5D Stuekelberg电动力学中的Abraham-Lorentz-Dirac方程
Abraham-Lorentz-Dirac Equation in 5D Stuekelberg Electrodynamics
摘要：
我们推导出亚伯拉罕-洛伦兹-狄拉克（ALD）方程的电动力学理论与Stueckelberg明显协变正则力学的框架。在这个框架中，粒子世界线是通过事件$x^μ(τ)$的演化来追踪的。通过承认位置和速度之间的无约束对易关系，相关的电磁规范场一般依赖于参数$τ$，它在牛顿力学中扮演时间的角色。标准麦克斯韦理论从这个系统作为一个$τ$独立的平衡极限出现。本文计算了由任意演化事件引起的$τ$相关场，并研究了长程辐射部分，它被看作是麦克斯韦型的壳层平面波。根据狄拉克的方法，我们得到了自相互作用的有限部分的表达式，这导致ALD方程的推广的洛伦兹力。然后将该三阶微分方程转换为积分微分方程，除了场的$τ$依赖性之外，该积分微分方程与标准麦克斯韦表达式相同。通过详细研究这种$τ$依赖性，我们表明，该领域可以从集成，使洛伦兹力只依赖于瞬时外场和事件演化的动力学变量的积分。在这种形式中，不存在通过字段的未来值对事件的预加速。
英文摘要：
We derive the Abraham-Lorentz-Dirac (ALD) equation in the framework of the electrodynamic theory associated with Stueckelberg manifestly covariant canonical mechanics. In this framework, a particle worldline is traced out through the evolution of an event $x^μ(τ)$. By admitting unconstrained commutation relations between the positions and velocities, the associated electromagnetic gauge fields are in general dependent on the parameter $τ$, which plays the role of time in Newtonian mechanics. Standard Maxwell theory emerges from this system as a $τ$-independent equilibrium limit. In this paper, we calculate the $τ$-dependent field induced by an arbitrarily evolving event, and study the long-range radiation part, which is seen to be an on-shell plane wave of the Maxwell type. Following Dirac's method, we obtain an expression for the finite part of the self-interaction, which leads to the ALD equation that generalizes the Lorentz force. This third-order differential equation is then converted to an integro-differential equation, identical to the standard Maxwell expression, except for the $τ$-dependence of the field. By studying this $τ$-dependence in detail, we show that field can be removed from the integration, so that the Lorentz force depends only on the instantaneous external field and an integral over dynamical variables of the event evolution. In this form, pre-acceleration of the event by future values of the field is not present.
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