# The Reeh-Schlieder theorem and quantum geometry

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There have been some very nice discussions recently centered around the question of whether gravity and the geometry and topology of the classical world we see about us, could be phenomena which emerge in the low-energy limits of a more fundamental microscopic theory.

Among these, @Tim Van Beek's reply to the question on "How the topology of space [time] arises from more fundamental notions" contains the following description of the Reeh-Schlieder theorem:

It describes "action at a distance" in a mathematically precise way. According to the Reeh-Schlieder theorem there are correlations in the vacuum state between measurements at an arbitrary distance. The point is: The proof of the Reeh-Schlieder theorem is independent of any axiom describing causality, showing that quantum entanglement effects do not violate Einstein causality, and don't depend on the precise notion of causality. Therefore a change in spacetime topology in order to explain quantum entanglement effects won't work.

which is also preceded with an appropriate note of caution, saying that the above paragraph:

... describes an aspect of axiomatic quantum field theory which may become obsolete in the future with the development of a more complete theory.

I had a bias against AQFT as being too abstract an obtuse branch of study to be of any practical use. However, in light of the possibility (recently discussed on physics.SE) that classical geometry arises due to the entanglement between the degrees of freedom of some quantum many-body system (see Swingle's paper on Entanglement Renormalization and Holography) the content of the Rees-Schilder theorem begins to seem quite profound and far-sighted.

The question therefore is: Does the Rees-Schlieder theorem provide support for the idea of building space-time from quantum entanglement? or am I jumping the gun in presuming their is some connection between what the theorem says and the work of Vidal, Evenbly, Swingle and others on "holographic entanglement"?

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user user346
retagged Apr 1, 2014

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No, as Peter said, you are jumping the gun: A profound precondition of Reeh-Schlieder theorems is that spacetime is a smooth Lorentzian manifold. You can find a lot of information on the nLab:

The most pedagogical exposition I know is the paper by Summers referenced there:

• Stephen J. Summers: "Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State" (arXiv.)

There is also a proof on that page for the vacuum representation in Minkowski spacetime based on the Haag-Kastler axioms, which shows that the proof is basically mathematical gynmnastics starting with the axioms, that invokes some heavy machinery like the SNAG-theorem, the edge of the wedge theorem and a Lesbesgue dominated convergence theorem for spectral integrals (as you can see: all tools from calculus). The proof is mainly there to illustrate that the axiom of locality is not needed in the first part of the proof, as claimed by Halvorson in the paper referenced on the page. So, this version of the Reeh-Schlieder follows directly from the axioms :-)

In curved spacetime you have to replace the axioms that use the Poincarè group with versions that make use of the local structure only, which can be done, but again this relies heavily on the structure of a smooth manifold. (BTW: QFT on curved spacetimes has produced a lot of results that are very important for quantum gravity research, beside beeing useful by providing a different context for QFT in flat spacetimes.)

If and how any of this will be useful for future theories time will tell, but I think that understanding the properties of the vacuum state in AQFT as explained by Summers in his paper should be very useful for anyone working in QFT or quantum gravity. For example this theorem is the clearest explanation I know of, of how and why Einstein causality and quantum entanglement are complementary concepts. Locality in QFT is one of the most profound, difficult and misunderstood concepts...

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user Tim van Beek
answered Apr 30, 2011 by (505 points)
Hi @Tim, thanks for the answer. I was expecting that you and @Peter would be able to shed more light on the issue, and you have done just that. Though I will keep looking for a better understanding of this topic. Thanks for that Summers' paper. It seems very readable. I'm on the second page so far and if I manage to take a bigger bite out of it then hopefully my next question on this topic will be slightly less ill-informed ;) The main sticking point seems to be the assumption of a flat background. In curved space the notion of positive/negative energy modes becomes an observer dependent ...

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user user346
... notion. This is the source of particle "production" in curved spacetimes. The vacuum state of observer A, becomes an excited state w.r.t. the vacuum of observer B. What is not clear to me is whether the negative energy modes w.r.t to the vacuum of A, can be become the positive energy modes w.r.t to B - for some choice of A and B. Back to basics I go ...

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user user346
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No. The Reeh-Schlieder theorem is a consequence of the Wightman axioms and of the Haag-Kastler axioms, both of which assume a Minkowski space, complete with the trivial metric structure, underlies everything. Variations of these axiomatic systems either allow or do not allow an analogue of the Reeh-Schlieder theorem to be derived. The idea that we might build space-time from quantum entanglement, while conceivable, presumably introduces either an axiomatic or some less well-defined mathematical structure, which will then either allow or not allow a Reeh-Schlieder-like theorem to be derived. Some thought this afternoon has suggested to me no way in which the Reeh-Schlieder theorem might of itself suggest a particular underlying mathematical structure. I take the lack of other Answers to suggest that no-one else immediately thought of such a thing either.

I personally take the Reeh-Schlieder theorem to say that there is a sense in which modulations of vacuum fluctuations are nonlocal, in a way that is weakly analogous to the nonlocality of analytic complex functions, for which the equivalent to the Reeh-Schlieder theorem is the possibility of analytic continuation. Analyticity, in a relatively sophisticated sense, is fundamental to the proof of the Reeh-Schlieder theorem. This is talk rather than mathematics, but to me it suggests the negative reply that I began with.

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user Peter Morgan
answered Apr 29, 2011 by (1,085 points)
@Peter thanks for that answer. I asked the question because I found the implications of the RS theorem very startling, even if only in the limited context of flat space. Also the building space-time from entanglement is NOT my idea. The first place I saw it was in Raamsdonk's paper. However, I was smitten with that possibility and still am. If AQFT can shed more light on this prospect - than plain-vanilla QFT - then it (AQFT) becomes something to put on my "to study" list. Also, even though your answer is in the negative, you do point out the best lines of attack - a comparison between ...

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user user346
... the mathematical structures underlying the RS theorem and the structures which arise from space-time entanglement; and the need for generalizing the RS theorem for curved space-times.

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user user346
As a general rule, I don't involve myself much with QFT on CSTs, because I think they add too much for too little, although I got a lot from Wald's book on the subject. I consider that building a space-time geometry in any serious way requires the geometry to be both stochastic and dynamical in some way or another, which is several steps beyond taking CST as a background. Although many people seem to find Reeh-Schlieder bizarre, I find it a rather ho-hum consequence of the axioms, particularly of the restriction to positive frequency, which was more-or-less the point of my 2nd paragraph.

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user Peter Morgan
By the way, have you heard of Hegerfeldt nonlocality? For example, "Causality, particle localization and positivity of the energy", arXiv:quant-ph/9806036, and references therein. The proof of Hegerfeldt nonlocality is much more transparent than proofs of the Reeh-Schlieder theorem.

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user Peter Morgan
@Peter your reply was the first. @Tim's is also nice, but since I have no further criterion to determine which is objectively better, the first one gets the bounty. Hooray!!! I did not know of Hegerfeldt's work until now. Thanks for that reference. Apparently the proof of the RST ends up depending only on the positivity of the Hamiltonian. Since this should be true in curved spacetimes, or at least in those which satisfy the conditions for the positive energy theorem to hold true, it should be possible to generalize the RST to curved space. Of course, if what Hegerfeldt is talking about ...

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user user346
... is physically distinct from the implications of the RST, then I'm jumping the gun (again!) in making this assertion. Please do educate me. I'm an infant when it comes to AQFT.

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user user346
I disagree with you account that Reeh-Schlieder is due to the Wightman axioms. That is in fact not true, because although situated on a Minikowski spacetime, Wightman axioms give the "field" a fundamental role. The Haag-Kastler axioms, on the other hand, give the algebra of observables a fundamental role ,which is how Reeh-Schlieder is derived. Furthermore, the vacuum is allowed to be an eigenstate in Wightman's framework; this is not true in the case of Reeh-Schlieder, where the vacuum cannot be an eigenstate .

This post imported from StackExchange Physics at 2014-04-01 16:45 (UCT), posted by SE-user r.g.

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