On the adiabatic limit of Hadamard states

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.Comment: 19 pages, Section 4.2 improved, accepted for publication on Letters in Mathematical Physic

On the Cauchy problem for the Fadaray tensor on globally hyperbolic manifolds with timelike boundary

We study the well-posedness of the Cauchy problem for the Faraday tensor on globally hyperbolic manifolds with timelike boundary. The existence of Green operators for the operator $\mathrm{d}+\delta$ and a suitable pre-symplectic structure on the space of solutions are discussed.Comment: 19 pages -- accepted in Rendiconti Lincei Matematica e Applicazion

Feynman path integrals on compact Lie groups with bi-invariant Riemannian metrics and Schr\"odinger equations

In this work we consider a suitable generalization of the Feynman path integral on a specific class of Riemannian manifolds consisting of compact Lie groups with bi-invariant Riemannian metrics. The main tools we use are the Cartan development map, the notion of oscillatory integral, and the Chernoff approximation theorem. We prove that, for a class of functions of a dense subspace of the relevant Hilbert space, the Feynman map produces the solution of the Schr\"odinger equation, where the Laplace-Beltrami operator coincides with the second order Casimir operator of the group.Comment: 50 page

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

International audienceThe aim of this paper is to prove the existence of Hadamard states for Dirac fields coupled with MIT boundary conditions on any globally hyperbolic manifold with timelike boundary. This is achieved by introducing a geometric Møller operator which implements a unitary isomorphism between the spaces of L^2-initial data of particular symmetric systems we call weakly-hyperbolic and which are coupled with admissible boundary conditions. In particular, we show that for Dirac fields with MIT boundary conditions, this isomorphism can be lifted to a *-isomorphism between the algebras of Dirac fields and that any Hadamard state can be pulled back along this *-isomorphism preserving the singular structure of its two-point distribution