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

Given a pair of normally hyperbolic operators over (possibnly different) globally hyperbolic spacetimes on a given smooth manifold, the existence of a geometric isomorphism, called Moller 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 paracausal relation. In particular, it is shown that the Moller 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 Moller map is lifted to a *-isomorphism between (generally offshell) CCR-algebras. It is shown that theWave Front set of a Hadamard bidistribution (and of a Hadamard state in particular) is preserved by the pull-back action of this *-isomorphism.