Dirac方程作为Cosserat弹性的特例
Dirac equation as a special case of Cosserat elasticity
摘要：
我们提出了一种替代的数学模型的电子，其中的动力学变量是一个coframe（领域的标准正交基）和密度。电子质量和外部电磁场被纳入我们的模型通过卡鲁扎-克莱因扩展。我们的拉格朗日密度与轴向挠率的平方成正比。我们的方法的优点是，它不需要使用旋量，泡利矩阵或协变微分。唯一的几何概念，我们使用的是那些度量，微分形式，楔形产品和外部衍生物。我们证明了在不依赖于第三空间坐标的特殊情况下，我们的模型等价于狄拉克方程。证明的关键要素是观察到我们的拉格朗日允许因子分解。
英文摘要：
We suggest an alternative mathematical model for the electron in which the dynamical variables are a coframe (field of orthonormal bases) and a density. The electron mass and external electromagnetic field are incorporated into our model by means of a Kaluza-Klein extension. Our Lagrangian density is proportional to axial torsion squared. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. We prove that in the special case with no dependence on the third spatial coordinate our model is equivalent to the Dirac equation. The crucial element of the proof is the observation that our Lagrangian admits a factorisation.
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