# The quantization of Proca fields on globally hyperbolic spacetimes: Hadamard states and Møller operators

October 17, 2022

This paper deals with several issues concerning the algebraic quantization of
the real Proca fields in a globally hyperbolic spacetime and the definition and
existence of Hadamard states for that field. In particular, extending previous
work, we construct the so-called M{\o}ller $*$-isomorphism between the algebras
of Proca observables on paracausally related spacetimes, proving that the
pullback of these isomorphisms preserves the Hadamard property of corresponding
quasifree states defined on the two spacetimes. Then, we pullback a natural
Hadamard state constructed on ultrastatic spacetimes of bounded geometry, along
this $*$-isomorphism, to obtain a Hadamard state on a general globally
hyperbolic spacetime. We conclude the paper, by comparing the definition of a
Hadamard state, here given in terms of wavefront set, with the one proposed by
Fewster and Pfenning, which makes use of a supplementary Klein-Gordon Hadamard
form. We establish an (almost) complete equivalence of the two definitions.

Keywords:

Proca Field, Hadamard states, M{\o}ller operator