Formation of singularities in relativistic theories of electromagnetism

本研究从经典和量子两个层面研究相对论电磁场非线性动力学理论中奇点的形成问题。经典部分主要涉及两个问题：(1)著名的相对论Vlasov-Maxwell方程的分析，该方程被认为是一个具有适当有限能量经典数据的柯西问题。这个项目将严格分析最近开发的一个反例的场景，该反例展示了一族解决方案的有限时间崩溃。(2)无点电荷电磁场的非线性Maxwell-Born-Infeld方程的分析。这个项目将研究最近发现的呈现有限时间爆破的空间周期平面波解，以确定相应的柯西数据集是否是一般坏集的一部分。研究项目的量子部分涉及代表粒子的具有点缺陷的Maxwell-Born-Infeld场方程的解。带电粒子根据量子速度场运动，量子速度场由耦合到一般电磁场的多体狄拉克公式获得。这个项目将研究系统的狄拉克哈密顿量是否可以变得无界。这个项目解决了电磁学理论中的基本问题，这是现代科学和工程的核心。奇点形成的研究长期以来一直处于广义相对论和流体力学研究的前沿。最近的发现表明，奇点也可能在非线性电磁模型中构成一个重大的概念挑战，这些模型被认为是没有人工正则化的电磁理论的一致公式的候选者。首席研究人员最近发展了一个一致的电磁理论公式，包含了粒子的本征自旋，在经典和量子水平上都是一致的。目前的项目调查了这一理论中可能形成的奇点。这项工作的另一部分研究了相对论Vlasov-Maxwell模型中的有限时间崩塌。这种塌缩将为非常小的天体的形成提供一种新的机制，这些天体的引力太弱，不能帮助它们的形成。该项目旨在建立这种可能性；其结果可能对行星系统形成的理论产生重大影响。

This research project studies the question of singularity formation in nonlinear dynamical theories of relativistic electromagnetic fields, both at the classical and the quantum level.The classical part is concerned with two main problems: (1) Analysis of the well-known relativistic Vlasov-Maxwell equations, which had been conjectured to be globally well-posed as a Cauchy problem with suitable finite-energy classical data. This project will rigorously analyze a recently-developed scenario for a counterexample exhibiting finite-time collapse for a family of solutions. (2) Analysis of the nonlinear Maxwell-Born-Infeld equations for electromagnetic fields in the absence of point charges. This project will investigate recently-discovered spatially periodic plane wave solutions that exhibit finite-time blow-up to determine whether the corresponding set of Cauchy data is part of a generic bad set.The quantum part of the research project concerns solutions of the Maxwell-Born-Infeld field equations with point defects that represent particles. The charged particles move according to a quantum velocity field obtained from a many body Dirac formalism coupled to generic electromagnetic fields. This project will study whether the Dirac Hamiltonian for the system can become unbounded below.This project addresses fundamental issues in the theory of electromagnetism, which is central to modern science and engineering. The study of singularity formation has long been at the forefront of research in general relativity and in fluid dynamics. Recent discoveries suggest that singularities may also pose a major conceptual challenge in the nonlinear electromagnetic models that have been proposed as candidates for a consistent formulation of an electromagnetic theory without artificial regularizers. The principal investigator recently developed a consistent formulation of electromagnetic theory, incorporating intrinsic spin of particles, that is consistent at both classical and quantum levels. The current project investigates possible singularity formation in this theory. Another part of this work examines finite-time collapse in the relativistic Vlasov-Maxwell model. Such collapse would provide a novel mechanism for the formation of very small celestial bodies whose gravitational self-attraction is too weak to aid in their formation. This project aims to establish that possibility; the results could have a major impact on theories of planetary system formation.