广义高阶场几何及其在经典线性电动力学中的应用
Geometry of generalized higher order fields and applications to classical linear electrodynamics
摘要：
为得到线性经典电动力学中点荷电粒子辐射反应的一致数学描述，引入了广义高阶张量和微分形式理论。对微分几何和微分形式理论中的一些基本概念进行了推广。特别地，发展了广义高阶形式的上同调和积分理论，包括Cartan演算，de Rham上同调的推广和Thom同构定理的一个版本。我们详细考虑了一种特殊类型的广义高阶张量与有界最大$n$ -加速度，并使用它作为一个模型的时空。介绍了高阶场电动力学理论的一种推广。我们证明，结合广义高阶场和最大加速度几何，点带电粒子与广义高阶场相互作用的演化可以用一个隐式二阶常微分方程的解来描述。在平坦空间中，这样的方程是洛伦兹不变的，没有狄拉克型的预加速解或逃逸解，它与牛顿第一动力学定律和协变拉莫尔功率辐射定律相容。还介绍了麦克斯韦-洛伦兹理论的推广。该理论是线性的领域部门和它减少到标准的麦克斯韦-洛伦兹电动力学时，最大加速度是无限的。最后，我们讨论了我们的框架的假设，除了一些理论的预测。
英文摘要：
Motivated by obtaining a consistent mathematical description for the radiation reaction of point charged particles in linear classical electrodynamics, a theory of generalized higher order tensors and differential forms is introduced. The generalization of some fundamental notions of the differential geometry and the theory of differential forms is presented. In particular, the cohomology and integration theories for generalized higher order forms are developed, including the Cartan calculus, a generalization of de Rham cohomology and a version of Thom's isomorphism theorem. We consider in detail a special type of generalized higher order tensors associated with bounded maximal $n$-acceleration and use it as a model of spacetime. A generalization of electrodynamic theory with higher order fields is introduced. We show that combining the generalized higher order fields with maximal acceleration geometry the evolution of a point charged particle interacting with the generalized higher order fields can be described by solutions of an implicit second order ordinary differential equation. In flat space such equation is Lorentz invariant, does not have pre-accelerated solutions of Dirac's type or run-away solutions, it is compatible with Newton's first law of dynamics and with the covariant Larmor's power radiation law. A generalization of the Maxwell-Lorentz theory is also introduced. The theory is linear in the field sector and it reduces to the standard Maxwell-Lorentz electrodynamics when the maximal acceleration is infinite. Finally, we discuss the assumptions of our framework in addition to some predictions of the theory.
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