论文标题
根据dirac代数与基本无关计算的注释
Notes on basis-independent computations with the Dirac algebra
论文作者
论文摘要
在这些注释中，我们首先回顾了保利的“基本定理”证明，该证明指出了任何两组狄拉克矩阵$ \ {γ^μ\} $的等效性。由于这个定理，不仅在狄拉克方程的上下文中的所有物理结果都必须独立于为dirac矩阵选择的基础，而且还应该有可能在计算过程中而无需求助于特定基础而获得结果。的确，我们在洛伦兹转换下的狄拉克纺纱器的行为，迪拉克场的量化，旋转操作员的期望值和其他几个主题的情况下证明了这一点。特别是，我们强调了矩阵$β$的完全不同的物理和数学背景，用于偶联的狄拉克旋转器的定义和$γ^0 $。最后，我们将独立的操作与在dirac矩阵的韦尔基础上进行的操作进行了比较。本注释仅利用Dirac代数的简单性和Pauli定理的力量，为狄拉克理论提供了独立的介绍。
In these notes we first review Pauli's proof of his `fundamental theorem' that states the equivalence of any two sets of Dirac matrices $\{ γ^μ\}$. Due to this theorem not only all physical results in the context of the Dirac equation have to be independent of the basis chosen for the Dirac matrices, but it should also be possible to obtain the results without resorting to a specific basis in the course of the computation. Indeed, we demonstrate this in the case of the behaviour of Dirac spinors under Lorentz transformations, the quantization of the Dirac field, the expectation value of the spin operator and several other topics. In particular, we emphasize the totally different physics and mathematics background of the matrix $β$, used in the definition of the conjugate Dirac spinor, and $γ^0$. Finally, we compare the basis-independent manipulations with those performed in the Weyl basis of Dirac matrices. The present notes provide a self-contained introduction to the Dirac theory by solely exploiting the simplicity of the Dirac algebra and the power of Pauli's Theorem.
