrase may contradict something that I said earlier), i.e. KSP should not be applied to the integro-differential equation of SFED, but, as I mentioned above, it should be applied directly to the Dirac-Maxwell equations (or their modification). Certainly, there may be additional difficulties with getting the fermion statistics correctly, maybe some other difficulties. However, it is clear that there is a possibility of eliminating nonlocal correlations. Indeed, KSP builds a bridge between NDE and linear equations in the Fock space, but obviously it does not introduce nonlocality on the set of solutions of NDE (and maybe this set is all we should use, not the entire Fock space).

Well if you want to ignore all the current Bell inequality experiments (even if they are nonideal, it doesn't matter) and their logical implications, as I have explained numerous times already, then you are free to do so; but then I think you are leading yourself down a potentially dead end.

Maybe you are right, and it does not matter that all the current Bell inequality experiments are nonideal, maybe you are wrong, I just don’t know. As for a potential dead end, you see, my goal is to understand something. If, as a result, I’ll have to accept those “logical implications”, so be it. All I am trying to say, the situation is not as clear-cut as it seems, and, strictly speaking, both theoretical and experimental grounds of those implications are shaky.

I mean you can apply KSP to a nonlinear differential equation (without radically changing the meaning of the equation) and then trim the resulting Fock space to the two-particle configuration space, thus radicaly changing the meaning of the equation and obtaining entangled particles, which will be an artifact of such trimming.


There are actually some ambiguities here in your argument, I noticed now. First off, the configuration space formalism can describe both entangled and non-entangled two particle systems. So just because you might be able to get a two-particle system in configuration space, it doesn't mean these two particles are necessarily entangled.

But where is the ambiguity? I cannot insist that this indeed happens, i.e. that entanglement arises as a result of “trimming” or “projecting” (though I strongly suspect it, as you radically expand the set of wavefunctions), as I did not check it, I am just saying that this is a possibility. And we can confidently say that the configuration space can indeed arise as a result of KSP.

Secondly, if you apply KSP to, say, Barut's nonlinear SFED equation for two particles, it sounds like you are then saying that the resultant "entangled particles" in configuration space, is just a mathematical artifact of KSP linearization, and does not correspond to a physically real situation that the nonlinear SFED equation also describes as distinct from the non-entangled two particle system. Is that correct?

I am afraid I don’t quite understand your question (especially the following part: “as distinct from the non-entangled two particle system”, however, maybe I don’t really need to understand it, if the following explanation could satisfy you.

You see, I don’t consider application of KSP to Barut's nonlinear SFED equation “for two particles”. I consider its application to the original 1st quantized SFED equation in 3+1 dimensions (or to the Dirac-Maxwell equations). I hypothesize that that equation may in fact describe not one, but an arbitrary number of particles (e.g. however intense is the electromagnetic field, it is still described by the Maxwell equations, with obvious caveats). There may also be some modifications along the lines of the Dirac’s “new electrodynamics”. And those particles become “explicit” as a result of linearization.

So why is he saying that the second action optimization condition is weaker? If this is not an underoptimization, what is?


Where does he say that again?

In the Google book, p. 130: “Now our second variational principle is that the action be stationary not with respect to the variations of the individual fields, but with respect to the total composite field only. This is a weaker condition than before and leads to an equation for \Phi in configuration space.”

The problem was you denied that our discussion had anything to do with reality and causality. As for how I understand those notions, I guess this is a long story, and not very relevant, so I am afraid I cannot give definitions right now without unnecessarily delaying the reply.


Reality is a red herring issue here. The implications of our discussion has to do with locality, and much less to do with Bell's causality assumption. Nevertheless, I still don't know if you really understand what these three terms mean, and I suspect you have a different understanding of the term locality, which is why you are so reluctant to see why Barut's SFED needs to include entanglement nonlocality.

Sorry, I just cannot discuss the definitions now. If you believe that makes our discussion meaningless, I am sorry. I just don’t have time.

So I guess the evidence is strong enough for you, but not strong enough for me. I guess this is perfectly normal.

I don't think your view is perfectly normal at all.

I mean what’s normal is the fact that we disagree.

You still have never given any specific physics-based reasons for why you think locally causal theories can plausibly account for all of QM phenomena. And this requires much more than just saying "well there has not yet been any true VBI yet".

I assume you believe that “locally causal theories cannot plausibly account for all of QM phenomena”. Maybe you’re absolutely right. I was just trying to say that this statement is, on the one hand, quite radical, on the other hand, it has not been reliably proven, at least not to my satisfaction. It was not my task to prove the opposite of that statement, which opposite can be true or false. And I believe I offered physics-based reasons in support of my opinion (at least I think those reasons are indeed physics-based, but you may disagree). My arguments were as follows (if you say that I just rephrase nightlight’s arguments, I could largely agree): 1). The Bell theorem uses self-contradictory assumptions; 2) No genuine VBI have been demonstrated experimentally; 3) Entanglement can be an artifact of linearization of NDE.

And, to be honest, with due respect, I am not even convinced yet that you understand exactly why in the current experimental situation VBI's have not been demonstrated. I, on the other hand, have explained my views quite thoroughly, especially in relation to Barut's SFED.

I understand VBI have not been demonstrated just because there are no VBI in nature. I may be dead wrong, of course. As for your views, I think I understand them, but I don’t feel I have to accept them.

 

[Maaneli]

I don’t agree that your explanation was qualitative. Of course, the wavefunctions of the protons overlap to much smaller extent than those of the electrons, but this is a quantitative difference.


Well then I don't understand what is your definition of qualitative. Qualitative to me means 'hand-waving' explanation with physical intuition and words, not mathematics. That's what I gave, and by this definiton it certainly is qualitative. I did not explicitly write down the wavefunction of the protons and electrons and show the entanglement or separability of the wavefunctions in configuration space. That kind of detailed, mathematical explanation would be quantitative.


For the same reason the difference between the two physical situations is quantitative only and therefore it only exists at the approximation level.


I don't understand what you mean. What do you mean by the "approximation level"?


I am afraid I don’t quite understand which of my statements you are referring to. Could you remind me the exact words? I searched in this thread, but failed to find something like this. Maybe what you had in mind was Kowalski’s words that a nonlinear dif. equation (NDE) is equivalent in some respect to a Shroedinger-like linear evolution equation? But THAT Schroedinger-like equation is 2nd-quantized.


Yes that's what I was referring to.


But the 1st-quantized equation in the configuration space is practically a 2nd-quantized equation, just in a different form (if you use the symmetry condition).


No, sorry, I completely disagree. The SFED equation and wavefunction is simply not 2nd quantized. The wavefunction is a c-number field, not an operator field. Moreover, the wavefunction has as its argument the position coordinate psi(x), NOT the field phi(x). In standard QED, the wavefunction is not a c-number field, but an operator field, and it satisfies the quantum commutation relations. Furthermore, the wavefunction is a function of the field phi(x), NOT the position coordinate, x, as in the first-quantized wavefunction. Also, the symmetry condition (I assume you're talkin about symmetric and anti-symmetric wavefunctions) exists in 1st quantized theory too, so I think it is simply incorrect to say this is a 2nd quantization condition.



I presume that you disagree with my statement that what Barut does has something to do with KSP.


Yes.


I did not say that what Barut does is in fact KSP. I said that it “has to do” with KSP.


Actually you said "Yes, Barut shows something, and this something (the arisal of the configuration space) has everything to do with the Kowalski-Steeb procedure (KSP), because the configuration space arises as a result of KSP as well."


Let me explain. Let us assume that we start from the Barut’s nonlinear equation in 3+1 dimensions. Then we can apply KSP to it and obtain a linear equation in the Fock space. Then we can “project” this equation onto the configuration space for two particles.


Yes I knew what you were trying to say here before.




I agree that this reasoning is not straight-forward, because Barut has a nonlinear equation in the configuration space, and the “projection” should be linear. However, that may have to do with the fact that the Barut’s equation is integro-differential, not just differential (maybe this was a source of some of your previous remarks).


Yes these are my objections. Barut has already constructed a nonlinear equation in configuration space which takes into account nonlocal entanglement correlations, as you saw in those papers. Indeed it has to do with the fact that Barut's equation is integro-differential; the integral term from the self-field is what makes the whole damn equation nonlinear in the first place. And since Barut already has a configuration space formulation which involves nonlocal entanglement correlations, the KSP procedure is not relevant to SFED for dealing with entanglement nonlocality.


Maybe KSP (or something similar) should be applied to the 1st-quantized Dirac-Maxwell equations (i.e. differential equations) before elimination of the electromagnetic field by Barut.


I'm sure you could apply it to the 1st quantied Dirac-Maxwell equations; but I don't see why this would get rid of the nonlinearity of the self-field, since you would have to include it inevitably after applying the KSP procedure.


But what is important is that the configuration space, which Barut introduces “manually” (as a new postulate or as an approximation, I just cannot understand from his texts), can naturally arise from KSP. In that sense what Barut does can have a lot to do with KSP.


No, I think you're comparison is superficial. Yes, KSP gives a configuration space wave equation, and so does Barut's approach. That doesn't at all mean that Barut's approach has "everything to do with KSP" or anything to do with KSP for that matter.

Also, I disagree with your characterization of Barut's approach to getting the configuration space formulation as introduced "manually" and as either an approximation or a new postulate. First off, it is a fact of the mathematics of 2 and 4 component spinor wavefunctions that one can take the tensor products of them. For two wavefunctions, this physically just means that those wavefunctions are overlapping in configuration space (and therefore have a common support). The variational derivation of the wave equations is as standard of an approach to getting the wave equation as any other. In getting the nonlinear configuration space wave equation, Barut just decides to take the variation with respect to this case of two overlapping wavefunctions, and therefore gets the configuration space version of his SFED equation. Indeed, this is no more of an approximation or postulate than is his variational derivation of the coupled Hartree-Fock SFED equations for the case of separable wavefunctions. So I just cannot understand what you're confused about or why you think there is an ambiguity here.


I agree, Barut does not seem to do it, at least not more than partially. As far as I know, this has not been done (it is difficult to understand if nightlight himself or somebody else connected to him developed this idea - as you know, he ties peculiarities of quantum theory to linearization along the KSP lines; furthermore, he is much interested in SFED).


After thinking about this issue with you, I honestly think nightlight's approach (at least to the extent that he presented it) is based on some misunderstandings of Barut's SFED, and the relation between 1st and 2nd quantization.


Furthermore, maybe it should not be done literally (and I appreciate that this phrase may contradict something that I said earlier), i.e. KSP should not be applied to the integro-differential equation of SFED, but, as I mentioned above, it should be applied directly to the Dirac-Maxwell equations (or their modification).

Well there you go. But, again, if you're intent on keeping within SFED, then I don't see how applying KSP first to the D-M equations, and then including the self-field is going to keep the linearity of the resultant equation.

Honestly, it looks to me like KSP might be an ingenious alternative method of 2nd quantization of a nonlinear classical field theory wave equation (like the nonlinear Schroedinger equation that describes classical soliton waves). In other words, it looks like a way of going to a linear 2nd quantized Schroedinger QED version of a nonlinear classical field theory. If you were to apply KSP to Barut's theory (which is not a nonlinear classical wave equation but is a nonlinear Schroedinger equation nonetheless), I suspect you might just get the linear, 2nd quantized Schroedinger equation and wavefunction of standard QED. In other words, KSP might be a way of formally relating Barut's 1st quantized nonlinear SFED to 2nd quantized linear Schroedinger and Dirac QED. That would be quite interesting to me.



Certainly, there may be additional difficulties with getting the fermion statistics correctly, maybe some other difficulties. However, it is clear that there is a possibility of eliminating nonlocal correlations. Indeed, KSP builds a bridge between NDE and linear equations in the Fock space, but obviously it does not introduce nonlocality on the set of solutions of NDE (and maybe this set is all we should use, not the entire Fock space).


No I don't agree that there is a "clear possibility of eliminating nonlocal correlations". The KSP method introduces the possibility of nonlocal correlations, and says nothing about nonlocal correlations in SFED; and as Barut demonstrated, SFED on its own already contains nonlocal correlations.


Maybe you are right, and it does not matter that all the current Bell inequality experiments are nonideal, maybe you are wrong, I just don’t know.


I certainly didn't mean to say (and I don't think this) that it doesn't matter that all the current Bell experiments are nonideal. I just meant that insofar as constructing a more fundamental physical theory of QM processes, if it is a locally causal theory, it better have a way of accounting for EVEN the nonideal correlations in these experiments, just as standard QM does. One would have to then implement a stochastic optics type of mechanism or some other ad-hoc mechanism to do this, in which case, there certainly are experimentally testable differences.


As for a potential dead end, you see, my goal is to understand something. If, as a result, I’ll have to accept those “logical implications”, so be it. All I am trying to say, the situation is not as clear-cut as it seems, and, strictly speaking, both theoretical and experimental grounds of those implications are shaky.


I agree the situation experimentally is not clear cut; but that doesn't mean one can't draw reliable conclusions yet. Theoretically speaking, the theories are quite unambiguous in their predictions. So I would have to totally disagree with you there.



But where is the ambiguity? I cannot insist that this indeed happens, i.e. that entanglement arises as a result of “trimming” or “projecting” (though I strongly suspect it, as you radically expand the set of wavefunctions), as I did not check it, I am just saying that this is a possibility. And we can confidently say that the configuration space can indeed arise as a result of KSP.


I of course agree the configuration space can bet obtained from KPS. And if KSP is just another form of 2nd quantization of a classical wave equation, then I am willing to agree that entanglement is possible in that derived configuration space.


I am afraid I don’t quite understand your question (especially the following part: “as distinct from the non-entangled two particle system”, however, maybe I don’t really need to understand it, if the following explanation could satisfy you.


Let me rephrase it. It sounds like you are saying Barut's nonlinear SFED equation for two particles never describes entanglement in configuration space (that the two particle wavefunctions are always sparable and local in 3-space), and it sounds like you are then saying that entangled particles in configuration space, is just a mathematical artifact of KSP linearization. Is that correct? If so, then I just don't agree with you. First off, you already agreed that the existence of a configuration space does necessarily imply entanglement. If that is the case, then the question becomes, suppose you have entanglement for two configuration space wavefunctions satisfying the linear equation derived by KSP - does that entanglement persist if you reverse the KSP and go back to the NDE? Besides that, Barut already gave a counterexample to the first case that does not involve linearization, and that the Shroedinger equation obtained from KSP linearization is not the linear 1st quantized Schroedinger equation of QM. Furthermore, in terms of your subsequent proposal, I fail to see how applying KSP to the D-M equations and then putting in the self-field interaction (If I understand you properly) will keep linearity.



You see, I don’t consider application of KSP to Barut's nonlinear SFED equation “for two particles”. I consider its application to the original 1st quantized SFED equation in 3+1 dimensions (or to the Dirac-Maxwell equations). I hypothesize that that equation may in fact describe not one, but an arbitrary number of particles (e.g. however intense is the electromagnetic field, it is still described by the Maxwell equations, with obvious caveats). There may also be some modifications along the lines of the Dirac’s “new electrodynamics”. And those particles become “explicit” as a result of linearization.


It seems that this KSP 2nd quantization applied to SFED might be just a 2nd quantized version of SFED. By the way, a 2nd quantized version of SFED developed by Babiker, Barut, and Dowling does already exist. The matter field is actually 2nd quantized (the wavefunction has the form psi[phi(x)]), and the self-field is sourced by the current from this 2nd quantized Schroedinger or Dirac equation. What might be interesting is if KSP applied to SFED is empirically equivalent to the approach developed by Babiker, Barut, and Dowling.


In the Google book, p. 130: “Now our second variational principle is that the action be stationary not with respect to the variations of the individual fields, but with respect to the total composite field only. This is a weaker condition than before and leads to an equation for \Phi in configuration space.”


OK. I don't exactly know what he means by a "weaker" condition except that maybe he means the variational degrees of freedom are greater for the composite field case. I don't think this word says anything though about whether there really is entanglement of wavefunctions in configuration space in SFED.



Sorry, I just cannot discuss the definitions now. If you believe that makes our discussion meaningless, I am sorry. I just don’t have time.


It doesn't necessarily make it meaningless, as long as you can agree for now to just stick to locality and nonlocality as the central issue.



I assume you believe that “locally causal theories cannot plausibly account for all of QM phenomena”.


No, I think it is implausible that a locally causal theory can account for all of QM phenomena.


I was just trying to say that this statement is, on the one hand, quite radical, on the other hand, it has not been reliably proven, at least not to my satisfaction.


I think you missed the point. There does not yet exist a locally causal theory that accounts for nonideal Bell correlations for both photons and massive particles, and which accounts for all other physical processes in QM and QED. There does not yet even exist a locally causal theory that that accounts for nonideal Bell correlations for both photons and massive particles (only photons thus far). There does not yet even exist a locally causal theory that accounts for all QM and QED phenomena except nonideal Bell correlations. Therefore, there is no reason currently to think that a locally causal theory that can do everything that QM can do, is plausible as opposed to merely possible. And the burden of proof is indeed on people like yourself who do believe it is plausible, to demonstrate this plausibility by constructing one of the types of locally causal theories I described.


It was not my task to prove the opposite of that statement, which opposite can be true or false. And I believe I offered physics-based reasons in support of my opinion (at least I think those reasons are indeed physics-based, but you may disagree). My arguments were as follows (if you say that I just rephrase nightlight’s arguments, I could largely agree): 1). The Bell theorem uses self-contradictory assumptions; 2) No genuine VBI have been demonstrated experimentally; 3) Entanglement can be an artifact of linearization of NDE.


Well the first argument we already discussed, and it is a red herring argument as far as I am concerned. As I explained before, the projection postulate is not the cause of VBI, it is entanglement in configuration space. Furthermore, Bell's theorem is perfectly compatible with the other empirically equivalent formulations of QM like deBB that do not have PP. So 1) really is just not a valid argument. Sorry.

The second argument does not prove the plausibility of a locally causal theory of QM, but only the possibility of it, as I have already explained.

The third argument is not relevant to entanglement nonlocality in standard QM or to Barut's SFED for reasons already discussed. Furthermore, KSP seems to me to be just another form of 2nd quantization.

So if those are your physics-based arguments, then I think they are not even valid for the above reasons. Sorry.



I understand VBI have not been demonstrated just because there are no VBI in nature. I may be dead wrong, of course. As for your views, I think I understand them, but I don’t feel I have to accept them.


Well you just made the a priori assumotion that VBI don't exist in nature, which you cannot possibly know. There may be true VBI's and we just have to wait until those loophole free experiments are done. So already you are starting from a position of unscientific belief and bias.

Finally, I would contend that you have not quite understood my views yet, which is why you have not felt inclined to accept them.

 

[akhmeteli]

Well then I don't understand what is your definition of qualitative. Qualitative to me means 'hand-waving' explanation with physical intuition and words, not mathematics. That's what I gave, and by this definiton it certainly is qualitative. I did not explicitly write down the wavefunction of the protons and electrons and show the entanglement or separability of the wavefunctions in configuration space. That kind of detailed, mathematical explanation would be quantitative.

Maybe it’s just my poor English. Actually, I meant the following: I could not see any qualitative, rather than quantitative, criteria of separable versus nonseparable particles, neither in Barut’s texts, nor in yours. You told me (maybe I’m cutting corners here) that protons should be treated as separable as their wavefunctions’ overlap is small, and the electrons must be treated as nonseparable, as their wavefunctions’ overlap is large. In my book, this is a quantitative, rather than qualitative, difference.

I don't understand what you mean. What do you mean by the "approximation level"?

I mean that it exists only as long as approximations, rather than a rigorous theory, are concerned. This difference, strictly speaking, does not exist in a rigorous theory.

No, sorry, I completely disagree. The SFED equation and wavefunction is simply not 2nd quantized. The wavefunction is a c-number field, not an operator field. Moreover, the wavefunction has as its argument the position coordinate psi(x), NOT the field phi(x). In standard QED, the wavefunction is not a c-number field, but an operator field, and it satisfies the quantum commutation relations. Furthermore, the wavefunction is a function of the field phi(x), NOT the position coordinate, x, as in the first-quantized wavefunction. Also, the symmetry condition (I assume you're talkin about symmetric and anti-symmetric wavefunctions) exists in 1st quantized theory too, so I think it is simply incorrect to say this is a 2nd quantization condition.

I thought we had been through it. I am not talking about the form now, I am talking about the physical contents. As soon as you introduce wavefunctions with certain symmetry properties in the configuration space, you can introduce 2nd quantization in a standard way without changing the physical contents of the theory (well, maybe some people would demand the Fock space instead of configuration space, but this is not so important).

Actually you said "Yes, Barut shows something, and this something (the arisal of the configuration space) has everything to do with the Kowalski-Steeb procedure (KSP), because the configuration space arises as a result of KSP as well."

Maybe again it’s my English. I tried to use “everything” as an emphatic expression.

Yes these are my objections. Barut has already constructed a nonlinear equation in configuration space which takes into account nonlocal entanglement correlations, as you saw in those papers. Indeed it has to do with the fact that Barut's equation is integro-differential; the integral term from the self-field is what makes the whole damn equation nonlinear in the first place. And since Barut already has a configuration space formulation which involves nonlocal entanglement correlations, the KSP procedure is not relevant to SFED for dealing with entanglement nonlocality.

I'm sure you could apply it to the 1st quantied Dirac-Maxwell equations; but I don't see why this would get rid of the nonlinearity of the self-field, since you would have to include it inevitably after applying the KSP procedure.

Not really. Again, KSP actually turns an NDE into a linear 2nd quantized equation (actually, KSP is a linearization procedure). Or maybe you do not accept this statement? Then please advise. The Dirac-Maxwell equations (DME) are nonlinear due to the interaction of the electromagnetic and Dirac field (which interaction generates self-field). Maybe we should not start from DME, but some similar equations.

No, I think you're comparison is superficial. Yes, KSP gives a configuration space wave equation, and so does Barut's approach. That doesn't at all mean that Barut's approach has "everything to do with KSP" or anything to do with KSP for that matter.

Again, “everything” was emphatic. And the comparison may be superficial, but still it’s a comparison. Maybe you’re right, maybe you’re wrong, but it’s just your opinion. I cannot offer a “final” theory, I am just trying to draw your attention to the fact that the configuration space naturally arises in a situation where there is no entanglement. If you dismiss this argument, it’s certainly your right, but I believe I also have a right to reserve my opinion when you tell me what the true source of nonlocality is.

Also, I disagree with your characterization of Barut's approach to getting the configuration space formulation as introduced "manually" and as either an approximation or a new postulate. First off, it is a fact of the mathematics of 2 and 4 component spinor wavefunctions that one can take the tensor products of them. For two wavefunctions, this physically just means that those wavefunctions are overlapping in configuration space (and therefore have a common support). The variational derivation of the wave equations is as standard of an approach to getting the wave equation as any other. In getting the nonlinear configuration space wave equation, Barut just decides to take the variation with respect to this case of two overlapping wavefunctions, and therefore gets the configuration space version of his SFED equation. Indeed, this is no more of an approximation or postulate than is his variational derivation of the coupled Hartree-Fock SFED equations for the case of separable wavefunctions. So I just cannot understand what you're confused about or why you think there is an ambiguity here.

As I said, I am confused when I try to understand in what cases Barut deems the particles separable and in what cases he does not. I believe there are no qualitative criteria for that. Therefore, his choice can be justified only by precision of his approximations, which makes them approximations, by the way, not a rigorous theory.

After thinking about this issue with you, I honestly think nightlight's approach (at least to the extent that he presented it) is based on some misunderstandings of Barut's SFED, and the relation between 1st and 2nd quantization.

I see your point of view.

Well there you go. But, again, if you're intent on keeping within SFED, then I don't see how applying KSP first to the D-M equations, and then including the self-field is going to keep the linearity of the resultant equation.

Again, maybe we should not keep within SFED, but, as I said above, applying KSP to DME or something similar does not require subsequent introduction of the self-field, as DME already contain self-field.

Honestly, it looks to me like KSP might be an ingenious alternative method of 2nd quantization of a nonlinear classical field theory wave equation (like the nonlinear Schroedinger equation that describes classical soliton waves). In other words, it looks like a way of going to a linear 2nd quantized Schroedinger QED version of a nonlinear classical field theory. If you were to apply KSP to Barut's theory (which is not a nonlinear classical wave equation but is a nonlinear Schroedinger equation nonetheless), I suspect you might just get the linear, 2nd quantized Schroedinger equation and wavefunction of standard QED. In other words, KSP might be a way of formally relating Barut's 1st quantized nonlinear SFED to 2nd quantized linear Schroedinger and Dirac QED. That would be quite interesting to me.

I am glad you might have some interest in KSP, but I’d like to emphasize that typically 2nd quantization radically changes the theory (if you start from a 3+1 – dimensional space, not configuration space), but KSP does not, so in this respect it is not an alternative method of 2nd quantization.

No I don't agree that there is a "clear possibility of eliminating nonlocal correlations". The KSP method introduces the possibility of nonlocal correlations

Could you explain this?

I certainly didn't mean to say (and I don't think this) that it doesn't matter that all the current Bell experiments are nonideal. I just meant that insofar as constructing a more fundamental physical theory of QM processes, if it is a locally causal theory, it better have a way of accounting for EVEN the nonideal correlations in these experiments, just as standard QM does. One would have to then implement a stochastic optics type of mechanism or some other ad-hoc mechanism to do this, in which case, there certainly are experimentally testable differences.

Again, I cannot offer a final theory. But I believe the experimentally testable differences will be due to the fact that the projection postulate is just an approximation (and you also said that it is ia approximation). So it is inevitable, in my opinion, that such differences will be demonstrated experimentally (no matter whether there will be an alternative locally causal theory or not).

I agree the situation experimentally is not clear cut; but that doesn't mean one can't draw reliable conclusions yet. Theoretically speaking, the theories are quite unambiguous in their predictions. So I would have to totally disagree with you there.

When a theory includes mutually contradicting postulates (unitary evolution and projection postulate), it cannot be unambiguous.

I of course agree the configuration space can bet obtained from KPS. And if KSP is just another form of 2nd quantization of a classical wave equation, then I am willing to agree that entanglement is possible in that derived configuration space.

KSP does not introduce entanglement on the set of solutions of the original equation, because the result of KSP is equivalent to the original equation on this set. Entanglement can occur either when we are considering a broader set of functions or when we project the result of KSP onto some configuration space (it is not clear to me when exactly it occurs).

Let me rephrase it. It sounds like you are saying Barut's nonlinear SFED equation for two particles never describes entanglement in configuration space (that the two particle wavefunctions are always sparable and local in 3-space), and it sounds like you are then saying that entangled particles in configuration space, is just a mathematical artifact of KSP linearization. Is that correct? If so, then I just don't agree with you. First off, you already agreed that the existence of a configuration space does necessarily imply entanglement. If that is the case, then the question becomes, suppose you have entanglement for two configuration space wavefunctions satisfying the linear equation derived by KSP - does that entanglement persist if you reverse the KSP and go back to the NDE? Besides that, Barut already gave a counterexample to the first case that does not involve linearization, and that the Shroedinger equation obtained from KSP linearization is not the linear 1st quantized Schroedinger equation of QM. Furthermore, in terms of your subsequent proposal, I fail to see how applying KSP to the D-M equations and then putting in the self-field interaction (If I understand you properly) will keep linearity.

I am afraid you’ve lost me. Again, I just don’t consider application of KSP to two-particle SFED, so maybe your question is irrelevant. I consider application of KSP to what looks like a “one-particle theory”, be it SFED, DME, or something similar (a theory in 3+1 dimensions). As a result, I obtain an equation in the Fock space. Whether there’ll be entanglement in this Fock space, I don’t know for sure (but I cannot exclude such a possibility). However, there will be no entanglement for the states in the Fock space that correspond to the solutions of the original NDE (such states are coherent states). And again, I am not going to introduce self-field interaction after application of KSP.

It seems that this KSP 2nd quantization applied to SFED might be just a 2nd quantized version of SFED. By the way, a 2nd quantized version of SFED developed by Babiker, Barut, and Dowling does already exist. The matter field is actually 2nd quantized (the wavefunction has the form psi[phi(x)]), and the self-field is sourced by the current from this 2nd quantized Schroedinger or Dirac equation. What might be interesting is if KSP applied to SFED is empirically equivalent to the approach developed by Babiker, Barut, and Dowling.

I don’t know.

OK. I don't exactly know what he means by a "weaker" condition except that maybe he means the variational degrees of freedom are greater for the composite field case. I don't think this word says anything though about whether there really is entanglement of wavefunctions in configuration space in SFED.

He means (I think; as I said, Barut’s texts are not always easy to understand, at least for me:-) ) that function \Phi(x_1,x_2)=\phi_1(x_1)\phi2(x_2) can be the same even when \phi_1(x_1) and \phi2(x_2) vary (if their product remains constant). Therefore, when you vary the action with respect to \phi_1(x_1) and \phi2(x_2), you vary more parameters than when varying only with respect to \Phi(x_1,x_2), so the second condition is weaker.

It doesn't necessarily make it meaningless, as long as you can agree for now to just stick to locality and nonlocality as the central issue.

As a central issue – yes.

I assume you believe that “locally causal theories cannot plausibly account for all of QM phenomena”.


No, I think it is implausible that a locally causal theory can account for all of QM phenomena.

Could you explain the difference between these statements? They pretty much look identical to me. Maybe it’s my English again.

I think you missed the point. There does not yet exist a locally causal theory that accounts for nonideal Bell correlations for both photons and massive particles, and which accounts for all other physical processes in QM and QED. There does not yet even exist a locally causal theory that that accounts for nonideal Bell correlations for both photons and massive particles (only photons thus far). There does not yet even exist a locally causal theory that accounts for all QM and QED phenomena except nonideal Bell correlations. Therefore, there is no reason currently to think that a locally causal theory that can do everything that QM can do, is plausible as opposed to merely possible. And the burden of proof is indeed on people like yourself who do believe it is plausible, to demonstrate this plausibility by constructing one of the types of locally causal theories I described.

I agree, there are no such theories. And I disagree, there is such a reason. And the reason is: rejection of either locality or causality is a very radical step, therefore, it should not be done without bullet-proof arguments. There are no such bullet-proof arguments, if you ask me.

Well the first argument we already discussed, and it is a red herring argument as far as I am concerned. As I explained before, the projection postulate is not the cause of VBI, it is entanglement in configuration space. Furthermore, Bell's theorem is perfectly compatible with the other empirically equivalent formulations of QM like deBB that do not have PP. So 1) really is just not a valid argument. Sorry.

So we disagree on this point.

The second argument does not prove the plausibility of a locally causal theory of QM, but only the possibility of it, as I have already explained.

As I have already explained, this is also a matter of opinion.

The third argument is not relevant to entanglement nonlocality in standard QM or to Barut's SFED for reasons already discussed.

It may be relevant to a possible locally causal theory, and that’s what matters.

Furthermore, KSP seems to me to be just another form of 2nd quantization.

As I said, it depends on whether you deem a theory as 2nd quantized based on its form or its physical content.

So if those are your physics-based arguments, then I think they are not even valid for the above reasons. Sorry.

So we disagree on this point.

Well you just made the a priori assumotion that VBI don't exist in nature, which you cannot possibly know. There may be true VBI's and we just have to wait until those loophole free experiments are done. So already you are starting from a position of unscientific belief and bias.

I am afraid I am somewhat at a loss. Should I take this paragraph seriously or the emoticon after the next paragraph covers this one as well?:-) If you are serious, then I’d say the following. I just expressed my opinion based on the failure of about 45 years of attempts to find genuine VBI. I made a caveat that I could be wrong. What’s unscientific about it? As for bias, perhaps I am biased. I believe I have a right to be biased as long as such bias does not contradict reliably established experimental facts.

Finally, I would contend that you have not quite understood my views yet, which is why you have not felt inclined to accept them.

:-)

 

[Maaneli]

Maybe it’s just my poor English. Actually, I meant the following: I could not see any qualitative, rather than quantitative, criteria of separable versus nonseparable particles, neither in Barut’s texts, nor in yours. You told me (maybe I’m cutting corners here) that protons should be treated as separable as their wavefunctions’ overlap is small, and the electrons must be treated as nonseparable, as their wavefunctions’ overlap is large. In my book, this is a quantitative, rather than qualitative, difference.


Ok, then this is just a terminological distinction without a difference. By the way, I don't think your English is poor.



I mean that it exists only as long as approximations, rather than a rigorous theory, are concerned. This difference, strictly speaking, does not exist in a rigorous theory.


I don't see any reason to think Barut's approach to entanglement nonlocality with wavefunctions is any less rigorous than the standard QM approach to entanglement nonlocality with wavefunctions. Do you think the treatment of entanglement nonlocality in standard QM is only an approximation?




I thought we had been through it. I am not talking about the form now, I am talking about the physical contents. As soon as you introduce wavefunctions with certain symmetry properties in the configuration space, you can introduce 2nd quantization in a standard way without changing the physical contents of the theory (well, maybe some people would demand the Fock space instead of configuration space, but this is not so important).


But the "form" does affect the physical contents. You can't second quantize the 1st quantized wavefunction without changing its physical contents, such as to allow for variable fermion particle numbers. Look at any of the textbooks on quantum field theory, and you will see that 2nd quantization of a wavefunction involves the transformation of the c-number wavefunction psi(x), to the wavefunction operator psi(phi(x)). which satisfies the quantum commutation relations. Also, I'm not sure (or forgot) what "symmetry" properties you are defining as 2nd quantization.



Maybe again it’s my English. I tried to use “everything” as an emphatic expression.


Well OK.


Not really. Again, KSP actually turns an NDE into a linear 2nd quantized equation (actually, KSP is a linearization procedure). Or maybe you do not accept this statement?


Yes of course I accept this statement. I don't know what I said to suggest that I didn't. I just don't think it is relevant for describing entanglement nonlocality in Barut's SFED, since SFED is fundamentally a 1st quantized matter theory and already has a nonlinear equation of motion for the case of entangled wavefunctions (and the nonlocality does indeed come in from the self-fields, as Barut shows). That being said, I think it is interesting to consider whether KSP might be another (perhaps more convenient) way to extend SFED into a 2nd quantized matter theory, and for that matter, one that is linear. But it is not clear to me what the linearization will do to the empirical predictions of the theory.


Then please advise. The Dirac-Maxwell equations (DME) are nonlinear due to the interaction of the electromagnetic and Dirac field (which interaction generates self-field). Maybe we should not start from DME, but some similar equations.


Not sure I understand. What other equations are similar to the DM and SFED equations? Are you now saying you don't think KSP should be applied to SFED, but to a different set of equations? I don't understand what your intentions are then.



I cannot offer a “final” theory, I am just trying to draw your attention to the fact that the configuration space naturally arises in a situation where there is no entanglement.


Once again, I don't dispute this. I just said this situation (obtained from KSP) has no relevance to describing nonlocality in SFED, which you seem to agree with now.



As I said, I am confused when I try to understand in what cases Barut deems the particles separable and in what cases he does not. I believe there are no qualitative criteria for that. Therefore, his choice can be justified only by precision of his approximations, which makes them approximations, by the way, not a rigorous theory.


For any entanglement situation with two-particle wavefunctions in standard QM, one can easily generalize it to SFED in a way I already showed before. What Barut does also clearly supports this. Let me show it again:

The wavefunctions for two particles in the separable case are factorizable, meaning that

psi(x1, x2) = psi(x1)psi(x2),

and so his currents will be given by

j(x1) + j(x2) = rho(x1)*v1 + rho(x2)*v2,

where

rho(x1) = |psi(x1)|^2 and rho(x2) = |psi(x2)|^2.

If he were to consider however the basic singlet-state for two electrons, and include the self-fields, then not only would the wavefunctions not be factorizable, but neither would the currents. In other words, he would only have

rho(x1, x2) = |psi(x1, x2)|^2

and

j_n(x1, x2) = rho(x1, x2)*v_n = rho(x1, x2)*(v1 + v2).

So the probability currents would necessarily be in configuration space. This also means that the self-fields would not be distinctly separable as being sourced by two separate electrons.

So, again, this is not any less rigorous than standard QM.




Again, maybe we should not keep within SFED, but, as I said above, applying KSP to DME or something similar does not require subsequent introduction of the self-field, as DME already contain self-field.


So do you want to start from a different approach because you want to now find a way to avoid Barut's variational approach which obtains nonlocality in the self-field interactions? I have a hard time understanding seeing what other sort of nonlinear self-field quantum theory you could possibly start from, which doesn't already contain the possibility of entanglement nonlocality. Maybe you could start from a nonlinear classical wave equation with classical soliton wave solutions, put in self-field interactions, and then apply KSP to 2nd quantize the whole thing. But it wouldn't be surprising that this nonlinear classical wave equation doesn't contain entanglement nonlocality, for the same reason that the classical field theory Hamiltonian equations of motion doesn't contain the possibility of entanglement nonlocality.


I am glad you might have some interest in KSP, but I’d like to emphasize that typically 2nd quantization radically changes the theory (if you start from a 3+1 – dimensional space, not configuration space), but KSP does not, so in this respect it is not an alternative method of 2nd quantization.


So then you do agree that changing the "form" of the theory changes its contents. I don't get how you could say that KSP does not radically change the theory it is applied to, especially when you already acknowledged that the linearized wave equation from KSP is a 2nd quantized equation.


Could you explain this?


When I said

"No I don't agree that there is a "clear possibility of eliminating nonlocal correlations". The KSP method introduces the possibility of nonlocal correlations",

I am simply pointing out that you said KSP introduces a configuration space in a linearized theory, and therefore the possibility of entanglement nonlocality, whereas the original NDE to which KSP is applied, does not contain it. Furthermore, I disagree with your assumption that the KSP DE is just an approximation to the original NDE. If the original NDE is fundamentally a theory in 3+1 space, and KSP is a form of 2nd quantization (which it has to be if the resultant equation is a second quantized linear wave equation, which you also acknowledged it is), then the original NDE is just a classical equation and the KSP DE is simply not an approximation but rather just the quantized version. So it would not be surprising that the KSP DE permits the possibility of entanglement nonlocality, while the classical NDE doesn't.



Again, I cannot offer a final theory. But I believe the experimentally testable differences will be due to the fact that the projection postulate is just an approximation (and you also said that it is ia approximation).

You still fail to grasp this most essential point. Yes, the PP is an approximation - but this approximation has *NOTHING* fundamental to do with locality vs nonlocality. The GRW and deBB theories, which replace the PP approximation with a rigorous measurement theory, are both nonlocal theories. And their nonlocality is due to the entanglement of wavefunctions in configuration space. This is NOT a matter of opinion either. I am in disbelief that you still try to cling to this idea that PP somehow is the cause of the appearance of nonlocality.


So it is inevitable, in my opinion, that such differences will be demonstrated experimentally (no matter whether there will be an alternative locally causal theory or not).


Come on Andy. I was obviously talking about empirical differences between locally causal models and QM.


When a theory includes mutually contradicting postulates (unitary evolution and projection postulate), it cannot be unambiguous.


Of course I agree that QM can be made less ambiguous by replacing PP with a GRW or deBB measurement theory. But I'll say it once more: the contradiction between unitary evolution and the PP approximation has NOTHING to do with the existence of entanglement nonlocality. Let me also explain once more the reason for this:

If a wavefunction psi(x1, x2, t) is not factorizable (it is entanglement in configuration space), then add in the PP and you get VBI. However, if a wavefunction psi(x1, x2, t) is factorizable (there is no entanglement in configuration space), then add in the PP (because you still have reduction of the state vector) and you do not get VBI. So it is obvious that in the first case, entanglement of wavefunctions plus PP necessarily implies VBI, and in the second case, there are factorizable wavefunctions plus PP, and no VBI is possible. The same is also true of GRW collapse QM. So which do you think is more directly relevant to the cause of VBI in standard QM? Entangled wavefunctions in configuration space or the PP? Also throw in the fact that you can eliminate PP with deBB, keep unitary evolution, and still get VBI. There is no doubt that PP is not the culprit of VBI. The PP is actually a deceptive, red herring.


I am afraid you’ve lost me. Again, I just don’t consider application of KSP to two-particle SFED, so maybe your question is irrelevant. I consider application of KSP to what looks like a “one-particle theory”, be it SFED, DME, or something similar (a theory in 3+1 dimensions). As a result, I obtain an equation in the Fock space.


Andy, are you aware that entanglement nonlocality is only possible in a two-particle case in a 1st quantized configuration space theory? This is also true in the 2nd quantized Schroedinger theory. You can't get entanglement in a one particle theory by introducing the Fock space. That has always been well-known in standard QED too.


However, there will be no entanglement for the states in the Fock space that correspond to the solutions of the original NDE (such states are coherent states).


Once again, that's not surprising since the original NDE must be a classical equation of motion.


And again, I am not going to introduce self-field interaction after application of KSP.


Then what you are suggesting can only be another approach to second quantized standard QED. And yet, it is not clear if it would be empirically equivalent to standard QED. Furthermore, it is clear that it would not accomplish the "elimination of nonlocal correlations".



He means (I think; as I said, Barut’s texts are not always easy to understand, at least for me:-) ) that function \Phi(x_1,x_2)=\phi_1(x_1)\phi2(x_2) can be the same even when \phi_1(x_1) and \phi2(x_2) vary (if their product remains constant). Therefore, when you vary the action with respect to \phi_1(x_1) and \phi2(x_2), you vary more parameters than when varying only with respect to \Phi(x_1,x_2), so the second condition is weaker.


Yes, that makes sense.


Could you explain the difference between these statements? They pretty much look identical to me. Maybe it’s my English again.


I am not sure either if there is a difference in these statements.


I agree, there are no such theories. And I disagree, there is such a reason. And the reason is: rejection of either locality or causality is a very radical step, therefore, it should not be done without bullet-proof arguments. There are no such bullet-proof arguments, if you ask me.


I think you're still confusing plausibility with possibility. Do you know the difference between these two words?


So we disagree on this point.


We may disagree, but that doesn't mean it is actually an open question.


As I said, it depends on whether you deem a theory as 2nd quantized based on its form or its physical content.


Sorry but there is no such distinction about what second quantization of the matter field is defined as. Have a look at every QFT textbook on what second quantization is defined as.



I am afraid I am somewhat at a loss. Should I take this paragraph seriously or the emoticon after the next paragraph covers this one as well?:-) If you are serious, then I’d say the following. I just expressed my opinion based on the failure of about 45 years of attempts to find genuine VBI. I made a caveat that I could be wrong. What’s unscientific about it? As for bias, perhaps I am biased. I believe I have a right to be biased as long as such bias does not contradict reliably established experimental facts.


You said

"I understand VBI have not been demonstrated just because there are no VBI in nature."

This statement implies that you already think nature does not VBI - but you cannot rationally make this claim unless you know what the future outcomes of further VBI experimental tests will be. So the fact that you do nevertheless make this claim (and cling to the belief that the contradiction of PP and unitary evolution may be the source of nonlocality) tells me that you are not looking at this issue objectively and honestly. There is a famous quote by Shakespeare: "This above all to thine own self be true".

 

[akhmeteli]

I don't see any reason to think Barut's approach to entanglement nonlocality with wavefunctions is any less rigorous than the standard QM approach to entanglement nonlocality with wavefunctions. Do you think the treatment of entanglement nonlocality in standard QM is only an approximation?

Within the framework of SQM – maybe not. Eventually (I mean in some more general theory) – maybe yes. Again, this is just my opinion, which may be correct or wrong. Anyway, my problem remains unsolved – I cannot understand if Barut has any rigorous criteria to tell separable particles from nonseparable. How does he choose what variation principle to apply?

I thought we had been through it. I am not talking about the form now, I am talking about the physical contents. As soon as you introduce wavefunctions with certain symmetry properties in the configuration space, you can introduce 2nd quantization in a standard way without changing the physical contents of the theory (well, maybe some people would demand the Fock space instead of configuration space, but this is not so important).


But the "form" does affect the physical contents. You can't second quantize the 1st quantized wavefunction without changing its physical contents, such as to allow for variable fermion particle numbers. Look at any of the textbooks on quantum field theory, and you will see that 2nd quantization of a wavefunction involves the transformation of the c-number wavefunction psi(x), to the wavefunction operator psi(phi(x)). which satisfies the quantum commutation relations. Also, I'm not sure (or forgot) what "symmetry" properties you are defining as 2nd quantization.

You see, you can introduce second quantization even if the number of particles is conserved. So whether the number of particle varies is not very relevant. By symmetry properties I mean either symmetry of the wavefuncion under permutations, or antisymmetry (in case of fermions). So if you have a wavefunction in configuration space with certain symmetry properties, you can introduce the operator wavefunction with the standard commutation properties without changing the contents of the theory, just its form. Let me assure you that I have seen my fair share of textbooks on quantum field theory. So in this case the form does not affect the physical contents.

Yes of course I accept this statement. I don't know what I said to suggest that I didn't. I just don't think it is relevant for describing entanglement nonlocality in Barut's SFED, since SFED is fundamentally a 1st quantized matter theory and already has a nonlinear equation of motion for the case of entangled wavefunctions (and the nonlocality does indeed come in from the self-fields, as Barut shows).

Again, if you have entangled wavefunctions (a wavefunction (with some symmetry properties) in configuration space), you have second quantization, for all intents and purposes. I fully agree that it is not 2nd quantized as far as the form is concerned, but it is fully equivalent to a second-quantized theory in its contents.

Not sure I understand. What other equations are similar to the DM and SFED equations? Are you now saying you don't think KSP should be applied to SFED, but to a different set of equations? I don't understand what your intentions are then.

I am on a shaky ground now. I don’t know a clear answer. In my work, I am trying the Dirac-Maxwell Lagrangian with some constraint. Maybe this specific approach is not as promising as I hope. The general idea, however, is that maybe it is possible to start with some NDE, apply KSP, and obtain an apparently second-quantized theory emulating current experimental results of QED. In the same time, such a theory will still remain equivalent to the original NDE on the set of solutions of the latter.

I cannot offer a “final” theory, I am just trying to draw your attention to the fact that the configuration space naturally arises in a situation where there is no entanglement.


Once again, I don't dispute this. I just said this situation (obtained from KSP) has no relevance to describing nonlocality in SFED, which you seem to agree with now.

It may be relevant. What I mean is maybe you just should not introduce the wavefunction in the configuration space manually, as Barut does (if you dislike my wording, you can say that he introduces a new postulate for several particles). Maybe the correct theory can be obtained by applying KSP to some NDE. Technically, you can say, of course, that this situation is not relevant to SFED, but it may be relevant to the actual content of the eventual theory. In other words, the apparent entanglement (in the “final” theory, rather than in SFED) may be a result of KSP and may have nothing to do with nonlocality. You may say that this is highly speculative, and I’ll have to agree. However, this may be an alternative to the radical conclusion of nonlocality or noncausality.

As I said, I am confused when I try to understand in what cases Barut deems the particles separable and in what cases he does not. I believe there are no qualitative criteria for that. Therefore, his choice can be justified only by precision of his approximations, which makes them approximations, by the way, not a rigorous theory.


For any entanglement situation with two-particle wavefunctions in standard QM, one can easily generalize it to SFED in a way I already showed before. What Barut does also clearly supports this. Let me show it again:

The wavefunctions for two particles in the separable case are factorizable, meaning that

psi(x1, x2) = psi(x1)psi(x2),

and so his currents will be given by

j(x1) + j(x2) = rho(x1)*v1 + rho(x2)*v2,

where

rho(x1) = |psi(x1)|^2 and rho(x2) = |psi(x2)|^2.

If he were to consider however the basic singlet-state for two electrons, and include the self-fields, then not only would the wavefunctions not be factorizable, but neither would the currents. In other words, he would only have

rho(x1, x2) = |psi(x1, x2)|^2

and

j_n(x1, x2) = rho(x1, x2)*v_n = rho(x1, x2)*(v1 + v2).

So the probability currents would necessarily be in configuration space. This also means that the self-fields would not be distinctly separable as being sourced by two separate electrons.

So, again, this is not any less rigorous than standard QM.

Maybe not. There are two circumstances though that I don’t quite understand: 1) what is the status of the underoptimization, which takes place in this case, and 2) why this procedure should not be applied to separable particles. I just don’t see any qualitative difference.

Again, maybe we should not keep within SFED, but, as I said above, applying KSP to DME or something similar does not require subsequent introduction of the self-field, as DME already contain self-field.


So do you want to start from a different approach because you want to now find a way to avoid Barut's variational approach which obtains nonlocality in the self-field interactions? I have a hard time understanding seeing what other sort of nonlinear self-field quantum theory you could possibly start from, which doesn't already contain the possibility of entanglement nonlocality. Maybe you could start from a nonlinear classical wave equation with classical soliton wave solutions, put in self-field interactions, and then apply KSP to 2nd quantize the whole thing. But it wouldn't be surprising that this nonlinear classical wave equation doesn't contain entanglement nonlocality, for the same reason that the classical field theory Hamiltonian equations of motion doesn't contain the possibility of entanglement nonlocality.

First, it is not obvious that you must add self-field interaction to the nonlinear classical solution, just because the nonlinearity of the latter can include such interaction. Second, I agree that such an equation will not contain entanglement nonlocality even after KSP. What’s important though is whether such an equation will emulate the results of current experiments, which are in agreement with QED. Again, there has been no genuine experimental demonstration of nonlocality so far.

I am glad you might have some interest in KSP, but I’d like to emphasize that typically 2nd quantization radically changes the theory (if you start from a 3+1 – dimensional space, not configuration space), but KSP does not, so in this respect it is not an alternative method of 2nd quantization.


So then you do agree that changing the "form" of the theory changes its contents. I don't get how you could say that KSP does not radically change the theory it is applied to, especially when you already acknowledged that the linearized wave equation from KSP is a 2nd quantized equation.

I don’t agree that “changing the "form" of the theory changes its contents”. I do agree that “the linearized wave equation from KSP is a 2nd quantized equation.” However, this 2nd quantized linear equation is completely equivalent to the banal NDE on the set of solutions of the latter. These equations have radically different forms, but they describe the same evolution of the solutions of NDE. I mean for each solution of NDE you can construct a coherent state that is a solution of the 2nd-quantized equation.

When I said

"No I don't agree that there is a "clear possibility of eliminating nonlocal correlations". The KSP method introduces the possibility of nonlocal correlations",

I am simply pointing out that you said KSP introduces a configuration space in a linearized theory, and therefore the possibility of entanglement nonlocality, whereas the original NDE to which KSP is applied, does not contain it. Furthermore, I disagree with your assumption that the KSP DE is just an approximation to the original NDE. If the original NDE is fundamentally a theory in 3+1 space, and KSP is a form of 2nd quantization (which it has to be if the resultant equation is a second quantized linear wave equation, which you also acknowledged it is), then the original NDE is just a classical equation and the KSP DE is simply not an approximation but rather just the quantized version. So it would not be surprising that the KSP DE permits the possibility of entanglement nonlocality, while the classical NDE doesn't.

I agree that KSP introduces a configuration space in a linearized theory, but I did not say that it introduces the possibility of entanglement nonlocality. It actually introduces an appearance of entanglement nonlocality. KSP is not a form of 2nd quantization, it just produces something that looks 2nd-quantized. As far as I remember, I did not say KSP DE is an approximation, I said it is a linearization of the original NDE, however in this case the linearization is not approximate, it is exact, however strange this may sound. KSP DE is not a quantized version of NDE, as it is equivalent to NDE on the set of solutions of the latter. There is an injection of the set of solutions of NDE into the set of solutions of KSP DE: for each solution of NDE there is a coherent state in the Fock space, which is a solution of KSP DE. So what happens is you tend to think that KSP DE demonstrates entanglement nonlocality, if you regard KSP DE as an equation on the set of states in the Fock space, whereas it is strictly equivalent to the NDE on the set of solutions of the latter. I beg you, just look at the KSP, say, as it is outlined in one of my previous posts, or directly in the Kowalski’s work.

You still fail to grasp this most essential point. Yes, the PP is an approximation - but this approximation has *NOTHING* fundamental to do with locality vs nonlocality. The GRW and deBB theories, which replace the PP approximation with a rigorous measurement theory, are both nonlocal theories. And their nonlocality is due to the entanglement of wavefunctions in configuration space. This is NOT a matter of opinion either. I am in disbelief that you still try to cling to this idea that PP somehow is the cause of the appearance of nonlocality.

I have to respectfully disagree:-( I believe PP introduces nonlocality directly and shamelessly:-). Just think about it: if you believe PP, as soon as you measure a projection of spin of one particle of a singlet, the projection of spin of the other particle of the singlet immediately acquires a certain value (becomes definite), no matter how far the second particle is from the first. If this is not nonlocality, then what is? I don’t know anything about GRW, but I guess in dBB you have to use something like decoherence to obtain equivalence to SQM, and this is not a trivial matter. It may well be that PP and entanglement of wavefunctions in configuration space are both responsible for nonlocality, but that does not mean that PP is innocent.

So it is inevitable, in my opinion, that such differences will be demonstrated experimentally (no matter whether there will be an alternative locally causal theory or not).


Come on Andy. I was obviously talking about empirical differences between locally causal models and QM.

I fully appreciate that. However, I also fully appreciate that when somebody tells people that there’ll be experimentally demonstrated deviations from SQM, those people may just suggest that he go and entertain himself, as SQM has been solidly tested experimentally. That’s why a suggestion that an experiment may produce results in favor of a locally causal model, as opposed to SQM, will probably fall on a deaf ear. However, the contradiction between PP and unitary evolution makes me confident that one of them is, strictly speaking, wrong (and that it is PP), and this will be eventually demonstrated experimentally. And that might give a stimulus to a search for a locally causal theory.

When a theory includes mutually contradicting postulates (unitary evolution and projection postulate), it cannot be unambiguous.


Of course I agree that QM can be made less ambiguous by replacing PP with a GRW or deBB measurement theory. But I'll say it once more: the contradiction between unitary evolution and the PP approximation has NOTHING to do with the existence of entanglement nonlocality. Let me also explain once more the reason for this:

If a wavefunction psi(x1, x2, t) is not factorizable (it is entanglement in configuration space), then add in the PP and you get VBI. However, if a wavefunction psi(x1, x2, t) is factorizable (there is no entanglement in configuration space), then add in the PP (because you still have reduction of the state vector) and you do not get VBI. So it is obvious that in the first case, entanglement of wavefunctions plus PP necessarily implies VBI, and in the second case, there are factorizable wavefunctions plus PP, and no VBI is possible. The same is also true of GRW collapse QM. So which do you think is more directly relevant to the cause of VBI in standard QM? Entangled wavefunctions in configuration space or the PP? Also throw in the fact that you can eliminate PP with deBB, keep unitary evolution, and still get VBI. There is no doubt that PP is not the culprit of VBI. The PP is actually a deceptive, red herring.

Again, maybe PP and entanglement are both guilty of nonlocality. That does not mean PP is innocent. And I tried to explain above how PP introduces nonlocality, shamelessly and brazenly.

Andy, are you aware that entanglement nonlocality is only possible in a two-particle case in a 1st quantized configuration space theory? This is also true in the 2nd quantized Schroedinger theory. You can't get entanglement in a one particle theory by introducing the Fock space. That has always been well-known in standard QED too.

I just don’t quite see what you’re trying to prove. Do you understand that I just don’t need entanglement nonlocality in the “final” theory, just because there is no experimental evidence of such entanglement nonlocality? What I mean is it may be possible to obtain something very close to QED from NDE using KSP. It may even fully coincide with QED, however the derivation would suggest that this “final” theory should be considered just on the set of solutions of NDE, not in the entire Fock space, so the apparent entanglement nonlocality just does not exist in nature. Again, I agree that this is highly speculative.

And again, I am not going to introduce self-field interaction after application of KSP.


Then what you are suggesting can only be another approach to second quantized standard QED. And yet, it is not clear if it would be empirically equivalent to standard QED. Furthermore, it is clear that it would not accomplish the "elimination of nonlocal correlations".

No, it is not clear that it would not accomplish the "elimination of nonlocal correlations" for reasons given above.

I agree, there are no such theories. And I disagree, there is such a reason. And the reason is: rejection of either locality or causality is a very radical step, therefore, it should not be done without bullet-proof arguments. There are no such bullet-proof arguments, if you ask me.


I think you're still confusing plausibility with possibility. Do you know the difference between these two words?

I know the difference. But I am not going to accept radical conclusions without bullet-proof arguments, just because such conclusions don’t look plausible to me without such arguments.

As I said, it depends on whether you deem a theory as 2nd quantized based on its form or its physical content.


Sorry but there is no such distinction about what second quantization of the matter field is defined as. Have a look at every QFT textbook on what second quantization is defined as.

If you stick to the book definitions of 2nd quantization, then KSP is not a form of 2nd quantization, because standard 2nd quantization provides a theory that is not equivalent to the 1st quantized theory, and KSP gives a theory that is equivalent to NDE on the set of solutions of the latter.

I am afraid I am somewhat at a loss. Should I take this paragraph seriously or the emoticon after the next paragraph covers this one as well?:-) If you are serious, then I’d say the following. I just expressed my opinion based on the failure of about 45 years of attempts to find genuine VBI. I made a caveat that I could be wrong. What’s unscientific about it? As for bias, perhaps I am biased. I believe I have a right to be biased as long as such bias does not contradict reliably established experimental facts.


You said

"I understand VBI have not been demonstrated just because there are no VBI in nature."

This statement implies that you already think nature does not VBI - but you cannot rationally make this claim unless you know what the future outcomes of further VBI experimental tests will be. So the fact that you do nevertheless make this claim (and cling to the belief that the contradiction of PP and unitary evolution may be the source of nonlocality) tells me that you are not looking at this issue objectively and honestly. There is a famous quote by Shakespeare: "This above all to thine own self be true".

Well, I tried to give you my reasons. Obviously, I failed.

 

[Maaneli]

Within the framework of SQM – maybe not. Eventually (I mean in some more general theory) – maybe yes. Again, this is just my opinion, which may be correct or wrong. Anyway, my problem remains unsolved – I cannot understand if Barut has any rigorous criteria to tell separable particles from nonseparable. How does he choose what variation principle to apply?

As far as how Barut decides what variational principle to apply, I think it is no different than in the variational formulation of standard QM. One would first write down the wavefunction of the physical system; if it is the wavefunction describes a single state, then that is the wavefunction with respect to which you apply the variational principle, as it is not factorizable. I don't see any confusion here.

You see, you can introduce second quantization even if the number of particles is conserved. So whether the number of particle varies is not very relevant. By symmetry properties I mean either symmetry of the wavefuncion under permutations, or antisymmetry (in case of fermions). So if you have a wavefunction in configuration space with certain symmetry properties, you can introduce the operator wavefunction with the standard commutation properties without changing the contents of the theory, just its form. Let me assure you that I have seen my fair share of textbooks on quantum field theory. So in this case the form does not affect the physical contents.

Sorry, it is simply not accurate to call a c-numbered wavefunction in configuration space satisfying the proper permutation symmetries, "second quantized". Furthermore, just because second quantization can be introduced when particle numbers are fixed, doesn't mean that the 1st-quantized wavefunction is also "second quantized", especially because the operator wavefunction form allows for the POSSIBILITY of variable particle number. This is why the Fock space is not something you can just dispense with, if you are going to call a matter theory second quantized.

Again, if you have entangled wavefunctions (a wavefunction (with some symmetry properties) in configuration space), you have second quantization, for all intents and purposes. I fully agree that it is not 2nd quantized as far as the form is concerned, but it is fully equivalent to a second-quantized theory in its contents.

Then you are just not using the terminology of second-quantization properly. By your logic, the original Dirac equation with the Dirac sea mechanism to allow for variable fermion numbers, is also a second quantized theory "for all intents and purposes". But this is just not true. The "form" of the theory does indeed matter in the definition of whether it is 1st or 2nd quantized.

I am on a shaky ground now. I don’t know a clear answer. In my work, I am trying the Dirac-Maxwell Lagrangian with some constraint. Maybe this specific approach is not as promising as I hope. The general idea, however, is that maybe it is possible to start with some NDE, apply KSP, and obtain an apparently second-quantized theory emulating current experimental results of QED. In the same time, such a theory will still remain equivalent to the original NDE on the set of solutions of the latter.

But in the DM Lagrangian, you already have self-field interactions and entanglement. How does KSP linearize this self-interaction? Also, it sounds like you are admitting that KSP is just a form of second quantization (otherwise I don't see why you would use it). Also, just because there are solutions to the linear equation that are not solutions to the nonlinear equation, doesn't mean those new solutions to the linear equation are artifacts. There is however a way I discovered to transform between the linear Schroedinger equation and a nonlinear Burger's equation (using the Nagasawa-Schroedinger and Cole-Hopf substitutions), and both equations describe the same physics. I have sent you my paper where I do this.

It may be relevant. What I mean is maybe you just should not introduce the wavefunction in the configuration space manually, as Barut does (if you dislike my wording, you can say that he introduces a new postulate for several particles). Maybe the correct theory can be obtained by applying KSP to some NDE. Technically, you can say, of course, that this situation is not relevant to SFED, but it may be relevant to the actual content of the eventual theory. In other words, the apparent entanglement (in the “final” theory, rather than in SFED) may be a result of KSP and may have nothing to do with nonlocality. You may say that this is highly speculative, and I’ll have to agree. However, this may be an alternative to the radical conclusion of nonlocality or noncausality.

See my comments in the previous post.

Maybe not. There are two circumstances though that I don’t quite understand: 1) what is the status of the underoptimization, which takes place in this case, and 2) why this procedure should not be applied to separable particles. I just don’t see any qualitative difference.

I think the point is that the first variational principle can only be applied to physical systems whose wavefunctions can be factorized (the variational principle does not determine whether a wavefunction can be factorizable). For the second variational principle, it can only be applied to physical systems whose wavefunctions are not factorizable (like the singlet state). That's all.

First, it is not obvious that you must add self-field interaction to the nonlinear classical solution, just because the nonlinearity of the latter can include such interaction. Second, I agree that such an equation will not contain entanglement nonlocality even after KSP. What’s important though is whether such an equation will emulate the results of current experiments, which are in agreement with QED.

You could also start from a Schroedinger equation with a nonlinear term proportional to the quantum potential. Such an equation describes soliton waves in classical fluid dynamics. But if you do not include either self-field interactions or else zero-point fields in your nonlinear theory, then I don't see how your theory could produce radiative effects like the Lamb shift or spontaneous emission.

Again, there has been no genuine experimental demonstration of nonlocality so far.

Again, you just keep on missing the point. Even though no genuine experimental demonstration of nonlocality has been demonstrated so far, your local alternative theory must still be able to predict, within experimental limits, the empirically observed nonideal statistical correlations for both electrons and photons. That within itself is still a nontrivial problem.

I don’t agree that “changing the "form" of the theory changes its contents”.

Really? So the wavefunction on Fock space doesn't have different content than the wavefunction on configuration space? Then it sounds to me like you have not understood 2nd quantization.

I do agree that “the linearized wave equation from KSP is a 2nd quantized equation.” However, this 2nd quantized linear equation is completely equivalent to the banal NDE on the set of solutions of the latter. These equations have radically different forms, but they describe the same evolution of the solutions of NDE. I mean for each solution of NDE you can construct a coherent state that is a solution of the 2nd-quantized equation.

Hmm, well, Steeb seems to disagree with you:

A note on Carleman linearization
W. -H. Steeb
Abstract: Finite dimensional nonlinear ordinary differential equations duj/dt = Vj(u) (j = 1, …, n) can be embedded into a associated infinite dimensional linear systems with the help of the Carleman linearization. We show that the linear infinite systems can admit solutions which are not a solution of the associated nonlinear finite system.
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVM-46SNJVY-H0&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_version=1&_urlVersion=0&_userid=10&md5=9bcc5fc2c0be15aa2bf2585add76bb3b

I agree that KSP introduces a configuration space in a linearized theory, but I did not say that it introduces the possibility of entanglement nonlocality. It actually introduces an appearance of entanglement nonlocality.

This is a semantic distinction without a physical difference. Honestly, what is the difference in your mind between the "appearance" of entanglement nonlocality in the mathematics of a theory, and the "possibility" of entanglement nonlocality in the mathematics of a theory?

KSP is not a form of 2nd quantization, it just produces something that looks 2nd-quantized.

Does it introduce the Fock space for the linearize equation and allow for variable particle numbers? Is the wavefunction solution a c-number field or an operator field? If so, then it has to be a second-quantized theory.

As far as I remember, I did not say KSP DE is an approximation, I said it is a linearization of the original NDE, however in this case the linearization is not approximate, it is exact, however strange this may sound. KSP DE is not a quantized version of NDE, as it is equivalent to NDE on the set of solutions of the latter. There is an injection of the set of solutions of NDE into the set of solutions of KSP DE: for each solution of NDE there is a coherent state in the Fock space, which is a solution of KSP DE.

Again, Steeb disagrees with you.

So what happens is you tend to think that KSP DE demonstrates entanglement nonlocality, if you regard KSP DE as an equation on the set of states in the Fock space, whereas it is strictly equivalent to the NDE on the set of solutions of the latter.

Aha, so then it does introduce the Fock space! Then why isn't it second quantization? What doesn't it have that the second-quantized Schroedinger theory does have.

I beg you, just look at the KSP, say, as it is outlined in one of my previous posts, or directly in the Kowalski’s work.

I did look at Kowalski's paper, and he doesn't really talk about these issues at all.

I have to respectfully disagree:-( I believe PP introduces nonlocality directly and shamelessly:-). Just think about it: if you believe PP, as soon as you measure a projection of spin of one particle of a singlet, the projection of spin of the other particle of the singlet immediately acquires a certain value (becomes definite), no matter how far the second particle is from the first. If this is not nonlocality, then what is?

Your example here indicates that you did bother to read or understand my example from earlier. Honestly, I'm baffled that you still don't understand this very basic point. And this isn't a matter of opinion at all; the nonlocality from the instantaneous collapse of the wavefunction is already present in the single particle version of SQM, but single particle SQM doesn't display VBI because the Bell theorem requires correlations between TWO particles. Also, the nonlocality from the instantaneous collapse of the wavefunction is already present in two particle SQM for separate wavefunctions - for example, if you make 100 spin measurements in the z-direction for a single electron on earth, and then someone makes 100 spin measurements in the z-direction for a single electron somewhere in the Andromeda galaxy, in both places there will be this instantaneous wavefunction collapse from the PP of SQM; but if someone then computes the correlations between these two spin measurements, they will clearly find NO VBI. On the other hand, if one makes 100 measurements on two electrons which originally came from a singlet state so that their wavefunctions are entangled in configuration space, then there clearly will be VBI. Therefore, it is entangelement in configuration space that is the source of VBI. If you do not bother to show me that you at least understand this example, then this is the last time I will discuss this with you because I am tired of having my words ignored.

I fully appreciate that. However, I also fully appreciate that when somebody tells people that there’ll be experimentally demonstrated deviations from SQM, those people may just suggest that he go and entertain himself, as SQM has been solidly tested experimentally. That’s why a suggestion that an experiment may produce results in favor of a locally causal model, as opposed to SQM, will probably fall on a deaf ear. However, the contradiction between PP and unitary evolution makes me confident that one of them is, strictly speaking, wrong (and that it is PP), and this will be eventually demonstrated experimentally. And that might give a stimulus to a search for a locally causal theory.

It sounds like you are confused about a number of issues here. A word of advice: it will probably help you more immediately to study and understand the already well-developed nonlocal hidden variable theories like deBB and GRW to understand why PP is not the crux of the issue here, as opposed to banking on a locally causal theory that you only have the vaguest idea about it would work.

I just don’t quite see what you’re trying to prove. Do you understand that I just don’t need entanglement nonlocality in the “final” theory, just because there is no experimental evidence of such entanglement nonlocality? What I mean is it may be possible to obtain something very close to QED from NDE using KSP. It may even fully coincide with QED, however the derivation would suggest that this “final” theory should be considered just on the set of solutions of NDE, not in the entire Fock space, so the apparent entanglement nonlocality just does not exist in nature. Again, I agree that this is highly speculative.

Your proposal is too vague for me to understand and productively comment on any further. Sorry.

I know the difference. But I am not going to accept radical conclusions without bullet-proof arguments, just because such conclusions don’t look plausible to me without such arguments.

Sigh. You still didn't understand my point. Forget it then.

If you stick to the book definitions of 2nd quantization, then KSP is not a form of 2nd quantization, because standard 2nd quantization provides a theory that is not equivalent to the 1st quantized theory, and KSP gives a theory that is equivalent to NDE on the set of solutions of the latter.

Again, Steeb seems to contradict what you say.

Well, I tried to give you my reasons. Obviously, I failed.

You didn't fail to give me your reasons. You gave them, but they just aren't well thought out at the moment or based on reliable premises, with all due respect. Sorry.

 

[akhmeteli]

Hmm, well, Steeb seems to disagree with you:

A note on Carleman linearization
W. -H. Steeb
Abstract: Finite dimensional nonlinear ordinary differential equations duj/dt = Vj(u) (j = 1, …, n) can be embedded into a associated infinite dimensional linear systems with the help of the Carleman linearization. We show that the linear infinite systems can admit solutions which are not a solution of the associated nonlinear finite system.

I fail to see how Steeb's words contradict mine. I was only talking about equivalence on the set of solutions of the NDE. Any extra solutions the linear infinite system may have are beyond that set, so their existence is in no contradiction with my words.

I'll try to reply to other points of your post later.

 

[Maaneli]

I fail to see how Steeb's words contradict mine. I was only talking about equivalence on the set of solutions of the NDE. Any extra solutions the linear infinite system may have are beyond that set, so their existence is in no contradiction with my words.

I'll try to reply to other points of your post later.

But if the KSP equation has solutions that do not exist in the original NDE, then that sounds to me like the former is fundamentally a different theory. And there seems to me no reason why you could just ignore the extra solutions and only consider the solutions that they share in common. And this difference of solution sets would seem to justify charaterizing KSP as indeed a form of 2nd quantization.

 

[Maaneli]

As far as how Barut decides what variational principle to apply, I think it is no different than in the variational formulation of standard QM. One would first write down the wavefunction of the physical system; if it is the wavefunction describes a single state, then that is the wavefunction with respect to which you apply the variational principle, as it is not factorizable. I don't see any confusion here.

Correction I meant to say "if it is the wavefunction describing a singlet state, then that is the wavefunction with respect to which you apply the variational principle, as it is not factorizable."

 

[akhmeteli]

But if the KSP equation has solutions that do not exist in the original NDE, then that sounds to me like the former is fundamentally a different theory. And there seems to me no reason why you could just ignore the extra solutions and only consider the solutions that they share in common. And this difference of solution sets would seem to justify charaterizing KSP as indeed a form of 2nd quantization.

I am not saying the theories are not different, if only because the post-KSP theory acts in a mind-bogglingly larger space (the Fock space, rather than in 3(+1)D space of the pre-KSP theory). However, your phrase "And there seems to me no reason why you could just ignore the extra solutions and only consider the solutions that they share in common" seems to imply that the post-KSP theory is the true one. I suggest, however, that we imagine a completely different situation for a moment. Specifically, let us imagine that: 1) the pre-KSP theory is the true one, i.e. nature is precisely described by the NDE; 2) in the course of progress of physics, the humanity adopted the post-KSP theory (KSPT). Then no experimental result will contradict the post-KSPT, as only solutions of pre-KSPT will exist, and on this set of solutions the theories are equivalent. It is not easy to experimentally check if indeed all solutions of post-KSPT are realizable. One way to prove that pre-KSPT is wrong is to experimentally demonstrate genuine VBI. You'll appreciate, however, that in our imaginary world no true VBI are possible. Nevertheless, as no experimental results will contradict the theory, many people would deem it reasonable to believe that post-KSPT is the last word in physics and that VBI will be inevitably demonstrated as technology advances. Are you absolutely sure this imaginary world is indeed just an imaginary one?

As for "charaterizing KSP as indeed a form of 2nd quantization"... Well, as for you the definition of 2nd quantization is limited to a certain form, I cannot really argue with definitions. I'll just reiterate that KSP is different from traditional 2nd quantization as we know it in that it does not radically change the contents of the theory it is applied to (unless you compare it to introduction of 2nd quantization for (anti)symmetric functions in configuration space, where the physical contents of the 2nd quantization, rather than its form, is already present).