# Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory

September 14, 2021

Given a pair of normally hyperbolic operators over (possibily different)
globally hyperbolic spacetimes on a given smooth manifold, the existence of a
geometric isomorphism, called {\em M{\o}ller operator}, between the space of
solutions is studied. This is achieved by exploiting a new equivalence relation
in the space of globally hyperbolic metrics, called {\em paracausal relation}.
In particular, it is shown that the M{\o}ller operator associated to a pair of
paracausally related metrics and normally hyperbolic operators also intertwines
the respective causal propagators of the normally hyperbolic operators and it
preserves the natural symplectic forms on the space of (smooth) initial data.
Finally, the M{\o}ller map is lifted to a $*$-isomorphism between (generally
off-shell) $CCR$-algebras. It is shown that the Wave Front set of a Hadamard
bidistribution (and of a Hadamard state in particular) is preserved by the
pull-back action of this $*$-isomorphism.

Keywords:

paracausal deformation, convex interpolation, Cauchy problem, Møller operators, normally hyperbolic operators, algebraic quantum field theory, Hadamard states, globally hyper- bolic manifolds