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Hegerfeldt's theorem holds under extremely general conditions, including special relativity, as shown, e.g., in his paper G.C. Hegerfeldt, Phys. Rev. D 10, 3320 (1974), so it is certainly not limited to the non-relativistic regime. The theorem basically states that in any first quantized theory an initially localized wave packet with frequency components bounded from below (e.g., an eigenstate of the Newton-Wigner position operator), will, under time evolution, spread out so that it has tails outside the light cone of the initial localized region; thus, a subsequent measurement outside the light cone has non-zero probability of finding that the system has propagated faster than light.[The difficulty with Hegerfeldt is that none of his many papers are definitively enough stated or proved. Have a look at his paper arXiv:quant-ph/9809030, for example, which I believe is his most recent. Because of the way Hegerfeldt does things, one hears it said that Hegerfeldt only applies to nonrelativistic QM, however it more seems that positive frequency issufficient, the (stronger) spectrum condition is not required. The connection of positive frequency with analyticity through the Hilbert transform is relatively more elementary than working with the spectrum condition.]

I had the impression that most theorists familiar with the Hegerfeldt theorem believe that this problem is overcome in quantum field theory. For instance,

On the other hand,It's the other way around, and this argument can be found in standard textbooks like Peskin&Schröder: Because time evolution with using √^→p2+m2p→^2+m2\sqrt{\hat{\vec{p}}^2+m^2} as an Hamiltonian in a putative 1st-quantization formulation of relativistic QT (which I call relativistic QM) leads to non-locality and breaks causality even for free fields, one concludes that one has to include the negative-frequency modes into the came, and then the observation of a stable world, i.e., the boundedness of the Hamiltonian of particles from below, forces us to use the 2nd-quantization formulation, i.e., QFT, which I'd call the only physically sensible relativistic QT we know of. It also allows for microcausality and validity of the linked-cluster theorem for the S-matrix, which clearly shows that interactions in QFT are indeed described as local interactions. There are no "spooky actions at a distance" as Einstein thought in the modern formulation of QFT, aka the "Standard Model".

I am trying to understand where this leaves us. Is the Reeh-Schlieder theorem an analog of the Hegerfeldt theorem for quantum field theory, implying that the causality problems suggested by Hegerfeldt don't go away after all? I have to admit to finding the paper referenced by the original poster, https://arxiv.org/pdf/1803.04993.pdf, difficult, at least on initial reading.I agree with all of this, which is an entirely consistent way of discussing QFT, but it's a perspective that I consider to be laden with conventions. What might be called the "Einstein conventions" are also entirely consistent, and we can rigorously transform from one to the other (arguably this is what is done in my arXiv:1709.06711 for the free EM, Dirac, and complex KG quantum and random fields, which is being not discussed here on PF; I'll propose that the math of the exact transformations there implicitly defines what the Einstein conventions might be), but within the Einstein conventions there is a precise kind of Lorentz invariant nonlocality and other properties are transformed (including that the positivity of the quantum Hamiltonian operator becomes the positivity of the Hamiltonian function). The conventions you are pressing for, almost insisting upon, which might be crudely stated as the Correspondence Principle and all its consequences, have been supremely successful for the last 90 years, but I suggest that a significant part of the progress in our understanding of and in our ability to engineer using quantum physics over the last 30 years, say, has been through considering alternative conventions, in some of which the effective nonlocality of a state can be considered something of a resource.