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 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