vanesch
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nightlight said:It is typical for a differential eq's linearization procedures to introduce vast quantities of redundant functions which evolve linearly, instead of a single function which evolves non-linearly.
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That is precisely the relation between the Maxwell-Dirac equations and the multiparticle QM formalism. The latter is a linearized approximation of the former. The entangled states are simply result of the coarse-graining of the nonlinear evolution, which introduces artificial indeterminism in the approximate linear evolution.
I know that you can solve a non-linear differential equation by going to a Hilbert space mechanism. However, what you have completely missed is that in the case of QFT, the Hilbert space is there BY POSTULATE.
Now, you (and Barut and others) can think that this is the same machinery at work, and that people are in fact, without knowing, using this "linearised hilbert space mechanism" to solve, without their knowing, a non-linear differential equation. But that idea is fundamentally flawed for an obvious reason:
the postulates of quantum theory ASSOCIATE A DIFFERENT PHYSICAL STATE to each element of the hilbert state. The non-linear PDE cannot do that. So it could be (it isn't, but the reasons are somewhat difficult to go into) that the linearised system ALSO ALLOWS SOLUTIONS TO THE PDE. But it contains immensely MORE solutions, and BY POSTULATE they are all true, physical states which are distinguishable one from another. This very fundamental postulate of quantum theory makes that it doesn't even matter if the Hilbert system is the result of a linearization or not of a NL PDE. We are now linked directly to the Hilbert space by postulate.
So in any case, the QFT contains many more physical situations than could ever be described by the non-linear PDE ; that's the FUNDAMENTAL CONTENT OF THE SUPERPOSITION PRINCIPLE I have been claiming you don't understand, and of which what you write above is again an illustration.
Now, that doesn't mean that QFT is the "correct" theory, and the NL PDE is the "wrong" theory or vice versa: only experiment can tell. But one thing is sure: the NL PDE doesn't describe the same physical theory as the QFT, which contains immensely more potential physical situations.
You can call them "spurious" but according to quantum theory, they are not. So that's a clear difference between both physical theories.
Now, you were making a claim about a prediction of QFT. If you do so, you should work with QFT, and not with the theory you think should replace it (the NL PDE). And the predictions of QFT are, for this setup, very clear: we have anti-correlation. This can be experimentally right, or it can be wrong. But one thing is sure: QFT predicts anti-correlation.
If you say that you work out a prediction of QFT, but:
- you do not accept the superposition principle
- you do not accept von Neumann's measurement theory
- you do not accept the usual links between systems and their mathematical representation in standard QFT
- you base yourself on another theory (NL PDE) of which you think erroneously that it is the superceding theory of QFT
...
well, then you're not working out a prediction of QFT :-)
If you claim that it does, it can only mean that you don't understand fundamental aspects of QFT, and those aspects are so fundamental, that it makes me conclude that you don't understand the basic postulates of quantum theory in its generality.
Otherwise you wouldn't claim that QFT makes these predictions: you would say that you have another theory, which contains the only "valid" predictions of QFT, and which theory does not predict anticorrelations. Even that would be wrong, but less so. The solutions of the NL PDE are not even in general the "converging solutions" of QFT. But it doesn't matter. The important point is that you recognize that what you are claiming is not a prediction of QFT.
That's all I'm saying.
cheers,
Patrick.
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