任意电磁背景下的Frenkel电子与磁Zitterbewegung
Frenkel electron on an arbitrary electromagnetic background and magnetic Zitterbewegung
摘要：
给出了相对论自旋二分之一粒子的位置和自旋的拉格朗日量，它包含了必要的约束条件和动力学方程。该模型对任意磁矩值$μ$和任意电磁背景都是一致的。我们的方程与Frenkel的方程在Frenkel的近似下相吻合，后者是由Frenkel得到的。从近似方程到精确方程的过渡产生了理论的两个结构修正。首先，自旋张量的Frenkel条件转化为Pirani条件。其次，正则动量不再与速度成正比。因此，即使当$μ=1$（Frenkel情况）时，完全方程和近似方程预测粒子的不同行为。动量和速度之间的差异意味着对自旋轨道相互作用的额外贡献。为了估计贡献，我们找到了精确解的情况下，均匀磁场的完整方程。当Frenkel电子绕圆运动时，我们的粒子经历磁{ \it Zitterbewegung}，即在磁场的方向上以快粒子的康普顿波长量级的振幅振荡。此外，该粒子具有偶极矩。
英文摘要：
We present Lagrangian which implies both necessary constraints and dynamical equations for position and spin of relativistic spin one-half particle. The model is consistent for any value of magnetic moment $μ$ and for arbitrary electromagnetic background. Our equations coincide with those of Frenkel in the approximation in which the latter have been obtained by Frenkel. Transition from approximate to exact equations yields two structural modifications of the theory. First, Frenkel condition on spin-tensor turns into the Pirani condition. Second, canonical momentum is no more proportional to velocity. Due to this, even when $μ=1$ (Frenkel case), the complete and approximate equations predict different behavior of particle. The difference between momentum and velocity means extra contribution into spin-orbit interaction. To estimate the contribution, we found exact solution to complete equations for the case of uniform magnetic field. While Frenkel electron moves around the circle, our particle experiences magnetic {\it Zitterbewegung}, that is oscillates in the direction of magnetic field with amplitude of order of Compton wavelength for the fast particle. Besides, the particle has dipole electric moment.
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