旋量

${\eta }_{mn}\sim (-1,1,1,1)$，与度规相同，反称张量在$\text{SL}(2,\mathbb{C})$下不变

$${\epsilon }_{\alpha \beta }=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right){\epsilon }^{\alpha \beta }=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$

$${\epsilon }_{\alpha \beta }=M_{\alpha }^{\gamma }{M}_{\beta }^{\delta }{\epsilon }_{\gamma \delta }{\epsilon }^{\alpha \beta }={\epsilon }^{\gamma \delta }{M}_{\gamma }^{\alpha }{M}_{\delta }^{\beta }$$

分别服从Lorentz群旋量表示$(1/2,0),(0,1/2)$下变换关系的 $\psi ,\overline{\psi }$ 称为Weyl旋量

$${\psi }_{\alpha }={\epsilon }_{\alpha \beta }{\psi }^{\beta }{\psi }^{\alpha }={\epsilon }^{\alpha \beta }{\psi }_{\beta }$$

$${\psi }_{\alpha }^{\prime }=M_{\alpha }^{\beta }{\psi }_{\beta }{\overline{\psi }}_{\dot{\alpha }}^{\prime }=(M^{\ast }{)}_{\dot{\alpha }}^{\dot{\beta }}{\overline{\psi }}_{\dot{\beta }}$$

$${\psi }^{\prime \alpha }={M^{-1}}_{\beta }^{\alpha }{\psi }^{\beta }{\overline{\psi }}_{\dot{\alpha }}^{\prime }={(M^{\ast }{)}^{-1}}_{\dot{\beta }}^{\dot{\alpha }}{\overline{\psi }}^{\dot{\beta }}$$

Weyl旋量直和可得Dirac旋量和Majorana旋量，服从Lorentz群四维表示下变换关系

$${\mathrm{\Psi }}_D=\left(\begin{array}{c}{\chi }_{\alpha }\\ {\overline{\psi }}^{\dot{\alpha }}\end{array}\right){\mathrm{\Psi }}_M=\left(\begin{array}{c}{\chi }_{\alpha }\\ {\overline{\chi }}^{\dot{\alpha }}\end{array}\right)$$

约定以下记号，默认旋量分量乘积服从反交换律

$$\psi \chi ={\psi }^{\alpha }{\chi }_{\alpha }=-{\psi }_{\alpha }{\chi }^{\alpha }={\chi }^{\alpha }{\psi }_{\alpha }=\chi \psi$$

$$\overline{\psi }\overline{\chi }={\overline{\psi }}_{\dot{\alpha }}{\overline{\chi }}^{\dot{\alpha }}=-{\overline{\psi }}^{\dot{\alpha }}{\overline{\chi }}_{\dot{\alpha }}={\overline{\chi }}_{\dot{\alpha }}{\overline{\psi }}^{\dot{\alpha }}=\overline{\chi }\overline{\psi }$$

$$(\chi \psi {)}^{†}=({\chi }^{\alpha }{\psi }_{\alpha }{)}^{†}={\overline{\psi }}_{\dot{\alpha }}{\overline{\chi }}^{\dot{\alpha }}=\overline{\psi }\overline{\chi }=\overline{\chi }\overline{\psi }$$

Sigma矩阵

$${\sigma }^0=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right){\sigma }^1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$

$${\sigma }^2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right){\sigma }^3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

$${\gamma }^m=\left(\begin{array}{cc}0& {\sigma }^m\\ {\overline{\sigma }}^m& 0\end{array}\right){\gamma }^5={\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3=\left(\begin{array}{cc}-i& 0\\ 0& i\end{array}\right)$$

$2\times 2$ Hermitian矩阵可分解为${\sigma }^m$的实系数线性组合，洛伦兹变换保持其行列式不变

$$P=P_m{\sigma }^m$$

$$P^{\prime }=M{P}_m{\sigma }^m{M}^{†}$$

由此可得$\sigma$指标结构与升降规律

$${\overline{\sigma }}^{m\dot{\alpha }\alpha }={\epsilon }^{\dot{\alpha }\dot{\beta }}{\epsilon }^{\alpha \beta }{\sigma }_{\beta \dot{\beta }}^m$$

双旋量和矢量可以通过$\sigma$相互转化

$$v_{\alpha \dot{\alpha }}={\sigma }_{\alpha \dot{\alpha }}^m{v}_m{v}^m=-\frac{1}{2}{\overline{\sigma }}^{m\dot{\alpha }\alpha }{v}_{\alpha \dot{\alpha }}$$

Lorentz群生成元在旋量表示下可写为

$${\sigma }_{\alpha }^{mn\beta }=\frac{1}{4}({\sigma }_{\alpha \dot{\alpha }}^m{\overline{\sigma }}^{n\dot{\alpha }\beta }-{\sigma }_{\alpha \dot{\alpha }}^n{\overline{\sigma }}^{m\dot{\alpha }\beta })$$

$${\overline{\sigma }}_{\dot{\beta }}^{mn\dot{\alpha }}=\frac{1}{4}({\overline{\sigma }}^{m\dot{\alpha }\alpha }{\sigma }_{\alpha \dot{\beta }}^n-{\overline{\sigma }}^{n\dot{\alpha }\alpha }{\sigma }_{\alpha \dot{\beta }}^m)$$

与$\gamma$相似，$\sigma$具有以下性质，式中 ${\epsilon }^{0123}=1$

$${\overline{\sigma }}^0={\sigma }^0{\overline{\sigma }}^{1,2,3}=-{\sigma }^{1,2,3}$$

$$\mathrm{tr}{\sigma }^m{\overline{\sigma }}^n=-2{\eta }^{mn}$$

$${\sigma }_{\alpha \dot{\alpha }}^m{\overline{\sigma }}_m^{\dot{\beta }\beta }=-2{\delta }_{\alpha }^{\beta }{\delta }_{\dot{\alpha }}^{\dot{\beta }}$$

$$({\sigma }^m{\overline{\sigma }}^n+{\sigma }^n{\overline{\sigma }}^m{)}_{\alpha }^{\beta }=-2{\eta }^{mn}{\delta }_{\alpha }^{\beta }$$

$$({\overline{\sigma }}^m{\sigma }^n+{\overline{\sigma }}^n{\sigma }^m{)}_{\dot{\beta }}^{\dot{\alpha }}=-2{\eta }^{mn}{\delta }_{\dot{\beta }}^{\dot{\alpha }}$$

$${\sigma }_{\alpha }^{mn\alpha }=0$$

$${\sigma }_{\alpha }^{mn\beta }{\epsilon }_{\beta \gamma }={\sigma }_{\gamma }^{mn\beta }{\epsilon }_{\beta \alpha }$$

$${\epsilon }^{abcd}{\sigma }_{cd}=-2i{\sigma }^{ab}$$

$${\epsilon }^{abcd}{\overline{\sigma }}_{cd}=2i{\overline{\sigma }}^{ab}$$

$${\sigma }_{\alpha \dot{\alpha }}^m{\sigma }_{\beta \dot{\beta }}^n-{\sigma }_{\alpha \dot{\alpha }}^n{\sigma }_{\beta \dot{\beta }}^m=2[({\sigma }^{mn}\epsilon {)}_{\alpha \beta }{\epsilon }_{\dot{\alpha }\dot{\beta }}+(\epsilon {\overline{\sigma }}^{mn}{)}_{\dot{\alpha }\dot{\beta }}{\epsilon }_{\alpha \beta }]$$

$${\sigma }_{\alpha \dot{\alpha }}^m{\sigma }_{\beta \dot{\beta }}^n+{\sigma }_{\alpha \dot{\alpha }}^n{\sigma }_{\beta \dot{\beta }}^m=-{\eta }^{mn}{\epsilon }_{\alpha \beta }{\epsilon }_{\dot{\alpha }\dot{\beta }}+4({\sigma }^{lm}\epsilon {)}_{\alpha \beta }(\epsilon {\overline{\sigma }}^{ln}{)}_{\dot{\alpha }\dot{\beta }}$$

$$\mathrm{tr}{\sigma }^{mn}{\sigma }^{kl}=-\frac{1}{2}({\eta }^{mk}{\eta }^{nl}-{\eta }^{ml}{\eta }^{nk})-\frac{i}{2}{\epsilon }^{mnkl}$$

$${\sigma }^a{\overline{\sigma }}^b{\sigma }^c+{\sigma }^c{\overline{\sigma }}^b{\sigma }^a=2({\eta }^{ac}{\sigma }^b-{\eta }^{bc}{\sigma }^a-{\eta }^{ab}{\sigma }^c)$$

$${\overline{\sigma }}^a{\sigma }^b{\overline{\sigma }}^c+{\overline{\sigma }}^c{\sigma }^b{\overline{\sigma }}^a=2({\eta }^{ac}{\overline{\sigma }}^b-{\eta }^{bc}{\overline{\sigma }}^a-{\eta }^{ab}{\overline{\sigma }}^c)$$

$${\sigma }^a{\overline{\sigma }}^b{\sigma }^c-{\sigma }^c{\overline{\sigma }}^b{\sigma }^a=2i{\epsilon }^{abcd}{\sigma }_d$$

$${\overline{\sigma }}^a{\sigma }^b{\overline{\sigma }}^c-{\overline{\sigma }}^c{\sigma }^b{\overline{\sigma }}^a=-2i{\epsilon }^{abcd}{\overline{\sigma }}_d$$

旋量代数

$${\theta }^{\alpha }{\theta }^{\beta }=-\frac{1}{2}{\epsilon }^{\alpha \beta }\theta \theta {\theta }_{\alpha }{\theta }_{\beta }=\frac{1}{2}{\epsilon }_{\alpha \beta }\theta \theta$$

$${\overline{\theta }}^{\dot{\alpha }}{\overline{\theta }}^{\dot{\beta }}=\frac{1}{2}{\epsilon }^{\dot{\alpha }\dot{\beta }}\overline{\theta }\overline{\theta }{\overline{\theta }}_{\dot{\alpha }}{\overline{\theta }}_{\dot{\beta }}=-\frac{1}{2}{\epsilon }_{\dot{\alpha }\dot{\beta }}\overline{\theta }\overline{\theta }$$

$$\theta {\sigma }^m\overline{\theta }\theta {\sigma }^n\overline{\theta }=-\frac{1}{2}\theta \theta \overline{\theta }\overline{\theta }{\eta }^{mn}$$

$$(\theta \varphi )(\theta \psi )=-\frac{1}{2}(\varphi \psi )(\theta \theta )(\overline{\theta }\overline{\varphi })(\overline{\theta }\overline{\psi })=-\frac{1}{2}(\overline{\varphi }\overline{\psi })(\overline{\theta }\overline{\theta })$$

$${\epsilon }^{\alpha \beta }\frac{\partial }{\partial {\theta }^{\beta }}=-\frac{\partial }{\partial {\theta }_{\alpha }}$$

$${\epsilon }^{\alpha \beta }\frac{\partial }{\partial {\theta }^{\alpha }}\frac{\partial }{\partial {\theta }^{\beta }}\theta \theta =4{\epsilon }_{\dot{\alpha }\dot{\beta }}\frac{\partial }{\partial {\overline{\theta }}_{\dot{\alpha }}}\frac{\partial }{\partial {\overline{\theta }}_{\dot{\beta }}}\overline{\theta }\overline{\theta }=4$$

$$\chi {\sigma }^m\overline{\psi }=-\overline{\psi }{\overline{\sigma }}^m\chi (\chi {\sigma }^m\overline{\psi }{)}^{†}=\psi {\sigma }^m\overline{\chi }$$

$$\chi {\sigma }^m{\overline{\sigma }}^n\psi =\psi {\sigma }^n{\overline{\sigma }}^m\chi (\chi {\sigma }^m{\overline{\sigma }}^n\psi {)}^{†}=\overline{\psi }{\overline{\sigma }}^n{\sigma }^m\overline{\chi }$$

$$(\varphi \psi ){\overline{\chi }}_{\dot{\alpha }}=-\frac{1}{2}(\varphi {\sigma }^m\overline{\chi })(\psi {\sigma }_m{)}_{\dot{\alpha }}$$