Paracausal deformations of Lorentzian metrics and M{\o}ller isomorphisms in algebraic quantum field theory

TL;DR

This paper introduces a new geometric relation called paracausal relation to study Møller isomorphisms between solutions of hyperbolic operators on different globally hyperbolic spacetimes, with implications for algebraic quantum field theory.

Contribution

It defines paracausal relation and demonstrates how Møller operators associated with it preserve causal structures and induce *-isomorphisms between CCR-algebras in quantum field theory.

Findings

Møller operators intertwine causal propagators.

Møller operators preserve the wave front set of Hadamard states.

The Møller map induces *-isomorphisms between CCR-algebras.

Abstract

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.