Global and microlocal aspects of Dirac operators: propagators and Hadamard states

用特征柯西问题构造Hadamard态

Construction of Hadamard states by characteristic Cauchy problem

我们从全局双曲时空$(M, g)$中的一个基点$p$出发，在一个等于未来光锥$C$内部的时空$M_{0}$中构造了Klein-Gordon场的Hadamard态.在$C$未来无穷远的某些正则性条件下，我们在$C$上确定了一个边界辛函数空间，它允许从与边界辛空间相关联的CCR代数上的态构造$M_{0}$中Klein-Gordon量子场的态.我们在$C$上的边界态上建立了自然微局域条件，保证了它在$M_{0}$中诱导的体态满足Hadamard条件。利用锥上的伪微分学$C$，我们构造了一大类具有伪微分协方差的边界上的Hadamard边界态，并刻画了其中的纯态.然后，我们表明，这些纯边界态诱导纯Hadamard态在$M_{0}$。

Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

We consider the Klein-Gordon operator on an $n$-dimensional asymptotically anti-de Sitter spacetime $(M,g)$ together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on $\partial M$ of order up to $2$. Using techniques from $b$-calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-soluti…

狄拉克-库仑传播子的奇异性

Singularities of Dirac-Coulomb propagators

本文研究含时库仑势下狄拉克场传播子与N点函数的奇异性，证明r≠0时狄拉克-库仑真空为哈达玛态，且任意两哈达玛态的相对电荷密度在r=0附近也可良好定义，推广了相关奇异性定理至含时势情形。

Minkowski空间扰动的Feynman传播子

The Feynman propagator on perturbations of Minkowski space

本文分析了洛伦兹散射空间上的费曼波动方程。我们证明了Feynman传播子作为由衰变和微局部Sobolev正则性定义的某些巴拿赫空间之间的映射存在。我们继续表明，在QFT中产生的某些非线性波动方程是适定性的小数据中的费曼设置。

Quantum fields from global propagators on asymptotically Minkowski and extended de Sitter spacetimes

We consider the wave equation on asymptotically Minkowski spacetimes and the Klein-Gordon equation on even asymptotically de Sitter spaces. In both cases we show that the extreme difference of propagators (i.e. retarded propagator minus advanced, or Feynman minus anti-Feynman), defined as Fredholm inverses, induces a symplectic form on the space of solutions with wave front set confined to the radial sets. Furthermore, we construct isomorphisms between the solution spaces and symplectic spaces of asymptotic data. As an application of this result we obtain distinguished Hadamard two-point funct…

A Gutzwiller Trace formula for Dirac Operators on a Stationary Spacetime

A Duistermaat-Guillemin-Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch [Adv. Math. \textbf{376}, 107434 (2021)] to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killin…

弱色散性下波动算子的损失导数的Lohartz估计

Strichartz estimates with loss of derivatives under a weak dispersion property for the wave operator

本文可以看作是Bernicot和Samoyeau的[BS 14]的续篇，在[BS 14]中，作者提出了一种从波传播子的色散性质导出Schr{<$}dinger方程的Schr{<$}dinger估计的一般方法.它通过减少的H1- BMO色散估计的薛定谔传播的L2-L2微局域（在空间和频率）色散不等式的波算子。本文的目的是启发我们理解波的色散如何意味着薛定谔方程的色散。更准确地说，我们的主要定理的假设编码色散的波动方程以统一的方式，关于光锥。在许多情况下，光锥边界附近出现的现象是更复杂的现象。我们提出的方法允许忘记那些我们还不太了解的现象。第二个主要步骤显示了在这些假设下我们可以得到的带有导数损失的哈茨估计。我们的工作设置是一般的，足以恢复各种各样的框架（无限度量空间，黎曼流形粗糙度量，一些群体，.）其中缺乏波传播子的知识限制了我们对色散现象的理解。

Lorentzian散射空间上波动算子的复幂与谱作用

Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces

本文在Lorentzian散射空间中用泛函演算定义波动算子复幂，证明迹密度的亚纯性并将极点联系于标量曲率，从而构建Chamseddine-Connes作用玻色部分的Lorentzian谱作用模型。

Lorentzian 度规的副因果形变与代数量子场论中的 Møller 同构

Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory

本文提出整体双曲度规间的副因果关系，据此构造正规双曲算子解空间的 Møller 同构，并将其提升为保持 Hadamard 波前集的 CCR 代数 *-同构。

Dirac算子的共振展开与局部能量衰减估计

Resonance expansions and local-energy decay estimates for Dirac operators

该文研究三维半经典Dirac算子的共振展开与局部能量衰减，通过预解式估计和FBI变换等方法，推导出快速衰减阈值及微局部障碍。