Using the well known complex representation of the proper Lorentz group \(\mathrm {SO}_+(3,1)\cong \mathrm {PSL}(2,{\mathbb {C}})\cong \mathrm {SO}(3,{\mathbb {C}})\) we study some Coriolis type effects in Special Relativity and Electromagnetism in close analogy with the more traditional kinematical treatment of the group of spatial rotations. Namely, we begin with the Clifford group of \({\mathbb {R}}^3\) viewed as a complexification of \({\mathbb {H}}^\times \) and consider the associated Maurer-Cartan form which yields a complex-valued analogue of the angular velocity characterizing the action of \(\mathrm {SO}(3)\) in rigid body kinematics. It appears in a linear ODE’s written in biquaternion form or a Ricatti equation if one works with its projective version instead, providing a far richer structure compared to the real case without imposing serious technical obstruction at least in the decomposable setting, which is our main emphasis due to its importance in physics. There are several distinct terms in the non-commutative part of the so obtained connection describing well known effects named after Coriolis, Thomas, Hall and Sagnac. We also consider a restriction to the so-called Wigner little groups \(\mathrm {SO}(3)\), \(\mathrm {SO}_+(2,1)\) and \(\mathrm {E}(2)\) discussing algebraic properties of the electromagnetic field. Some familiar constructions such as geometric phases, Hopf bundles and the Fubini-Study form appear naturally with this approach.

使用正确洛伦兹群的众所周知的复表示\(\mathrm {SO}_+(3,1)\cong \mathrm {PSL}(2,{\mathbb {C}})\cong \mathrm {SO} (3,{\mathbb {C}})\)我们研究了狭义相对论和电磁学中的一些科里奥利型效应，与更传统的空间旋转组运动学处理非常相似。即，我们从\({\mathbb {R}}^3\)的 Clifford 群开始，将其视为\({\mathbb {H}}^\times \)的复化，并考虑关联的 Maurer-Cartan 形式产生表征\(\mathrm {SO}(3)\)作用的角速度的复值模拟在刚体运动学中。如果使用其投影版本，它会出现在以双四元数形式或 Ricatti 方程编写的线性 ODE 中，与实际情况相比，它提供了更丰富的结构，至少在可分解环境中不会造成严重的技术障碍，这是我们的主要重点由于它在物理学中的重要性。在如此获得的连接的非交换部分中有几个不同的术语，描述了以 Coriolis、Thomas、Hall 和 Sagnac 命名的众所周知的效应。我们还考虑对所谓的 Wigner 小群\(\mathrm {SO}(3)\)、\(\mathrm {SO}_+(2,1)\)和\(\mathrm {E} (2)\)讨论电磁场的代数性质。一些熟悉的结构，如几何相位、Hopf 束和 Fubini-Study 形式在这种方法中自然出现。