Formation of singularities in relativistic theories of electromagnetism

本课题研究相对论电磁场非线性动力学理论中的奇点形成问题，包括经典和量子两个层面。经典部分主要涉及两个问题：(1)对著名的相对论性Vlasov-Maxwell方程进行分析，该方程被推测为具有合适有限能量经典数据的柯西问题。本项目将严格分析最近开发的一个反例场景，以展示一系列解决方案的有限时间崩溃。(2)无点电荷时电磁场非线性Maxwell-Born-Infeld方程的分析。该项目将研究最近发现的空间周期平面波解，这些平面波解表现出有限时间爆炸，以确定相应的柯西数据集是否属于一般坏集的一部分。该研究项目的量子部分涉及具有代表粒子的点缺陷的麦克斯韦-玻恩-因菲尔德场方程的解。带电粒子按照一个量子速度场运动，这个量子速度场是由一个多体狄拉克形式与一般电磁场耦合得到的。这个项目将研究系统的狄拉克哈密顿量是否可以在以下无界。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。奇点形成的研究一直是广义相对论和流体力学研究的前沿。最近的发现表明，在非线性电磁模型中，奇点也可能构成一个主要的概念挑战，这些模型已被提出作为没有人工正则化器的电磁理论的一致表述的候选者。首席研究员最近开发了一个电磁理论的一致公式，包括粒子的固有自旋，在经典和量子水平上都是一致的。目前的项目研究了这个理论中奇点形成的可能性。这项工作的另一部分研究了相对论弗拉索夫-麦克斯韦模型中的有限时间坍缩。这种坍缩将为非常小的天体的形成提供一种新的机制，这些天体的自我引力太弱，无法帮助它们形成。本项目旨在确立这种可能性；这一结果可能对行星系统形成理论产生重大影响。

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.