In particular, if RS were false, then there would exist a state orthogonal to all with all supported in , Witten’s proof of this is by contradiction.

The Reeh-Schlieder theorem (RS) is the remarkable statement that even if we restrict to an arbitrarily small open set, and the support of the functions to the corresponding neighborhood of in spacetime, the states still suffice to generate the vacuum sector of the full theory! This is often phrased as the statement that by acting on the vacuum with operators localized within some small region - say, the room in which you’re reading this - we can create the Moon!

Now consider the vacuum sector on, that is, the space of states (1) with the support of in. However, the Cauchy problem can still be made well-posed within the Einstein static universe, which only covers half the spacetime). Hence while any spacelike slice does cover the whole space, there nevertheless exist null geodesics from timelike infinite that do not intersect any point thereupon. (As a side-comment for holography, note that there are no Cauchy slices in AdS, since data can always sneak in from infinity! This follows from the fact that the boundary is timelike rather than null or spacelike. Quantum mechanically, this is the “time-slice axiom” of QFT, and encodes the physical expectation that there exists a dynamical law that enables one to compute fields at arbitrary time given the set of fields at some time slice (e.g., if is taken to be the surface, one could take the functions to have support in some open neighborhood of, say for some small positive ). This defines the vacuum sector of the theory.Ĭlassically, one formulates the initial data for a theory on a Cauchy (i.e., complete spacelike) hypersurface.

Hence states of the formĪre sufficient to generate, i.e., any state in can be approximated arbitrarily well by linear combinations of. The smearing ensures that the states have finite norm and are thus well-defined members of Hilbert space. By “local”, we mean that a field operator is smeared against some smooth function with finite spacetime support to create the smeared operator, where is the spacetime dimension and. Note that is not necessarily the full Hilbert space, which can generically contain other superselection sectors. The Reeh-Schlieder theorem is perhaps the most notorious result in algebraic quantum field theory, simultaneously one of the least intuitive and most fundamental.ĭenote the vacuum sector of Hilbert space, which consists of all states that can be created from the vacuum by local field operators.

Timelike infinity