The Reeh-Schlieder theorem is a result of axiomatic quantum field theory concerning observables localized in arbitrarily small open
sets of Minkowski space. In the usual relativistic
vacuum formulation, including locality, translation and Lorentz covariance, and the
positive-energy condition, the vacuum vector 
is a cyclic vector for
the local operator algebra 
of every nonempty open set 
(Reeh and Schlieder 1961).
Thus, in the vacuum Hilbert space 
,
 |
where the bar denotes set closure in the Hilbert space norm. Equivalently, applying local operators
to 
gives a dense subset of 
.
If the spacelike complement of 
contains a nonempty open set,
is also a separating vector for 
:
 |
To see this, locality makes 
commute with every operator 
in an algebra localized in the complement. Then 
, and cyclicity in that region implies that
the bounded operator 
vanishes on a dense subset and hence
everywhere. The local algebras here are algebras of bounded
operators; individual quantum fields need not be bounded.
