发表机构
Leibniz University Hannover（汉诺威莱布尼茨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明带类时边界时空中Dirichlet边界条件下Klein-Gordon场纯拟自由Hadamard态的存在性，通过b-波前集条件与传播定理构造此类态。
AI 中文摘要
我们阐述并证明了在具有类时边界的全局双曲时空中，带有Dirichlet边界条件的Klein-Gordon场的纯拟自由Hadamard态的存在性。微局域条件施加在单粒子Hilbert空间值分布$p\circ G_D$上，而非两点函数上。这通过压缩未来锥上的$b$-波前集条件来表达。对带角流形$\bar M\times\bar M$上的分析因此几乎完全被避免。我们证明了我们的Hadamard条件版本在拟自由Hadamard态类中是普适的。我们还证明了正能微局域分裂的传播定理，该定理仅使用小$b$-演算中的微局域分裂和Dirichlet问题的适定性，特别地，不需要反射射线或滑行射线的传播定理；结合超静态情形下的显式计算和Fulling--Narcowich--Wald形变论证，这产生了纯拟自由Dirichlet--Hadamard态的构造。
英文摘要
We formulate and prove existence of pure quasifree Hadamard states for the Klein--Gordon field with Dirichlet boundary condition on a globally hyperbolic spacetime with timelike boundary. The microlocal condition is imposed on the one-particle Hilbert-space-valued distribution $p\circ G_D$ rather than on the two-point function. This is expressed by a $b$-wave-front set condition on the compressed future cone. The analysis on the manifold with corners $\bar M\times\bar M$ is then almost completely avoided. We prove that our version of the Hadamard condition is universal in the class of quasi-free Hadamard states. We also prove a propagation theorem for positive-energy microlocal splittings which uses only a microlocal splitting in the small $b$-calculus and well-posedness of the Dirichlet problem, and in particular requires no propagation theorem for reflected or gliding rays; combined with an explicit computation in the ultrastatic case and a Fulling--Narcowich--Wald deformation argument this yields the construction of a pure quasi-free Dirichlet--Hadamard state.
