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

Abstract

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 Møller 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 Møller 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 Møller 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.

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

“…Proof of Theorem 1.1. Let t be a Cauchy temporal function for g and define g u := −dt 2 + h, where h is a complete Riemannian metric on t −1 (s) for every s ∈ R. On account of [64,Proposition 2.23], there exists a globally hyperbolic metric g such that J + g ⊂ J + gu ∩ J + g . Denote with SM g the spinor bundle over (M, g) and consider the linear isometries…”