# 相对论角动量

## 狭义相对论

### 轨道三维角动量

#### 叉积定义：赝矢量

${\displaystyle \mathbf {L} =\mathbf {x} \times \mathbf {p} }$

${\displaystyle L_{3}=x_{1}p_{2}-x_{2}p_{1}}$
${\displaystyle L_{1}=x_{2}p_{3}-x_{3}p_{2}}$
${\displaystyle L_{2}=x_{3}p_{1}-x_{1}p_{3}\,.}$

#### 楔积定义：反对称张量

${\displaystyle \mathbf {L} =\mathbf {x} \wedge \mathbf {p} }$

${\displaystyle L^{ij}=x^{i}p^{j}-x^{j}p^{i}=2x^{[i}p^{j]}}$

{\displaystyle {\begin{aligned}\mathbf {L} &={\begin{pmatrix}L^{11}&L^{12}&L^{13}\\L^{21}&L^{22}&L^{23}\\L^{31}&L^{32}&L^{33}\\\end{pmatrix}}={\begin{pmatrix}0&L_{xy}&L_{xz}\\L_{yx}&0&L_{yz}\\L_{zx}&L_{zy}&0\end{pmatrix}}={\begin{pmatrix}0&L_{xy}&-L_{zx}\\-L_{xy}&0&L_{yz}\\L_{zx}&-L_{yz}&0\end{pmatrix}}\\&={\begin{pmatrix}0&xp_{y}-yp_{x}&-(zp_{x}-xp_{z})\\-(xp_{y}-yp_{x})&0&yp_{z}-zp_{y}\\zp_{x}-xp_{z}&-(yp_{z}-zp_{y})&0\end{pmatrix}}\end{aligned}}}

## 参考文献

1. ^ D.S.A. Freed, K.K.A. Uhlenbeck. Geometry and quantum field theory 2nd. Institute For Advanced Study (Princeton, N.J.): American Mathematical Society. ISBN 0-821-886-835.
2. ^ 罗杰·彭罗斯. The Road to Reality. Vintage books. 2005: 433. ISBN 978-00994-40680. 彭罗斯在楔积使用了2的因子，其他作者可能沿用。