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 $M$ 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 $\phi$ and $\square_g\phi$ in order to ensure that $\square_g \circ G^\pm$ and $G^\pm \circ \square_g$ 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 $\ker \square_g$. This enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local $C^*$-algebras.