Maaneli
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akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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
akhmeteli said: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?
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.
akhmeteli said: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.