标题： 伪微分Weyl演算在向量丛上 显示英文标题

标题： Pseudodifferential Weyl calculus on vector bundles

摘要： 我们为伪黎曼流形上的Weyl量子化发展了一个几何框架，在这个框架中，伪微分算子作用于带有任意连接的向量丛的截面。 我们构造了相关的乘积，并计算了其在展开参数中的三阶近似下的半经典展开式。 我们方法的一个核心特征是形式自伴符号与形式自伴算子之间的一一对应关系，将平空间中的已知结果扩展到了弯曲几何中。 此外，我们分析了在此设置下Wigner函数满足的Moyal方程，并提供了几个物理上有重要意义的算子的Weyl符号的显式计算，包括狄拉克算子、麦克斯韦算子、线性化的杨-米尔斯算子和线性化的爱因斯坦算子。 我们的结果为未来在弯曲时空上的量子场论、半经典分析和手征动力学理论的发展奠定了基础。 显示英文摘要

摘要： We develop a geometric framework for Weyl quantization on pseudo-Riemannian manifolds, in which pseudodifferential operators act on sections of vector bundles equipped with arbitrary connections. We construct the associated star product and compute its semiclassical expansion up to third order in the expansion parameter. A central feature of our approach is a one-to-one correspondence between formally self-adjoint symbols and formally self-adjoint operators, extending known results from flat space to curved geometries. In addition, we analyze the Moyal equation satisfied by the Wigner function in this setting and provide explicit computations of Weyl symbols for several physically significant operators, including the Dirac, Maxwell, linearized Yang-Mills, and linearized Einstein operators. Our results lay the foundation for future developments in quantum field theory on curved spacetimes, semiclassical analysis, and chiral kinetic theory.