Green operators in low regularity spacetimes and quantum field theory

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 phi on a globally hyperbolic spacetime M withC(1,1) metricg. 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 phi and rectangle(g)phi in order to ensure that rectangle(g)degrees G(+/-) and G(+/-)degrees rectangle(g) are the identity maps on those spaces. The causal propagator G=G(+)-G(-) is then used to define a symplectic form omega on a normed space V(M) which is shown to be isomorphic to ker(rectangle(g)). This enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local C*-algebras.