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Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. (English) Zbl 0861.53074
In this paper the authors study in some detail quasifree Hadamard states on spacetimes with bifurcate Killing horizons. These manifolds are globally hyperbolic spacetimes possessing the structure necessary to naturally generalize the ideas used in previous work by B. S. Kay [Commun. Math. Phys. 100, No. 1, 57-81 (1985; Zbl 0578.46062)] to study the thermal properties of quasifree states on Schwarzschild spacetimes.
After establishing necessary geometrical properties of this class of spacetimes in Chapter 2 and then necessary functional analytic properties of quasifree (KMS) states on the CCR-algebra (defined in the usual manner on these globally hyperbolic spacetimes) in Chapter 3 (where a rigorous definition of Hadamard states finally appears in the literature), the authors prove that any quasifree Hadamard state that is invariant under the flow determined by the Killing vector field must be unique and pure. Moreover, if the spacetime admits a certain discrete reflection symmetry (which obtains in analytic spacetimes) such states must be KMS with respect to the Killing flow at the Hawking temperature \(T=\kappa/2\pi\), when restricted to the right or left wedge subalgebras of the global CCR-algebra. (\(\kappa\) is the surface gravity of the horizon.) The authors also prove that Killing-flow-invariant Hadamard states do not exist on (the CCR-algebra on) Schwarzschild-de Sitter spacetimes and on the globally hyperbolic region of Kerr spacetimes. Indeed, for each of these two classes of spacetimes, two independent, rigorous arguments (which have natural generalizations to different classes of spacetimes) for the nonexistence of such states are given, providing additional insight into the underlying reasons for the absence of such states. The question of existence in the case of other classes of spacetimes with bifurcate Killing horizons is left open.
This paper is an important contribution to the small body of mathematically rigorous results known about quantum field theories on curved spacetimes.

MSC:
53Z05 Applications of differential geometry to physics
81T20 Quantum field theory on curved space or space-time backgrounds
81T05 Axiomatic quantum field theory; operator algebras
83C47 Methods of quantum field theory in general relativity and gravitational theory
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