vanhees71 said:
The first cited paper investigates relativistic classical fields interpreting them in terms of first-quantized wave mechanics a la Schrödinger in the non-relativistic case. I don't think that in the year 2018 we still have to discuss why this doesn't work and why one has to employ relativistic quantum field theory to precisely cure this problem with apparent acausality. It's discussed in any textbook (see, e.g., Peskin-Schroeder).
One can't talk about relativistic quantum field theory "precisely", at least in 3+1-dimensions, except about free quantum fields, because interacting relativistic QFTs, again in 3+1-dimensions, only exist as asymptotic expansions, for which discussion is necessarily imprecise. In 1+1- or 2+1-dimensions, where there are models of the Wightman axioms, the Reeh-Schleider theorem is effectively the same as Hegerfeldt nonlocality.
To discuss free Wightman fields in 3+1-dimensions, one can consider as a simplest example the variance ##\hat\phi_f^2## of an observable ##\hat\phi_f=\hat\phi_f^\dagger## in the state ##\frac{\langle 0|\hat\phi_g^\dagger\hat A\hat\phi_g|0\rangle}{\langle 0|\hat\phi_g^\dagger\hat\phi_g|0\rangle}##, that is, the expression ##\frac{\langle 0|\hat\phi_g^\dagger\hat\phi_f^2\hat\phi_g|0\rangle}{(g,g)}=(f,f)+2\frac{(g,f)(f,g)}{(g,g)}##, where ##(f,g)=\langle 0|\hat\phi_f^\dagger\hat\phi_g|0\rangle## is a vacuum expectation value (which is enough to fix the Gaussian free field.)
This expression shows that the variance of the observable ##\hat\phi_f## is modified by the absolute value ##|(f,g)|^2## in the vector state ##\hat\phi_g|0\rangle/\sqrt{(g,g)}##. Of course it is the case that measurements ##\hat\phi_f## and ##\hat\phi_g## commute if ##f## and ##g## are at space-like separation, but ##|(f,g)|^2## in general is non-zero. Another way to state this is that ##[\hat\phi_f,\hat\phi_g|0\rangle\langle 0|\hat\phi_g]\not=0## even if ##f## and ##g## are at space-like separation. This simple computation shows that the relationship of state preparation to measurement is different from the relationship between two measurements; it can be dismissed as about free fields, which can be said to be not physically relevant, and the Reeh-Schlieder theorem (which subsumes this simple computation) can be dismissed as about Wightman fields, which can also be said to be not physically relevant, however interacting QFT would agree that ##[\hat\phi_f,\hat\phi_g|0\rangle\langle 0|\hat\phi_g]\not=0## in general, so there seems to me to be a
prima facie case for there being some value in identifying and characterizing different kinds of nonlocality, not only repeating "microcausality", powerful though that indubitably is.
Finally, you're right about the first Hegerfeldt paper I cited; in future I will cite only the second paper, which I think enough applies to the relativistic case as well as to the nonrelativistic case to be at least of historical interest to anyone who wishes to understand nonlocality/locality in QFT.