On the adiabatic limit of Hadamard states

TL;DR

This paper studies how Hadamard states for Klein-Gordon fields evolve under slow changes in background parameters, proving the vacuum state's stability in the adiabatic limit and extending results to massless cases.

Contribution

It extends the adiabatic theorem to the symplectic framework for Klein-Gordon fields and analyzes the adiabatic limit for varying mass and background metrics.

Findings

Adiabatic limit of vacuum states remains vacuum when the field stays massive.

The method applies to cases with mass variation, including massless scenarios.

The approach generalizes to thermal and other Hadamard states.

Abstract

We consider the adiabatic limit of Hadamard states for free quantum Klein-Gordon fields, when the background metric and the field mass are slowly varied from their initial to final values. If the Klein-Gordon field stays massive, we prove that the adiabatic limit of the initial vacuum state is the (final) vacuum state, by extending to the symplectic framework the adiabatic theorem of Avron-Seiler-Yaffe. In cases when only the field mass is varied, using a abstract version of the mode decomposition method we can also consider the case when the initial or final mass vanishes, and the initial state is either a thermal state or a more general Hadamard state.