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

Valter Moretti, Simone Murro, Daniele Volpe
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.

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