Green Operators in Low Regularity Spacetimes and Quantum Field Theory

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 $\varphi$ 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 $\varphi$ and ${\square }_g\varphi$ 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 $C^{\ast }$-algebras.

arXiv comment44 pages, 1 figure

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Cited by in corpus (1)

• Adiabatic Ground States in Non-Smooth Spacetimes