Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems
Alexander Strohmaier, Rainer Verch, Manfred Wollenberg
TL;DR
The paper develops a rigorous framework connecting analytic microlocal analysis to the Reeh-Schlieder property for quantum fields on real analytic spacetimes. By formulating an analytic microlocal spectrum condition (aμSC) and interpreting n-point functions as Hilbert-space valued distributions with analytic wavefront sets, it proves that aμSC implies the Reeh-Schlieder property in the curved-spacetime setting without presuming a specific equation of motion. For the Klein-Gordon field, the authors establish precise equivalences between μSC, Hadamard conditions, and their analytic counterparts, and show that quasifree ground- or KMS-states on real analytic stationary spacetimes satisfy aμSC. The results extend the utility of microlocal methods to curved backgrounds, enabling robust local-to-global state properties and enhancing the understanding of Hadamard states in quantum field theory on curved spacetimes.
Abstract
We show in this article that the Reeh-Schlieder property holds for states of quantum fields on real analytic spacetimes if they satisfy an analytic microlocal spectrum condition. This result holds in the setting of general quantum field theory, i.e. without assuming the quantum field to obey a specific equation of motion. Moreover, quasifree states of the Klein-Gordon field are further investigated in this work and the (analytic) microlocal spectrum condition is shown to be equivalent to simpler conditions. We also prove that any quasifree ground- or KMS-state of the Klein-Gordon field on a stationary real analytic spacetime fulfills the analytic microlocal spectrum condition.