We study Maxwell’s equation as a theory for smooth k-forms on globally hyperbolic spacetimes with timelike boundary as defined by Aké et al. (Structure of globally hyperbolic spacetimes with timelike boundary. arXiv:1808.04412 [gr-qc]). In particular, we start by investigating on these backgrounds the D’Alembert–de Rham wave operator \(\Box _k\) and we highlight the boundary conditions which yield a Green’s formula for \(\Box _k\). Subsequently, we characterize the space of solutions of the associated initial and boundary value problems under the assumption that advanced and retarded Green operators do exist. This hypothesis is proven to be verified by a large class of boundary conditions using the method of boundary triples and under the additional assumption that the underlying spacetime is ultrastatic. Subsequently we focus on the Maxwell operator. First we construct the boundary conditions which entail a Green’s formula for such operator and then we highlight two distinguished cases, dubbed \(\delta \mathrm {d}\)-tangential and \(\delta \mathrm {d}\)-normal boundary conditions. Associated to these, we introduce two different notions of gauge equivalence and we prove that in both cases, every equivalence class admits a representative abiding to the Lorenz gauge. We use this property and the analysis of the operator \(\Box _k\) to construct and to classify the space of gauge equivalence classes of solutions of the Maxwell’s equations with the prescribed boundary conditions. As a last step and in the spirit of future applications in the framework of algebraic quantum field theory, we construct the associated unital \(*\)-algebras of observables proving in particular that, as in the case of the Maxwell operator on globally hyperbolic spacetimes with empty boundary, they possess a non-trivial center.

中文翻译：

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具有时界的全局双曲时空的麦克斯韦方程

我们将麦克斯韦方程作为Aké等人定义的具有时间边界的全局双曲时空上的光滑k形式的理论进行研究。（具有类似时间边界的全局双曲时空的结构。arXiv：1808.04412 [gr-qc]）。特别是，我们首先研究这些背景下的D'Alembert-de Rham波算子\（\ Box _k \），然后突出显示产生Green （\ Box_k \）公式的边界条件。。随后，我们假设存在确实存在先进和弱化的格林算子，并描述了相关的初值和边值问题的解的空间。事实证明，使用边界三元组方法并在基础时空是超静态的附加假设下，这一假设已通过一大类边界条件得到了验证。随后，我们关注Maxwell运算符。首先，我们为此类算子构造了一个包含格林公式的边界条件，然后我们突出了两种不同的情况，称为\（\ delta \ mathrm {d} \）-切线和\（\ delta \ mathrm {d} \）-正常边界条件。与此相关的是，我们引入了两种不同的规范等效概念，并且我们证明了在两种情况下，每个等效类均接纳一个遵守洛伦兹规范的代表。我们使用此属性以及对运算符\（\ Box _k \）的分析来构造和分类具有指定边界条件的麦克斯韦方程组解的规范等价类的空间。作为最后一步，并且本着在代数量子场论框架中未来应用的精神，我们构造了可观察物的关联单位\（* \）-代数，特别是证明了这一点，例如全局双曲的Maxwell算子具有空边界的时空，它们拥有一个非平凡的中心。