The Reeh-Schlieder theorem and quantum geometry

Question

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"?

Answer 1

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:

Reeh-Schlieder theorem (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 gymnastics 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 being 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...

Answer 2

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.