The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes

Abstract

We consider the Klein-Gordon equation (m≧0) on the double Schwarzschild wedge of the Kruskal spacetime, and construct the Hartle-Hawking stateω H as a thermal state relative to the Boulware quantization. We prove that, on the double wedge,ω H is a pure state, and in the corresponding representation, the left- and right-wedgeC* algebras each have the Reeh-Schlieder property, while the corresponding von-Neumann algebras are typeIII 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 cased=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.