This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time ${\mathbb{R}}^4$ and its subgroup SO(3) of the rotations of three-dimensional Euclidean space ${\mathbb{R}}^3$. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.

中文翻译：

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旋转子的几何意义为理解量子力学提供了途径

本文旨在解释，理解量子力学（QM）的关键是对在其公式中使用的自旋代数的完美几何理解。在某些对称群的表示理论中，旋转子自然地出现。与QM相关的旋转子是Minkowski时空中均质Lorentz群SO（3,1）的旋转子${\mathbb{[R}}^4$ 欧几里得空间的旋转的子集及其子集SO（3） ${\mathbb{[R}}^3$。在三维旋转组中，旋转轴出现在其表示SU（2）中。我们将为读者提供有关Spinor代数背后的动态的完美直观见解。然后，我们将利用获得的理解从头开始推导出自由空间Dirac方程，证明这完全是Ballentine对QM的统计解释的精髓，它是对匀速运动的旋转电子的统计集合的描述。这是数学上严格的证明。进一步发展，我们允许存在电磁场。我们可以将结果视为基于对旋子代数的几何理解的QM重建。通过讨论传统方法的解释中的许多问题，