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キーワード:
General Relativity and Quantum Cosmology, Mathematics, Differential Geometry
要旨:
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.
