Green operators in low regularity spacetimes and quantum field theory

Abstract

Abstract In this paper we develop the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. In particular we consider the case of a massless scalar field ϕ on a globally hyperbolic spacetime M with C 1,1 metric g. This first entails showing that the (classical) Cauchy problem for the wave equation is well-posed for initial data and sources in Sobolev spaces and then constructing low-regularity advanced and retarded Green operators as maps between suitable function spaces. In specifying the relevant function spaces we need to control the norms of both ϕ and □ g ϕ in order to ensure that □ g ◦G ± and G ±◦□ g are the identity maps on those spaces. The causal propagator G = G + − G − is then used to define a symplectic form ω on a normed space V(M) which is shown to be isomorphic to ker(□ g ). This enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local C*-algebras.