Felix Finster, Maximilian Jokel
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.
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An elementary analytical fluid flow is composed by a geometric domain, a list of analytical constraints and by the function which depends on the physical properties, as Reynolds number, of the considered fluid. For this object, notions of laminar or weakly turbulent behavior are described using a simple mathematical model.
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…
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 …
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。
Elementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a $G$-skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map.
This paper collects and extends the lectures I gave at the "XXIV International Fall Workshop on Geometry and Physics" held in Zaragoza (Spain) August 31 - September 4, 2015. Within these lectures I review the formulation of Quantum Mechanics, and quantum theories in general, from a mathematically advanced viewpoint, essentially based on the orthomodular lattice of elementary propositions, discussing some fundamental ideas, mathematical tools and theorems also related to the representation of physical symmetries. The final step consists of an elementary introduction the so-called (C*-) algebrai…
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.
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…
On regularizations of the integral representation of Dirac delta function -- elementary approach
本文从施瓦茨泛函方法出发，以初等方式严谨且直观地阐述狄拉克δ函数积分表示的正则化，还介绍了米库辛斯基和西科尔斯基的序列方法及与佐藤分布方法相关的第三种正则化。
Causal perturbation theory and scattering amplitudes
本文在因果微扰理论框架下推导四维时空$\lambda\phi^4$模型的散射振幅，以通俗方式介绍该严谨形式体系，并将发散积分重新解释为分布定义域问题。
One of the goals of the present paper is to propose an elementary method to find a general formula for the Fourier transform containing a pair of complex gamma functions with a monomial sm in terms of Gauss's hypergeometric functions 2F1. We further collect some mathematical results that follow by means of this transform. Physical applications requiring the expectation values of position and momentum operators for a quantum system endowed with position-dependent effective mass are presented.
In this paper such Riemann metrics are established whose Laplace-Beltrami operators are identical to familiar Hamilton operators of elementary particle systems. Such metrics are the natural positive definite invariant metrics defined on two-step nilpotent Lie groups. The corresponding wave and Schroedinger operators emerge in the Laplacians of the static resp. solvable extensions of these nilpotent groups. The latter manifolds are endowed with natural invariant indefinite metric of Lorentz signature. Thus, these new exact mathematical models provide a relativistic theory for elementary particl…
The basic ideas and the important role of gauge principles in modern elementary particle physics are outlined. There are three theoretically consistent gauge principles in quantum field theory: the spin-1 gauge principle of electromagnetism and the standard model, the spin-2 gauge principle of general relativity, and the spin-3/2 gauge principle of supergravity. (Dirac Prize lecture, November 1993. Two figures are unavailable, but should not be essential.)
This article is concerned with a mathematical tool, the Associated Transfer Matrix T, which proves useful in the study of a wide class of physical problems involving multilayer heterostructures. General properties of linear, second order differential matrix Sturm Liouville operators are discussed as a basis for establishing general properties of T, which is also generally related to the Green function G. Some identities satisfied by T are derived, which prove useful in practice to monitor the numerical quality of computational processes.
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…
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…
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。
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