The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes
type: Item
instance of: scholarly article
title: The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes
zbMATH Open document ID: 0578.46062
DOI: 10.1007/BF01212687
author: Bernard S. Kay
published in: Communications in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 81T20
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 83C47
zbMATH DE Number: 3925600
zbMATH Keywords: Klein-Gordon equation
zbMATH Keywords: double Schwarzschild wedge of the Kruskal spacetime
zbMATH Keywords: Hartle-Hawking state
zbMATH Keywords: thermal state relative to the Boulware quantization
zbMATH Keywords: left- and right-wedge \(C^*\) algebras
zbMATH Keywords: Reeh-Schlieder property
zbMATH Keywords: Rindler-Fulling-Unruh analogy
zbMATH Keywords: Minkowski vacuum on a spacelike double wedge
zbMATH Keywords: thermal state relative to the Fulling quantization
zbMATH Keywords: Schwarzschild metric
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3925600