Sufficient conditions for quasifree states and an improved uniqueness theorem for quantum fields on space–times with horizons

Abstract

Let ω be a state on the Weyl algebra over a symplectic space. We prove that if either (i) the ‘‘liberation’’ of ω is pure or (ii) the restriction of ω to each of two generating Weyl subalgebras is quasifree and pure, then ω is quasifree and pure [and, in case (i) is equal to its liberation, in case (ii) is uniquely determined by its restrictions]. [Here, we define the liberation of a (sufficiently regular) 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 fields on space–times with bifurcate Killing horizons and thus to conclude 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 Hadamard form. The paper contains a number of further related developments including: (a) (i) implies a uniqueness result, e.g., for the usual free field in Minkowski space. We compare and contrast this with other known uniqueness results for this system. (b) A similar pair of results to (i) and (ii) is proven for ‘‘quasiFree’’ states and ‘‘libeRations’’ 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 the same one and two point functions. (c) We derive similar results for the canonical anticommutation relations.