# A Is there a local interpretation of Reeh-Schlieder theorem?

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#### Elemental

[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 is sufficient, 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.]
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.

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

#### Peter Morgan

Gold Member
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?
A restatement, then: from a QFT PoV in which microcausality between measurements being satisfied is the only kind of dynamical locality worth talking about, QFT is dynamically local and there isn't a problem. The nonlocal 2-point and higher correlations that there are in the vacuum (and in modulated form in all states) are just a property of the state, which is not a dynamical nonlocality. This shades us towards superdeterminism, in a quantum mechanical form, but in the QFT context no-one, AFAIK, objects to that in the physics or philosophy of physics literature. If you're willing to adopt that "minimal" QFT PoV, stop now, because it's pretty much adequate.
If, however, we adopt a PoV in which superdeterminism is somewhat objectionable even in a quantum mechanical form (in a classical form superdeterminism is most often taken to be objectionable, but with some exceptions, 't Hooft perhaps being the most well-known), then some or all of the nonlocal correlations presumably have to be accounted for dynamically, albeit whatever we introduce to give that account must, empirically, be Lorentz invariant as an emergent stochastic theory. Empirical considerations further modify this, however, because it's at least for now unclear what experiments we could do to split the nonlocal correlations into superdeterministic and stochastically Lorentz invariant FTL parts — so there's arguably no point worrying about it, we may as well just adopt the "minimal" QFT PoV.
That's not as definitive as you wanted, I'm sure! They do go away, or they don't, depending on what your other commitments are. I'm sure other commitments you might have, not just my choice here of anti-quantum-superdeterminism, could also interact with a commitment to a minimal QFT PoV.

#### A. Neumaier

a subsequent measurement outside the light cone has non-zero probability of finding that the system has propagated faster than light.
This is a problem with Born's rule, not with quantum field theory. It shows that at finite times (i.e., outside its use to interpret asymptotic S-matrix elements), Born's rule cannot be strictly true in relativistic QFT. The main reason may be that for interacting QFTs, the particle concept (and hence the Newton-Wigner operator) is only asymptotically valid - i.e., under conditions where particles are essentially free.
https://www.physicsforums.com/posts/5923754/
https://www.physicsforums.com/posts/5926655/
discussing limitations of Born's rule.

#### Elemental

If you're willing to adopt that "minimal" QFT PoV, stop now, because it's pretty much adequate.
If, however, we adopt a PoV in which superdeterminism is somewhat objectionable even in a quantum mechanical form (in a classical form superdeterminism is most often taken to be objectionable, but with some exceptions, 't Hooft perhaps being the most well-known), then some or all of the nonlocal correlations presumably have to be accounted for dynamically....
Thanks for the clarifications! Well, I don't think I'll "stop now" because I "know" I have free will, so instead I'll keep plowing through these intractable papers.

#### Demystifier

2018 Award
Some interesting papers on physical and conceptual aspects of Reeh-Schlieder theorem:

M. Redhead, The Vacuum in Relativistic Quantum Field Theory, PSA 1994, Volume 2, pp. 77-87

G.N. Fleming, Reeh-Schlieder Meets Newton-Wigner (free pdf can be found by google)

H. Halvorson, Reeh-Schlieder Defeats Newton-Wigner: On alternative localization schemes in relativistic quantum field theory, https://arxiv.org/abs/quant-ph/0007060v1

#### Elemental

Thanks for the references, Demystifier! So I do see a lot of similarities with Hegerfeldt's theorem. It looks like G. N. Fleming attempts to overcome Reeh-Schlieder localization problems by postulating Newton-Wigner fields, which appear local very much like Newton-Wigner states are local in relativistic quantum mechanics (but only at one instant of time in only one inertial frame). But H. Halvorson disputes their usefulness. If I recall, G. N. Fleming held the view that dismissing the localization problems via resort to quantum field theory was just "sweeping them under the rug" (can't find the reference right now), and I think I see why.

#### Demystifier

2018 Award
If I recall, G. N. Fleming held the view that dismissing the localization problems via resort to quantum field theory was just "sweeping them under the rug" (can't find the reference right now), and I think I see why.
I don't know whether Fleming said that, but I said something similar in https://arxiv.org/abs/quant-ph/0609163 , last sentence in Sec. 8.3.

"Is there a local interpretation of Reeh-Schlieder theorem?"

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