一阶偏微分方程组的分析
Analysis of first order systems of partial differential equations
摘要：
研究了作用在n维无界流形上复值半密度m列上的形式自伴一阶线性微分算子。我们研究了特征值的分布在椭圆设置和传播在双曲设置，推导出两项渐近公式。然后，我们把注意力转向四维中2乘2算子的特殊情况。我们发现，在我们的分析过程中，Lorentzian度规、Pauli矩阵、旋量场、旋量场的连接系数、电磁余矢势、Dirac方程和Dirac作用量等几何概念自然地产生出来。
英文摘要：
The paper deals with a formally self-adjoint first order linear differential operator acting on m-columns of complex-valued half-densities over an n-manifold without boundary. We study the distribution of eigenvalues in the elliptic setting and the propagator in the hyperbolic setting, deriving two-term asymptotic formulae for both. We then turn our attention to the special case of a two by two operator in dimension four. We show that the geometric concepts of Lorentzian metric, Pauli matrices, spinor field, connection coefficients for spinor fields, electromagnetic covector potential, Dirac equation and Dirac action arise naturally in the process of our analysis.
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