Letters in Mathematical Physics

, Volume 106, Issue 11, pp 1587–1615 | Cite as

Constructing Hadamard States via an Extended Møller Operator

Article

Abstract

We consider real scalar field theories, whose dynamics is ruled by normally hyperbolic operators differing only by a smooth potential V. By means of an extension of the standard definition of Møller operator, we construct an isomorphism between the associated spaces of smooth solutions and between the associated algebras of observables. On the one hand, such isomorphism is non-canonical, since it depends on the choice of a smooth time-dependant cut-off function \({\chi}\). On the other hand, given any quasi-free Hadamard state for a theory with a given V, such isomorphism allows for the construction of another quasi-free Hadamard state for a different potential. The resulting state preserves also the invariance under the action of any isometry, whose associated Killing field \({\xi}\) is complete and fulfilling both \({\mathcal{L}_\xi V=0 \,\, {\rm and} \,\, \mathcal{L}_\xi\chi=0}\). Eventually, we discuss a sufficient condition to remove on static spacetimes, the dependence on the cutoff via a suitable adiabatic limit.

Keywords

quantum field theory on curved backgrounds Hadamard sates Møller operator 

Mathematics Subject Classification

81T20 81T05 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Dipartimento di FisicaUniversità degli Studi di PaviaPaviaItaly
  2. 2.Istituto Nazionale di Fisica Nucleare-Sezione di PaviaPaviaItaly
  3. 3.Dipartimento di MatematicaUniversità di GenovaGenoaItaly
  4. 4.Istituto Nazionale di Fisica Nucleare-Sezione di GenovaGenoaItaly

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