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

Nicoló Drago (1) , Nicolas Ginoux (2) , Simone Murro (3)

1 UNITN - Università degli Studi di Trento = University of Trento
2 IECL - Institut Élie Cartan de Lorraine
3 LMO - Laboratoire de Mathématiques d'Orsay

The 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.

Mots clés: Møller operators; deformation arguments; Cauchy problem; symmetric weaklyhyperbolic systems; algebraic quantum field theory; Hadamard states; globally hyperbolic manifolds with timelike boundary

Domaines: Equations aux dérivées partielles [math.AP]; Géométrie différentielle [math.DG]; Physique mathématique [math-ph]

Nicoló Drago, Nicolas Ginoux, Simone Murro. Møller operators and Hadamard states for Dirac fields with MIT boundary conditions. 2021. ⟨hal-03333715v1⟩