This paper derives a linear, first-order, partial differential field equation (a Dirac-like equation) in the geometric calculus of the geometric algebra \({\mathcal {G}}_{4,1}\) that has free plane-wave solutions distinct from one another that correspond to the left and right chiral states of the electron and the neutrino. Besides the usual spacetime dependence of plane waves, the solutions have a multivector structure yielding a ladder of states with raising and lowering operators appropriate to electroweak theory and having an \(SU(2)_L\) relationship among the chiral electron and neutrino states. The required Dirac-like equation in \({\mathcal {G}}_{4,1}\) results from a systematic review of Dirac-like equations (i.e., first-order field equations whose solutions also satisfy the Klein–Gordon equation) in geometric algebras of lower dimension.

本文在具有自由平面的几何代数\({\mathcal {G}}_{4,1}\)的几何微积分中推导了一个线性、一阶、偏微分场方程（类狄拉克方程）-wave 解彼此不同，对应于电子和中微子的左右手征状态。除了平面波通常的时空相关性之外，这些解具有多向量结构，产生具有适合电弱理论的升降算子的状态阶梯，并且在手征电子和中微子状态之间具有\(SU(2)_L\)关系。\({\mathcal {G}}_{4,1}\) 中所需的类狄拉克方程 这是对低维几何代数中类狄拉克方程（即，其解也满足 Klein-Gordon 方程的一阶场方程）的系统回顾的结果。