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.

中文翻译：

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低规则时空和量子场论中的绿色算符

在本文中，我们开发了所需的数学知识，以便描述低正则性时空中的量子场的可观测量。特别地，我们考虑在具有 $C^{1,1}$ 度量 $g$ 的全局双曲时空 $M$ 上的无质量标量场 $\phi$ 的情况。这首先需要证明波动方程的（经典）柯西问题对于 Sobolev 空间中的初始数据和源是适定的，然后构建低正则性的高级和延迟格林算子作为合适函数空间之间的映射。在指定相关函数空间时，我们需要控制 $\phi$ 和 $\square_g\phi$ 的范数，以确保 $\square_g \circ G^\pm$ 和 $G^\pm \circ \square_g $ 是这些空间上的身份映射。然后，因果传播子 $G=G^+-G^-$ 用于在赋范空间 $V(M)$ 上定义辛形式 $\omega$，它被证明与 $\ker \square_g$ 同构。这使人们能够根据准局部 $C^*$-代数的元素提供量子场的局部协变描述。