Constructing Hadamard states via an extended M{\o}ller operator

TL;DR

This paper develops an extended Møller operator to construct isomorphisms between solution spaces and algebras of scalar fields with different potentials, enabling the transfer of Hadamard states while preserving invariance properties.

Contribution

It introduces a non-canonical extension of the Møller operator to relate scalar field theories with different potentials and constructs Hadamard states accordingly.

Findings

Constructed isomorphisms between solution spaces and algebras for different potentials.

Enabled transfer of Hadamard states between theories with different potentials.

Provided conditions for removing cut-off dependence via adiabatic limits on static spacetimes.

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 M{\o}ller 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. 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 commutes with the vector field built out of the normal vectors to a family of Cauchy surfaces, foliating the underlying manifold. Eventually we discuss a sufficient condition to remove on static spacetimes the dependence on the cut-off via a suitable adiabatic limit.