Causal Variational Principles in the $σ$-Locally Compact Setting: Existence of Minimizers

低维情形下因果作用原理的数值分析

Numerical Analysis of the Causal Action Principle in Low Dimensions

本文利用可微编程系统分析低维因果作用原理，提出离散 Dirac 球候选极小元，并通过大 m 数值实验验证其逼近性与可视化方法。

六边形格点在局部上是普遍最优的

The hexagonal lattice is universally locally optimal

本文证明在完全单调的对势（包括高斯相互作用和幂律）下，六边形格点在所有点配置中是单位体积相互作用能的局部极小化点。

Causal Variational Principles on Measure Spaces

We introduce a class of variational principles on measure spaces which are causal in the sense that they generate a relation on pairs of points, giving rise to a distinction between spacelike and timelike separation. General existence results are proved. It is shown in examples that minimizers need not be unique. Counter examples to compactness are discussed. The existence results are applied to variational principles formulated in indefinite inner product spaces.

Minkowski space is locally the Noldus limit of a Poisson process causet

A poisson process $P_λ$ on $\mathbb{R}^{d}$ with causal structure inherited from the the usual Minkowski metric on $\mathbb{R}^{d}$ has a normalised discrete causal distance $D_λ(x,y)$ given by the height of the longest causal chain normalised by $λ^{1/d}c_{d}$. We prove that $P_λ$ restricted to a compact set $Q$ converges in probability in the sense of Noldus to $Q$ with the Minkowksi metric.

Causal Fermion Systems: A Quantum Space-Time Emerging from an Action Principle

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…

因果变分原理的一个正拟局部质量

A Positive Quasilocal Mass for Causal Variational Principles

本文针对因果变分原理极小化子证明新不等式，用于正质量定理新证明，并引入正拟局部质量与标量曲率定义，在超静态及史瓦西时空示例中阐明与ADM及Brown-York质量的联系。

Causal Fermion Systems: Discrete Space-Times, Causation and Finite Propagation Speed

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…

On a system of multi-component Ginzburg-Landau vortices

We study the asymptotic behavior of solutions for $n$-component Ginzburg-Landau equations as $\ve \to 0$. We prove that the minimizers converges locally in any $C^k$-norm to a solution of a system of generalized harmonic map equations.

Defects of liquid crystals with variable degree of orientation

We prove that the zero set of a minimizer of the modified Ericksen energy for nematic liquid crystals with variable degree of orientation consists locally of isolated points and a finite union of Hölder continuous curves with only finitely many crossings.

A Hamiltonian Formulation of Causal Variational Principles

Causal variational principles, which are the analytic core of the physical theory of causal fermion systems, are found to have an underlying Hamiltonian structure, giving a formulation of the dynamics in terms of physical fields in space-time. After generalizing causal variational principles to a class of lower semi-continuous Lagrangians on a smooth, possibly non-compact manifold, the corresponding Euler-Lagrange equations are derived. In the first part, it is shown under additional smoothness assumptions that the space of solutions of the Euler-Lagrange equations has the structure of a sympl…

On differentiability of volume time functions

We show differentiability of a class of Geroch's volume functions on globally hyperbolic manifolds. Furthermore, we prove that every volume function satisfies a local anti-Lipschitz condition over causal curves, and that locally Lipschitz time functions which are locally anti-Lipschitz can be uniformly approximated by smooth time functions with timelike gradient. Finally, we prove that in stably causal spacetimes Hawking's time function can be uniformly approximated by smooth time functions with timelike gradient.

Fermion Systems in Discrete Space-Time

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.

Linearized Fields for Causal Variational Principles: Existence Theory and Causal Structure

The existence theory for solutions of the linearized field equations for causal variational principles is developed. We begin by studying the Cauchy problem locally in lens-shaped regions, defined as subsets of space-time which admit foliations by surface layers satisfying hyperbolicity conditions. We prove existence of weak solutions and show uniqueness up to vectors in the orthogonal complement of the jets used for testing. The connection between weak and strong solutions is analyzed. Global solutions are constructed by exhausting space-time by lens-shaped regions. We construct advanced and …