Using the nonrelativistic approximation in the curved-space Dirac equation, the analog of the Pauli equation is derived for a weak gravitational field with a gauge fixing condition related to the synchronous gauge, in the presence of an electromagnetic field. Different from the previous works, which were employing either the exact or conventional Foldy-Wouthuysen transformations, here we perform calculations by directly performing a nonrelativistic approximation, which reduced in the power series expansion in the inverse mass of the spinning particle. On top of that, the equations of motion for the massive spin-1/2 charged particle are obtained. The two particular cases of the previously explored backgrounds, namely (a) plane gravitational wave and (b) homogeneous static gravitational field are considered for control. In the case (a), we meet correspondence with the previous results. On the other hand, in case (b), there is no correspondence with neither perturbative nor with exact Foldy-Wouthuysen transformations, which we also recalculate and agree with the previous works. The disagreement is a kind of a theoretical challenge and most likely occurs because the potential energy, in the particular case of Newtonian approximation, is proportional to the mass of the particle.

中文翻译：

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泡利方程和弱引力场中的带电自旋 1/2 粒子


使用曲面空间狄拉克方程中的非相对论近似，在电磁场存在的情况下，推导出了具有与同步应变片相关的规范固定条件的弱引力场的泡利方程的模拟。与之前采用精确或常规 Foldy-Wouthuysen 变换的工作不同，这里我们通过直接执行非相对论近似来执行计算，该近似在旋转粒子的反质量中减少了幂级数膨胀。最重要的是，得到了大质量自旋 1/2 带电粒子的运动方程。考虑对先前探索的背景的两个特殊情况，即 （a） 平面引力波和 （b） 均匀静态引力场进行控制。在情况 （a） 中，我们满足与先前结果的对应。另一方面，在情况 （b） 中，与微扰变换和精确 Foldy-Wouthuysen 变换都没有对应关系，我们也重新计算并与以前的工作一致。这种分歧是一种理论挑战，很可能是因为势能（在牛顿近似的特殊情况下）与粒子的质量成正比。