In this article, we give all the Weitzenböck-type formulas among the geometric first-order differential operators on the spinor fields with spin \(j+1/2\) over Riemannian spin manifolds of constant curvature. Then, we find an explicit factorization formula of the Laplace operator raised to the power \(j+1\) and understand how the spinor fields with spin \(j+1/2\) are related to the spinors with lower spin. As an application, we calculate the spectra of the operators on the standard sphere and clarify the relation among the spinors from the viewpoint of representation theory. Next we study the case of trace-free symmetric tensor fields with an application to Killing tensor fields. Lastly we discuss the spinor fields coupled with differential forms and give a kind of Hodge–de Rham decomposition on spaces of constant curvature.

在本文中，我们给出了在常曲率黎曼自旋流形上自旋为\(j+1/2\)的旋量场上的几何一阶微分算子中的所有 Weitzenböck 型公式。然后，我们找到了拉普拉斯算子的显式分解公式\(j+1\)的幂，并理解了自旋\(j+1/2\)与自旋较低的旋翼有关。作为一个应用，我们计算了标准球面上算子的谱，从表示论的角度阐明了旋量之间的关系。接下来，我们研究无迹对称张量场的情况，并将其应用于杀死张量场。最后，我们讨论了与微分形式耦合的旋量场，并给出了一种关于常曲率空间的 Hodge-de Rham 分解。