标题： 5维Stuekelberg电动力学中的Abraham-Lorentz-Dirac方程 显示英文标题

标题： Abraham-Lorentz-Dirac Equation in 5D Stuekelberg Electrodynamics

摘要： 我们在与Stueckelberg显式协变正则力学相关的电动力学理论框架内推导了Abraham-Lorentz-Dirac（ALD）方程。 在这个框架中，粒子的世界线通过事件 $x^\mu(\tau)$的演化来描绘。 通过允许位置和速度之间不受约束的对易关系，所关联的电磁规范场通常依赖于参数 $\tau$，它在牛顿力学中起到时间的作用。 标准的麦克斯韦理论从这个系统中以 $\tau$-无关的平衡极限形式出现。 本文中，我们计算由任意演化的事件诱导出的 $\tau$-相关场，并研究长程辐射部分，发现其是一种Maxwell类型的驻波。 按照狄拉克的方法，我们得到了自相互作用有限部分的表达式，从而得到推广了洛伦兹力的ALD方程。 该三阶微分方程随后被转化为积分-微分方程，除了场的 $\tau$-依赖性外，它与标准麦克斯韦表达式相同。 通过详细研究这种 $\tau$-依赖性，我们表明场可以从积分中移除，因此洛伦兹力仅依赖于瞬时外部场以及事件演化的动力学变量的积分。 在这种形式下，事件不会因场的未来值而产生前加速。 显示英文摘要

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 $x^\mu(\tau)$. By admitting unconstrained commutation relations between the positions and velocities, the associated electromagnetic gauge fields are in general dependent on the parameter $\tau$, which plays the role of time in Newtonian mechanics. Standard Maxwell theory emerges from this system as a $\tau$-independent equilibrium limit. In this paper, we calculate the $\tau$-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 $\tau$-dependence of the field. By studying this $\tau$-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.