On the Mathematical Foundations of Causal Fermion Systems in Minkowski Space
Marco Oppio
备注：
67 pages, new document class (amsart), typos corrected, title changed, introduction expanded, reference added, proofs of Section 2 moved to appendix, Sections 2.3 and 2.4 improved, proof of Lemma 5.2 added (published version)
The emergence of the concept of a causal fermion system is revisited and further investigated for the vacuum Dirac equation in Minkowski space. After a brief recap of the Dirac equation and its solution space, in order to allow for the effects of a possibly nonstandard structure of spacetime at the Planck scale, a regularization by a smooth cutoff in momentum space is introduced, and its properties are discussed. Given an ensemble of solutions, we recall the construction of a local correlation function, which realizes spacetime in terms of operators. It is shown in various situations that the local correlation function maps spacetime points to operators of maximal rank and that it is closed and homeomorphic onto its image. It is inferred that the corresponding causal fermion systems are regular and have a smooth manifold structure. The cases considered include a Dirac sea vacuum and systems involving a finite number of particles and antiparticles.
The Fermionic Entanglement Entropy of Causal Diamonds in Two-Dimensional Minkowski Space
Fakultät für Mathematik Universität Regensburg（雷根斯堡大学数学系）；Institut für Theoretische Physik Universität Leipzig（莱比锡大学理论物理研究所）；Dipartimento di Matematica, Università di Genova（热那亚大学数学系）
The Fock Space Dynamics of Causal Fermion Systems: Non-Abelian Gauge Fields
Università degli Studi di Pavia（帕维亚大学）；INFN, Sezione di Pavia（意大利国家核物理研究所帕维亚分部）；Universität Regensburg（雷根斯堡大学）；McGill University（麦吉尔大学）；City University of Hong Kong SAR（香港城市大学）
Causality theory for closed cone structures with applications
我们发展了流形上锥的上半连续分布的因果关系理论，在两个方向上推广了数学相对论的结果：非圆锥和非正则可微性假设。利用光滑时间函数证明了正则Lorentzian因果关系理论的大多数结果的有效性，包括因果阶梯、Fermat原理、因果公式中的显著奇异性定理、Avez-Seifert定理、稳定因果关系的刻画和全局双曲性.例如，我们给这些结构的等价性之间的稳定的因果关系，$K$ -因果关系和存在的时间函数的第一个证明。这一结果意味着允许连续增函数的闭锥结构也允许光滑的闭锥结构。我们还研究了真锥的结构，即真锥的纤维束模拟。对于他们，我们得到的大多数结果的依赖域。此外，我们证明了horismos和柯西视界产生的类光测地线，后者被定义通过的非时性属性。因果测地线和陡峭的时间函数得到了一个强大的产品技巧。本文还研究了Lorentz-Minkowski空间在非常弱的正则性条件下的性质。最后，我们引入了稳定距离和稳定时空的概念，解决了两个著名的问题：（a）可嵌入Minkowski时空的Lorentz流形的特征，它们是稳定时空;（b）证明拓扑、序和距离（用一个公式a la Connes）可以用光滑陡峭的时间函数表示。本文自成体系，实际上没有使用任何数学相对论的先进结果。
Holographic Mixing and Fock Space Dynamics of Causal Fermion Systems
Università degli Studi di Pavia（帕维亚大学）；INFN & INDAM, Sezione di Pavia（意大利国家核物理研究所与数学、动力学和应用数学研究所帕维亚分部）；Universität Regensburg（雷根斯堡大学）；McGill University（麦吉尔大学）；City University of Hong Kong（香港城市大学）
