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Background

So I recently asked a question about relativistic quantum mechanics and the answerer invoked micro-causality (from QFT) to show me that the assumption the information would propagate infinitely fast due to the sudden approximation was false. After talking to a friend working in quantum information theory he said that even in regular quantum mechanics there are Lieb-Robinson velocity bounds:

The existence of such a finite speed was discovered mathematically by Lieb and Robinson, (1972). It turns the locality properties of physical systems into the existence of, and upper bound for this speed.

When I compare this to locality statement from QFT (microcausality):

A basic characteristic of physics in the context of special relativity and general relativity is that causal influences on a Lorentzian manifold spacetime propagate in timelike or lightlike directions but not spacelike.

The fact that any two spacelike-separated regions of spacetime thus behave like independent subsystems is called causal locality ... Microcausality condition a requirement that the causality condition (which states that cause must precede effect) be satisfied down to an arbitrarily small distance and time interval.

Question

Since both are statements of locality I'm curious if there is any relation between the two? Like would change into the other if you take some kind of limit as in the comment section?

... I would guess that the Lieb-Robinson bound in lattice QFT does become the usual causality property in the continuum limit. We could probably study this in a simple exactly-solvable example, like the lattice QFT of a free scalar field. ...

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    $\begingroup$ Can you make the question more precise? I mean, what would be a good answer: "Yes, a speed limit for propagation of any cause implies micro-causality. Note, however, that Lieb-Robinson-bounds only give an 'almost always' speed limit, in the sense that there is an exponential tail which propagates faster." Would that answer your question? If not, what are you looking for? Other than that, the two are not related in the sense that one would change into the other if you take some kind of limit. $\endgroup$
    – Norbert Schuch
    Commented Jan 17, 2021 at 14:57
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    $\begingroup$ I haven't analyzed this in detail, but I would guess that the Lieb-Robinson bound in lattice QFT does become the usual causality property in the continuum limit. We could probably study this in a simple exactly-solvable example, like the lattice QFT of a free scalar field. But again, I've never checked this in detail, so I might be missing some interesting subtleties. $\endgroup$
    – Chiral Anomaly
    Commented Jan 17, 2021 at 15:57
  • $\begingroup$ @NorbertSchuch Maybe you can counter the guess by Chiral Anomaly? In your answer? $\endgroup$
    – More Anonymous
    Commented Jan 17, 2021 at 16:09
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    $\begingroup$ I just found the recent paper Lieb-Robinson bounds imply locality of interactions (arxiv.org/abs/2006.10062). That paper stops short of establishing my guess, because they don't take a strict continuum limit. They show that the exponentially-small tails (which @NorbertSchuch's comment correctly highlights) are small enough to be unimportant on scales much larger than the lattice spacing, with some technical definition of "unimportant," but they leave the strict continuum limit as an exercise for the reader. $\endgroup$
    – Chiral Anomaly
    Commented Jan 17, 2021 at 16:37
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    $\begingroup$ @ChiralAnomaly You can perfectly well set up a lattice model in relativistic quantum mechanics which has a Lieb-Robinson velocity different from the speed of light. That's why I don't think it generically gives rise to the speed of light in a QFT-type limit. It is likely possible to set up a continuum limit which gives the speed of light, but it is unclear to me whether such a setup would be the natural one. (Similarly, I don't know of a way to obtain relativistic continuum quantum mechanics as a limit of non-relativistic lattice qm, and this would suggest such a limit would exist.) $\endgroup$
    – Norbert Schuch
    Commented Jan 17, 2021 at 16:52

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