AI 中文总结
研究\(U(1)\times U(1)\)双荷量子电动力学，用施温格固有时间方法推导单圈有效拉格朗日量，分析电磁响应，得到有效介电常数等，模型有真空双折射，磁荷消失时结果与标准欧拉 - 海森堡预测一致。
AI 中文摘要
利用施温格固有时间方法推导了具有\(U(1) \times U(1)\)规范对称性的双荷量子电动力学（dQED）的单圈有效拉格朗日量。确定了施温格对产生极限的类似物并计算了相应的非线性运动方程。通过在纯磁背景附近将运动方程线性化来分析模型的电磁响应。从所得平面波解中，获得了有效介电常数和磁导率张量，以及平行和垂直极化模式的相关色散关系和折射率。在对称情况\(q_{(1)} = q_{(2)}\)下，折射率涉及两个规范扇区的混合叠加。在此背景下，模型表现出真空双折射，表明量子真空表现为各向异性介质。当磁荷消失时，所有结果都一致地简化为QED的标准欧拉 - 海森堡预测。
英文摘要
The one-loop effective Lagrangian of quantum electrodynamics for dyons (dQED) with a $U(1) \times U(1)$ gauge symmetry is derived using the Schwinger proper-time method. We identify the analog of the Schwinger pair-production limit and compute the corresponding nonlinear equations of motion. The electromagnetic response of the model is analyzed by linearizing the equations of motion around a purely magnetic background. From the resulting plane-wave solutions, we obtain the effective permittivity and permeability tensors, along with the associated dispersion relations and refractive indices for parallel and perpendicular polarization modes. The refractive indices involve hybrid superpositions of both gauge sectors, as illustrated in the symmetric case $q_{(1)} = q_{(2)}$. In this background, the model exhibits vacuum birefringence, indicating that the quantum vacuum behaves as an anisotropic medium. All results consistently reduce to the standard Euler-Heisenberg predictions of QED when the magnetic charge vanishes.
