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arXiv 1012.2327gr-qchep-thmath-phmath.MP
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狄拉克波函数的四矢量和四标量表示

Four-vector vs. four-scalar representation of the Dirac wave function

Mayeul Arminjon, Frank Reifler

备注：
30 pages (standard 12pt). v2: version accepted for publication in Int. J. Geom. Meth. Mod. Phys. Some emphasis and a clarification in Sect. 2.1. The Appendix now proves that the complex tangent bundle is a spinor bundle according to precisely the definition given in Sect. 2.1. Proof of the main Theorem 2 made easier to follow
发表信息：
Int. J. Geom. Meth. Mod. Phys., Vol. 9, 1250026 (2012)
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摘要：

在闵可夫斯基时空中，我们可以在自旋群下变换狄拉克波函数，就像在庞加莱群下变换坐标一样。这在弯曲的时空中是不可能的。因此，在Fock和Weyl独立提出的方程中，Dirac波函数的四个复数分量在一般坐标变换下变换为标量。最近的工作表明，协变复四向量表示也是可能的。使用向量束理论的概念，我们描述这两个表示在一个统一的框架。我们证明定理，涉及到一起的不同表示和不同的选择，在每个表示的连接。因此，这两种表示中的任何一种都可以解释弯曲时空中的各种不等价的、线性的、协变的狄拉克方程，这些狄拉克方程可以简化为闵可夫斯基时空中的原始狄拉克方程。特别是，我们表明，标准狄拉克方程在弯曲的时空中，与任何选择的四元组字段，是等价的协变狄拉克方程的一个特定的实现为一个复杂的四矢量波函数。

英文摘要：

In a Minkowski spacetime, one may transform the Dirac wave function under the spin group, as one transforms coordinates under the Poincaré group. This is not an option in a curved spacetime. Therefore, in the equation proposed independently by Fock and Weyl, the four complex components of the Dirac wave function transform as scalars under a general coordinate transformation. Recent work has shown that a covariant complex four-vector representation is also possible. Using notions of vector bundle theory, we describe these two representations in a unified framework. We prove theorems that relate together the different representations and the different choices of connections within each representation. As a result, either of the two representations can account for a variety of inequivalent, linear, covariant Dirac equations in a curved spacetime that reduce to the original Dirac equation in a Minkowski spacetime. In particular, we show that the standard Dirac equation in a curved spacetime, with any choice of the tetrad field, is equivalent to a particular realization of the covariant Dirac equation for a complex four-vector wave function.

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