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The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes. (English) Zbl 0578.46062
We consider the Klein-Gordon equation (\(m\geq 0\)) on the double Schwarzschild wedge of the Kruskal spacetime, and construct the Hartle–Hawking state \(\omega_ H\) as a thermal state relative to the Boulware quantization. We prove that, on the double wedge, \(\omega_ H\) is a pure state, and in the corresponding representation, the left- and right-wedge \(C^*\) algebras each have the Reeh–Schlieder property, while the corresponding von Neumann algebras are type \(III_ 1\) factors which are dual to (i.e. commutants of) each other. We discuss the extent to which these properties may generalize to non-quasi-free field theories. Pursuing the Rindler–Fulling–Unruh analogy with the Klein–Gordon equation \((m>0)\) in (\(d\)-dimensional) flat spacetime, we establish an explicit formula for the Minkowski vacuum on a spacelike double wedge as a thermal state relative to the Fulling quantization. We also treat the case \(d=2\), \(m=0\) of this formula since this is essential input for a paper with Dimock on scattering theory for the quantum Klein–Gordon equation on the Schwarzschild metric.
Note that the proof of Theorem 3.2 (but not its statement) contains an error. An erratum will appear on this and on the paragraph immediately following Theorem 3.2.
Reviewer: Bernard S. Kay

MSC:
81T20 Quantum field theory on curved space or space-time backgrounds
46L60 Applications of selfadjoint operator algebras to physics
81T05 Axiomatic quantum field theory; operator algebras
83C47 Methods of quantum field theory in general relativity and gravitational theory
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