绝热矢量开普勒问题中几何电磁学的失效
Failure of geometric electromagnetism in the adiabatic vector Kepler problem
摘要：
绕直线载流导线运行的粒子的磁矩可能在导线磁场中足够快地进动，以证明绝热近似的合理性，消除了磁矩的快速时间依赖性，只留下粒子位置作为缓慢的自由度。在绝热膨胀的零级，粒子在垂直于金属丝的平面内的轨道是开普勒椭圆。高阶后绝热修正使轨道进动，但最近的分析表明，这个“矢量开普勒问题”的有效哈密顿量纳入后绝热标量位（“几何电磁学”）未能正确预测的进动，而一个启发式的替代成功。在本文中，我们解决了后绝热近似的明显失败，指出正确的二阶分析产生第三哈密顿量，其中几何电磁补充张量势。启发式的Schmiedmayer和Scrinzi的哈密顿量，然后示出是一个典型的变换的正确的绝热哈密顿量，二阶。该变换具有去除$1/r^3$奇异性的重要优点，该奇异性是绝热近似的伪影。
英文摘要：
The magnetic moment of a particle orbiting a straight current-carrying wire may precess rapidly enough in the wire's magnetic field to justify an adiabatic approximation, eliminating the rapid time dependence of the magnetic moment and leaving only the particle position as a slow degree of freedom. To zeroth order in the adiabatic expansion, the orbits of the particle in the plane perpendicular to the wire are Keplerian ellipses. Higher order post-adiabatic corrections make the orbits precess, but recent analysis of this `vector Kepler problem' has shown that the effective Hamiltonian incorporating a post-adiabatic scalar potential (`geometric electromagnetism') fails to predict the precession correctly, while a heuristic alternative succeeds. In this paper we resolve the apparent failure of the post-adiabatic approximation, by pointing out that the correct second-order analysis produces a third Hamiltonian, in which geometric electromagnetism is supplemented by a tensor potential. The heuristic Hamiltonian of Schmiedmayer and Scrinzi is then shown to be a canonical transformation of the correct adiabatic Hamiltonian, to second order. The transformation has the important advantage of removing a $1/r^3$ singularity which is an artifact of the adiabatic approximation.
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