vanesch The "piecewise linearised evolution" is correct when you consider the PERTURBATIVE APPROXIMATION[/color] of the functional integral (you know, in particle physics everybody calls it the path integral) using Feynman diagrams. Feynman diagrams (or, for that matter, Wick's theorem) are a technique to express each term in the series expansion of the FULL CORRELATION FUNCTION as combinations of the FREE CORRELATION FUNCTIONS, which are indeed the exact solutions to the linear quantum field parts.
Re-reading that comment, I think you too ought to pause for a moment and step back a bit and connect the dots you already have in front of you[/color]. I will just list them since there are several longer messages in this thread, with an in depth disscussion on each one with a hard fought battle back and forth, where each point took your and other contributors' best shots. And you can read them back, in the light of the totality listed here (and probably as many lesser dots I didn't list), and see what the outcome was.
a)[/color] You recognize here that the QED perturbative expansion is a 'piecewise linear approximation'. The QED field amplitude propagation can't be approximating another piecewise linear approximation, or linear fields. And each new order of the perturbation invalidates the previous order approximation as a possible candidate for the last and the exact evolution. Therefore, there must be a limiting nonlinear evolution, not equal to any of the orders of QED, thus an "underlying" (at the bottom) nonlinear evolution (of QED amplitudes in coordinate basis) being approximated by the QED perturbative expansion, which is (since nothing else is left of all the finite orders of QED expansion, each is invalid, thus not the last word) "some" local classical nonlinear field theory (we're not interpreting what this limit-evolution means, but just establishing its plausible existence).
b)[/color] You also read within the last ten days at least some of the Barut's results on radiative corrections, including the Lamb shift, the crown jewel of QED. There ought to be no great mystery then, what could it be, what kind of nonlinear field evolution is it that QED amplitudes are actually piecewise linearizing.
c)[/color] What is the most natural thing, say a classical mathematician, would have considered if someone had handed him Dirac and Maxwell equations and told him: here are our basic equations, what ought to be done. Would he consider EM field potential A occurring in Dirac equation external, or the Dirac currents in the Maxwell's equations as external? Or would he do exactly what Schroedinger, Einstein, Jaynes, Barut... thought needs to be done and tried to do -- see them as set of coupled nonlinear PDEs to be solved?
d)[/color] Oddly that this most natural and conceptually the simplest way to look at these classical field equations, nonlinear PDEs, fully formed in 1920s, already had the high precision result of Lamb shift in it, with nothing that had to be added or changed in them (all it needed was someone to carry out the calculations for a problem already posed along with the equations, the problem of H atom) -- it had the experimental fact which would come twenty years later.
e)[/color] In contrast, the Dirac-Jordan-Heisenberg-Pauli Quantum theory of EM fields at the time of the Lamb shift discovery (1947), a vastly more complex edifice, failed the prediction, and had to be reworked thorughly in the years after the result, until Dyson's QED finally had managed to put all the pieces together.
f)[/color] The Quantum Optics may not have the magic it shows off with. Even if you haven't got yet to the detector papers, or the Glauber's QO foundation papers, you at least should have the grounds for rational doubt and more questions, what is really being shown by these folks? Especially in view of dots (a)-(e) which imply the Quantum Opticians may be wrong, overly enthusiastic with their claims.
g)[/color] The point (f), considering that the plain classical nonlinear fields, the Maxwell-Dirac equations, already had the correct QED radiative corrections right from the start, and these are much closer to the core of QED, they're its crown jewels, couple orders beyond the orders at which the Quantum Optics operates (which is barely at the level of the Old QED).
h)[/color] In fact, the toppled Old QED (e), has already all that Quantum Optics needs, including the key of Quantum Magic, the remote state collapse. Yet that Old QED flopped and the Maxwell-Dirac worked. Would it be plausible that for the key pride of QED predictions, the Lamb shift, the one which toppled the Old QED, the Maxwell-Dirac just got lucky. What about (b) & (c), lucky again that QED appears to be approximating it, in the formalism through several orders, and well beyond the Quantum Optics level, as well? All just luck? And that Maxwell-Dirac nonlinear fields are the simplest and the most natural approach (d)?
i)[/color] The Glauber's correlations <[G]>, the standard QO "correlations" may not be what they are claimed to be (correlating something being literally and actually counted). Its flat-out vacuum removal procedures are over-subtracting whenever there is a vacuum generated mode. And this is the way the Bell test results are processed.
j)[/color] The photo-detectors are not counting "photons" but are counting photo-electrons and the established non-controversial detector theory neither shows nor claims any non-classical photo-electron count results (it is defined as a standard correlation function with non-negative p-e counts). Entire nonclassicality in QO comes from the reconstructed <[G]>, which doesn't correlate anything literally and actually being counted (the <[C]> is what is being counted). Keep also in mind dots (a)-(i)
k)[/color] The Bell theorem tests seem to be stalled for over three decades. So far they have managed to exclude only the "fair sampling" type local theories (none of such theories ever existed). The Maxwell-Dirac, the little guy from (a)-(j) which does happen to exist, is not a "fair sampling" theory, and more embarrassingly, it even predicts perfectly well what the actual counting data show and agrees perfectly with the <[C]> correlations, the photo-electron counts correlations, which is what the detectors actually and literally produce (within the photo-current amplification error limits).
l)[/color] The Bell's QM prediction is not a genuine prediction, in the sense of giving the range of its own accuracy. It is a toy derivation, a hint for someone to go and do full QO/QED calculation. Indeed such calculations exist, and if one removes <[G]> and simply leaves it at its bare quantitative prediction of detector counts (which would match fairly well the photo-electron counts obtained), the QO/QED predicts the obtained raw counts well and does not predict violation either. There is no prediction of detector counts (the photo-electron counts, the stuff that detectors actually count), not even in principle and not with the most ideal photo-electron counter conceivable, even with 100% QE, which would violate Bell's inequality. No such prediction and it cannot be deduced even in principle for anything that can be actually counted.
m)[/color] The Bell's QM "prediction" does require remote non-local projection/collapse. Without it can't be derived.
n)[/color] The only reason we need collapse at all is the allegedly verified Bell's QM prediction saying we can't have LHVs. Otherwise variables could have values (such as Maxwell-Dirac fields), just not known, but local and classical. The same lucky Maxwell-Dirac of the other dots.
o)[/color] Without the general 'global non-local collapse' postulate, Bell could not get state of particle (2) to become |+> for the sub-ensemble of particle (2) instances, for which the particle (1) gave (-1) result (and he does assume he can get that state, by Gleason's theorem by which the statistics determines the state; he assumes statistics of |+> state on the particle 2 sub-ensemble for which the particle 1 gave -1). Isn't it a bit odd that to deduce non-local QM prediction one needs to use non-local collapse as a premise? How could any other conclusion be reached, but the exclusion of locality, with the starting premise of non-locality?
p)[/color] Without collapse postulate, no Bell's QM prediction, thus no measurement problem (the Bell's no-go for LHV), thus no reason to keep collapse at all. The Born rule as an approximate operational rule would suffice (e.g. the way a 19th century physicist might have defined it for the light measurements: the incident energy is proportional to photocurrent, which is correct empirically and theoretically, the square law detection).
q)[/color] The Poissonian counts of photo-electrons in all Quantum Optics experiments preclude Bell's inequality violation ever in photon experiments. The simple classical model, the same Maxwell-Dirac from (a)...(p) points, the lucky one, predicts somehow exactly what you actually measure, the detector counts, with no need for untested conjectures or handwaving or euphemisms, all it uses is the established detector theory (QED based or Maxwell-Dirac based) and a model of an unknown but perfectly existent polarization. And it gets lucky again. While the QO needs to appologize and promise yet again, it will get it just as soon as the detectors which count Glauber's <[G]> get constructed, soon, no question about it.
r)[/color] Could it be all luck for Maxwell-Dirac? All the points above, just dumb luck? And the non-linear fields actually don't contradict any QED quantitative prediction, in fact agree to an astonishing precision with QED. QED amplitude evolution even converges to Maxwell-Dirac, as far as anyone can see and as precisely as anything that gets measured in this area. The Maxwell-Dirac disagrees only with the collapse postulate (the general non-local projection postulate), for which there is no empirical evidence, and for which there is no theoretical need of any sort, other than the conclusions it creates by itself (such as Bell's QM prediction or various QO non-classicalities based on <[G]> and collapse).
s)[/color] Decades of physicists have banged their heads to figure out how can QM work like that. And no good way out, but multiple universes or observers mind when all shots have been fired and all other QM "measurement theory" defense lines have fallen. It can't be defended with a serious face. That means, no way to solve it as long as the collapse postulate is there, otherwise someone would have thought it up. And the only thing that is holding up the whole of the puzzle is the general non-local collapse postulate. Why do we need it? As an approximate operational rule (as Bell himself advocated in his final QM paper) with only local validity it would be perfectly fine, no puzzle. What does it do, other than to uphold the puzzle of its own making, to earn its pricey keeps? What is that secret invaluable function it serves, the purpose so secret and invaluable that no one knows for sure what it is, to be able to explain it as plainly and directly as 2+2, but everyone believes that someone else knows exactly what it is? Perhaps, just maybe, if I were to dare to wildly conjecture here, there is none?
t)[/color] No single or even a few dots above may be decisive. But all of it? What are the odds?
PS: I will have to take at least a few weeks break from this forum. Thanks again to you 'vanesch' and all the folks who contributed their challenges to make this a very stimulating discussion.[/color]