对称平行引力中广义狄拉克方程的Foldy-Wouthuysen变换
Foldy--Wouthuysen Transformation of the Generalized Dirac Equation in Symmetric Teleparallel Gravity
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中文总结 AI 辅助
研究对称平行引力中广义狄拉克方程的非相对论极限，推导其到 $1/m^2$ 阶的Foldy-Wouthuysen变换后的有效哈密顿量，发现新的低能相互作用通道，为约束广义旋量联络的额外耦合提供框架。
中文摘要 AI 辅助
我们研究对称平行引力弱静态球对称背景下广义狄拉克方程的非相对论极限。该广义旋量联络包含完整克利福德代数基，引入了超出传统狄拉克理论的非度量 sector 额外耦合。在重合规范下采用各向同性坐标的弱场史瓦西几何，我们推导了对应的广义狄拉克哈密顿量，并执行了到 $1/m^2$ 阶的逐次Foldy-Wouthuysen变换，保留引力势及其空间导数的一阶项。所得块对角哈密顿量不仅包含预期的动能、自旋-轨道和达尔文相互作用的引力对应项，还包含广义旋量联络产生的额外算子结构。特别地，直接自旋-引力、各向异性自旋-动量-引力及潮汐自旋-动量耦合自然产生于广义度量-仿射相互作用。我们还对地球弱引力场中的电子进行了量级分析，以证明逆质量展开截断的合理性。这些结果表明，对称平行背景下的广义狄拉克方程产生了涉及费米子自旋、动量及引力场空间导数的新低能相互作用通道，所得有效哈密顿量为探索广义旋量联络中额外耦合的唯象约束提供了框架。
英文摘要
We investigate the non-relativistic limit of the generalized Dirac equation in a weak, static, and spherically symmetric background of symmetric teleparallel gravity. The underlying generalized spinor connection incorporates the complete Clifford-algebra basis and introduces additional couplings to the non-metricity sector beyond those of the conventional Dirac theory. Working in the coincident gauge and adopting the weak-field Schwarzschild geometry in isotropic coordinates, we derive the corresponding generalized Dirac Hamiltonian and perform successive Foldy--Wouthuysen transformations up to order $1/m^2$, retaining terms to first order in the gravitational potential and its spatial derivatives. The resulting block-diagonal Hamiltonian contains not only the expected gravitational counterparts of the kinetic, spin--orbit, and Darwin interactions, but also additional operator structures generated by the generalized spinor connection. In particular, direct spin--gravity, anisotropic spin--momentum--gravity, and tidal spin--momentum couplings arise naturally from the generalized metric-affine interaction. We further perform an order-of-magnitude analysis for an electron in the Earth's weak gravitational field to justify the adopted truncation of the inverse-mass expansion. These results demonstrate that the generalized Dirac equation in a symmetric teleparallel background gives rise to new low-energy interaction channels involving the fermion spin, momentum, and spatial derivatives of the gravitational field. The resulting effective Hamiltonian provides a framework for exploring phenomenological constraints on the additional couplings entering the generalized spinor connection.
发表机构
- Pamukkale University（帕穆克卡莱大学）
- Erciyes University（埃尔吉耶斯大学）
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