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.

In this paper we present well-posedness results for H 1 solutions of the wave equation for spacetimes that contain string-like singularities. These results extend a framework in which one characterises gravitational singularities as obstruction to the dynamics of test fields rather than point particles. In particular, we discuss spacetimes with cosmic strings.

The idea of defining a gravitational singularity as an obstruction to the dynamical evolution of a test field (described by a PDE) rather than the dynamical evolution of a particle (described by a geodesics) is explored. In particular, the concept of field regularity is introduced which serves to describe the well-posedness of the local initial value problem for a given field.In particular this is applied to (classical) scalar fields in the class of curve integrable spacetimes to show that the classical singularities do not interrupt the well-posedness of the wave equation.Comment: 28 pages, 1 figur

In this paper we obtain general conditions under which the wave equation is wellposed in spacetimes with metrics of Lipschitz regularity. In particular, the results can be applied to spacetimes where there is a loss of regularity on a hypersurface such as shell-crossing singularities, thin shells of matter, and surface layers. This provides a framework for regarding gravitational singularities not as obstructions to the world lines of point-particles, but rather as obstruction to the dynamics of test fields. Published by AIP Publishing. [http://dx

In this paper we define and construct advanced and retarded Green operators for the wave operator on spacetimes with low regularity. In order to do so we require that the spacetime satisfies the condition of generalised hyperbolicity which is equivalent to wellposedness of the classical inhomogeneous problem with zero initial data where weak solutions are properly supported. Moreover, we provide an explicit formula for the kernel of the Green operators in terms of an arbitrary eigenbasis of H 1 and a suitable Green matrix that solves a system of second order ODEs.