M{\o}ller operators and Hadamard states for Dirac fields with MIT boundary conditions

TL;DR

This paper proves the existence of Hadamard states for Dirac fields with MIT boundary conditions on globally hyperbolic manifolds with timelike boundary, using a geometric Møller operator to relate initial data spaces.

Contribution

Introduces a geometric Møller operator that establishes a unitary isomorphism between initial data spaces, enabling the construction of Hadamard states for Dirac fields with MIT boundary conditions.

Findings

Existence of Hadamard states for Dirac fields with MIT boundary conditions.

Construction of a $*$-isomorphism between Dirac field algebras.

Preservation of the singular structure of two-point distributions.

Abstract

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 M{\o}ller 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.