Felix Finster
A notion of entropy is introduced for causal fermion systems. This entropy is a measure of the state of disorder of a causal fermion system at a given time compared to the vacuum. The definition is given both in the finite and infinite-dimensional settings. General properties of the entropy are analyzed.
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This paper is dedicated to give a concise introduction to the theory of causal fermion systems. After putting the theory of causal fermion systems into the historical context, we recall fundamental physical preliminaries. Afterwards, we enter the underlying ideas of the principle of the fermionic projector and clarify its connection to causal fermion systems. We finally outline the main structures of the theory of causal fermion systems.
Causal Fermion Systems: Spacetime as the web of correlations of a many-body quantum system
本文论证因果费米系统中的时空是多体量子系统的关联网，揭示其完全关系性理论本质，并阐明在状态数趋于无穷大时背景感知的涌现机制及因果作用最小化的物理意义。
The theory of causal fermion systems is an approach to describe fundamental physics. We here introduce the mathematical framework and give an overview of the objectives and current results.
After reviewing the theory of "causal fermion systems" (CFS theory) and the "Events, Trees, and Histories Approach" to quantum theory (ETH approach), we compare some of the mathematical structures underlying these two general frameworks and discuss similarities and differences. For causal fermion systems, we introduce future algebras based on causal relations inherent to a causal fermion system. These algebras are analogous to the algebras previously introduced in the ETH approach. We then show that the spacetime points of a causal fermion system have properties similar to those of "events", a…
Causal Fermion Systems and Octonions
本文比较因果费米子理论与基于八元数的可除代数方法，发现八元数及其张量积可自然描述因果费米子真空对称性，两者通过因果作用原理互补。
A background independent notion of causality
该研究提出了独立于流形微分和拓扑结构的因果序概念，证明正则因果序可由特殊集合代数恢复，并探讨了因果集代数与自然测度的关联，为因果性研究提供了背景无关的新视角。
We give an elementary introduction to the theory of causal fermion systems, with a focus on the underlying physical ideas and the conceptual and mathematical foundations.
In this short review, we explain how and in which sense the causal action principle for causal fermion systems gives rise to classical gravity and the Einstein equations. Moreover, methods are presented for going beyond classical gravity, with applications to a positive mass theorem for static causal fermion systems, a connection between area change and matter flux and the construction of a quantum state.
The notions of two-dimensional area, Killing fields and matter flux are introduced in the setting of causal fermion systems. It is shown that for critical points of the causal action, the area change of two-dimensional surfaces under a Killing flow in null directions is proportional to the matter flux through these surfaces. This relation generalizes an equation in classical general relativity due to Ted Jacobson to the setting of causal fermion systems.
We give a brief introduction to causal fermion systems with a focus on the geometric structures in space-time.
The Fock Space Dynamics of Causal Fermion Systems: Non-Abelian Gauge Fields
研究因果费米子系统在特定极限下，其因果作用原理产生含非阿贝尔规范场和狄拉克场的微扰量子场论的线性福克空间动力学。
It is shown for causal fermion systems describing Minkowski-type spacetimes that an interacting causal fermion system at time $t$ gives rise to a distinguished state on the algebra generated by fermionic and bosonic field operators. The proof of positivity of the state is given and representations are constructed.
Notions of Fermionic Entropies of a Causal Fermion System
本文针对因果费米子系统定义了三种费米子熵，利用准自由态熵公式结合约化单粒子密度算符，通过多场景例子阐释定义，回顾面积定律并关联模理论计算相对熵。
The theory of causal fermion systems is a new physical theory which aims to describe a fundamental level of physical reality. Its mathematical core is the causal action principle. In this thesis, we develop a formalism which connects the causal action principle to a suitable notion of fields on space-time. We derive field equations from the causal action principle and find that the dynamics induced by the field equations conserve a symplectic form which gives rise to an Hamiltonian time evolution if the causal fermion system admits a notion of time. In this way, we establish the dynamics of ca…
Causal fermion systems are introduced as a general mathematical framework for formulating relativistic quantum theory. By specializing, we recover earlier notions like fermion systems in discrete space-time, the fermionic projector and causal variational principles. We review how an effect of spontaneous structure formation gives rise to a topology and a causal structure in space-time. Moreover, we outline how to construct a spin connection and curvature, leading to a proposal for a "quantum geometry" in the Lorentzian setting. We review recent numerical and analytical results on the support o…
The Continuum Limit Analysis of Causal Fermion Systems for Curved Spacetimes
本文在全局双曲时空中从代数量子场论出发构造因果费米子系统，通过准自由Hadamard态的单粒子密度算子识别费米子投影子，并发展连续极限分析，证明因果作用原理的Euler-Lagrange方程等价于耦合的Einstein-Dirac方程。
Causal Fermion Systems, Non-Commutative Geometry and Generalized Trace Dynamics
因果费米系统通过广义两点相关子编码时空点间关系，与非交换几何和广义迹动力学进行比较，揭示其在连续极限下恢复几何结构的核心创新。
Construction of Currents in Causal Fermion Systems
本文提出在因果费米子系统中推导经典场方程的系统化形式体系，通过波函数探测与泰勒展开将线性化场方程重构为秩递增的张量方程，秩一恢复麦克斯韦方程，二阶预期编码爱因斯坦方程，并揭示高阶新物理修正。
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 …
A Canonical Construction of the Extended Hilbert Space for Causal Fermion Systems
本文通过分解因果作用的二阶变分，提出近似解耦的动态波方程和玻色场方程，构建了包含这些解的正定内积希尔伯特空间。
A Mechanism of Baryogenesis for Causal Fermion Systems
本文提出因果费米子系统理论中的一种新型重子生成机制，并在全局双曲时空中通过计算详细阐明，使其能够用于具体宇宙学情形的定量研究。
Holographic Mixing and Fock Space Dynamics of Causal Fermion Systems
研究因果费米系统在闵可夫斯基空间中的极限情况，揭示量子玻色场的本源，并提出全息混合概念。
The theory of causal fermion systems is a recent approach to fundamental physics. Giving quantum mechanics, general relativity and quantum field theory as limiting cases, it is a candidate for a unified physical theory. The dynamics is described by a novel variational principle, the so-called causal action principle. The causal action principle does not rely on a presupposed space-time structure. Instead, it is a variational principle for space-time itself as well as for all structures in space-time (like particles, fields, etc.). After a general motivation and introduction, we report on mathe…
This monograph introduces the basic concepts of the theory of causal fermion systems, a recent approach to the description of fundamental physics. The theory yields quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for describing and analyzing non-smooth geometries and "quantum geometries." The dynamics is described by a novel variational principle, called the causal action principle. In addition to the basics, the book…
Modified Measures as an Effective Theory for Causal Fermion Systems
本文比较因果费米子系统与修正测度理论，指出后者可作为前者连续统极限下的有效引力描述，并为筛选相容理论奠定基础。
The theory of causal fermion systems is an approach to describe fundamental physics. Giving quantum mechanics, general relativity and quantum field theory as limiting cases, it is a candidate for a unified physical theory. We here give a non-technical introduction.
The dynamics of spinorial wave functions in a causal fermion system is studied. A so-called dynamical wave equation is derived. Its solutions form a Hilbert space, whose scalar product is represented by a conserved surface layer integral. We prove under general assumptions that the initial value problem for the dynamical wave equation admits a unique global solution. Causal Green's operators are constructed and analyzed. Our findings are illustrated in the example of the regularized Minkowski vacuum.
The Fermionic Entanglement Entropy of Causal Diamonds in Two-Dimensional Minkowski Space
该研究针对二维闵可夫斯基时空中因果钻石的费米子Rényi纠缠熵展开，选取带紫外正则化的闵可夫斯基真空对应的准自由态，推导出对数增强的面积律。
A mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable only in certain directions. A chain rule is proven for Hölder continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermion syste…
Causal Fermion Systems as an Effective Collapse Theory
该研究证明非相对论极限下因果费米子系统可形成有效坍缩理论，从因果作用量原理推导薛定谔方程修正项，明确其动力学形式及与连续自发局域化模型的差异，为量子坍缩机制提供新的理论视角。
As toy models for space-time on the Planck scale, we consider examples of fermion systems in discrete space-time which are composed of one or two particles defined on two up to nine space-time points. We study the self-organization of the particles as described by a variational principle both analytically and numerically. We find an effect of spontaneous symmetry breaking which leads to the emergence of a discrete causal structure.
Causal fermion systems and Riemannian fermion systems are proposed as a framework for describing non-smooth geometries. In particular, this framework provides a setting for spinors on singular spaces. The underlying topological structures are introduced and analyzed. The connection to the spin condition in differential topology is worked out. The constructions are illustrated by many simple examples like the Euclidean plane, the two-dimensional Minkowski space, a conical singularity, a lattice system as well as the curvature singularity of the Schwarzschild space-time. As further examples, it …
Causal fermion systems incorporate local gauge symmetry in the sense that the Lagrangian and all inherent structures are invariant under local phase transformations of the physical wave functions. In the present paper it is explained and worked out in detail that, despite this local gauge freedom, the structures of a causal fermion system give rise to distinguished gauges where the local gauge freedom is fixed completely up to global gauge transformations. The main method is to use spectral and polar decompositions of operators on Hilbert spaces and on indefinite inner product spaces. We also …
It is shown that the theory of causal fermion systems gives rise to a novel mechanism for dark matter and dark energy. This mechanism is first worked out for cubical subsets of Minkowski space with periodic boundary conditions. Then it is studied in Friedmann-Lemaître-Roberson-Walker spacetimes. The magnitude of the effect scales like one over the lifetime of the universe squared. In contrast to most models of dark matter and dark energy, our mechanism does not postulate any new particles. Instead, it is a result of the collective behavior of all the wave functions which form the Dirac sea, ne…
A Positive Mass Theorem for Static Causal Fermion Systems
本文引入渐近平坦静态因果费米子系统，通过表面层积分极限定义总质量，不依赖正则性假设，证明正质量定理，揭示因果作用原理如何纳入引力非线性效应。
A notion of local algebras is introduced in the theory of causal fermion systems. Their properties are studied in the example of the regularized Dirac sea vacuum in Minkowski space. The commutation relations are worked out, and the differences to the canonical commutation relations are discussed. It is shown that the spacetime point operators associated to a Cauchy surface satisfy a time slice axiom. It is proven that the algebra generated by operators in an open set is irreducible as a consequence of Hegerfeldt's theorem. The light cone structure is recovered by analyzing expectation values o…
Causal Coverage in Ordered Locales and Spacetimes
本文在无点拓扑中提出有序 locale 的因果覆盖，将其联系于广义 Grothendieck 拓扑与确定性时间演化，并证明其诱导的依赖域严格包含经典曲线定义。
Causal Fermion Systems: An Introduction to Fundamental Structures, Methods and Applications
本书系统介绍因果费米子系统理论的基本概念、数学框架与因果作用原理，涵盖量子力学、广义相对论及量子场论作为极限情形，面向研究生并作为研究者参考。
Entangled Quantum States of Causal Fermion Systems and Unitary Group Integrals
本文通过渐近计算酉群积分，分析了因果费米子系统的量子态，证明在特定极限下局域化精细预态为正，可描述一般纠缠态。
The connection between symmetries and conservation laws as made by Noether's theorem is extended to the context of causal variational principles and causal fermion systems. Different notions of continuous symmetries are introduced. It is proven that these symmetries give rise to corresponding conserved quantities, expressed in terms of so-called surface layer integrals. In a suitable limiting case, the Noether-like theorems for causal fermion systems reproduce charge conservation and the conservation of energy and momentum in Minkowski space. Thus the conservation of charge and energy-momentum…
Drawing from the theory of optimal transport we propose a rigorous notion of a causal relation for Borel probability measures on a given spacetime. To prepare the ground, we explore the borderland between causality, topology and measure theory. We provide various characterisations of the proposed causal relation, which turn out to be equivalent if the underlying spacetime has a sufficiently robust causal structure. We also present the notion of the 'Lorentz-Wasserstein distance' and study its basic properties. Finally, we discuss how various results on causality in quantum theory, aggregated a…
Dirac and Weyl Fermions -- the Only Causal Systems
本文证明在实质量有限旋量系统中，狄拉克与魏尔费米子是仅有的不可约因果系统，并揭示了其波函数的洛伦兹收缩行为及向非类时超平面的局域化扩展。
This thesis proposes a framework based on a notion of combinatorial cell complex (cc) whose cells are defined simply as finite sets of vertices. The cells of a cc are subject to four axioms involving a rank function that assigns a rank (or a dimension) to each cell. Our framework focuses on classes of cc admitting an inclusion-reversing duality map. We introduce a combinatorial notion of cobordism that allows us to single out a category whose morphisms are cobordisms having a causal structure. Our aim is to offer an approach to look for a combinatorial notion of quantum field theory having a b…
In the theory of causal fermion systems, the physical equations are obtained as the Euler-Lagrange equations of a causal variational principle. Studying families of critical measures of causal variational principles, a class of conserved surface layer integrals is found and analyzed.
We propose a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive the objects of our quantum geometry: the spin space, the tangent space endowed with a Lorentzian metric, connection and curvature. In order to get the correspondence to differential geometry, we construct examples of causal fermion systems by regularizing Dirac sea configurations i…
Drawing from our earlier works on the notion of causality for nonlocal phenomena, we propose and study the extension of the Sorkin--Woolgar relation $K^+$ onto the space of Borel probability measures on a given spacetime. We show that it retains its fundamental properties of transitivity and closedness. Furthermore, we list and prove several characterizations of this relation, including the `nonlocal' analogue of the characterization of $K^+$ in terms of time functions. This generalizes and casts new light on our earlier results concerning the causal precedence relation $J^+$ between measures.
A Geometric Derivation of the Einstein Equations from the Causal Action Principle
研究从因果作用原理推导爱因斯坦方程，通过对因果费米子系统的特定分析，引入密切真空概念，利用拉格朗日量诱导洛伦兹度量，得出里奇张量满足爱因斯坦方程及引力耦合常数，还提供了推导方程修正的系统程序。
Fermion systems in discrete space-time are introduced as a model for physics on the Planck scale. We set up a variational principle which describes a non-local interaction of all fermions. This variational principle is symmetric under permutations of the discrete space-time points. We explain how for minimizers of the variational principle, the fermions spontaneously break this permutation symmetry and induce on space-time a discrete causal structure.
The causal action principle is analyzed for a system of relativistic fermions composed of massive Dirac particles and neutrinos. In the continuum limit, we obtain an effective interaction described by classical gravity as well as the strong and electroweak gauge fields of the standard model.
Ollivier-Ricci curvature for causal sets
利用洛伦兹最优传输引入因果集的Ollivier-Ricci曲率，通过离散化方法恢复时间型Ricci曲率，并证明局部到全局传播结果和类时Bonnet-Myers定理。
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