SUFFICIENT CONDITIONS FOR QUASI-FREE STATES AND AN IMPROVED UNIQUENESS THEOREM FOR QUANTUM-FIELDS ON SPACE-TIMES WITH HORIZONS

Let omega be a state on the Weyl algebra over a symplectic space. We p rove that if either (i) the ''liberation'' of omega is pure or (ii) th e restriction of omega to each of two generating Weyl subalgebras is q uasifree and pure, then co is quasifree and pure [and, in case (i) is equal to its liberation, in case (ii) is uniquely determined by its re strictions]. [Here, we define the liberation of a (sufficiently regula r) state to be the quasifree state with the same two point function.] Results (i) and (ii) permit one to drop the quasifree assumption in a result due to Wald and the author concerning linear scalar quantum fie lds on space-times with bifurcate Killing horizons and thus to conclud e that, on a large subalgebra of the field algebra for such a system, there is a unique stationary state whose two point function has the Ha damard form. The paper contains a number of further related developmen ts including: (a) (i) implies a uniqueness result, e.g., for the usual free field in Minkowski space. We compare and contrast this with othe r known uniqueness results for this system. (b) A similar pair of resu lts to (i) and (ii) is proven for ''quasiFree'' states and ''libeRatio ns'' where the definition of quasiFree differs from what we call here quasifree in that nonvanishing one point functions are permitted, and the libeRation of a state is defined to be the quasiFree state with th e same one and two point functions. (c) We derive similar results for the canonical anticommutation relations.