无迹矩阵和伽马矩阵
一个矩阵$A$的迹\(\text{Tr}A = 0\)则称为无迹矩阵,容易验证,
\[\text{Tr}(A \cdot B) = \text{Tr}(B \cdot A),\text{Tr}(A \pm B) = \text{Tr}A \pm \text{Tr}B\]
\(\begin{pmatrix}
a & b \\
c & - a \\
\end{pmatrix}^{2} = \begin{pmatrix}
a^{2} + bc & 0 \\
0 & a^{2} + bc \\
\end{pmatrix}\),所以,若$A$是二阶无迹矩阵,则\(A^{2} = - |A|E\).
引入三个特殊的无迹矩阵,称之为伽马矩阵\(\gamma^{1} = \begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix},\gamma^{2} = \begin{pmatrix}
0 & - i \\
i & 0 \\
\end{pmatrix},\gamma^{3} = \begin{pmatrix}
1 & 0 \\
0 & - 1 \\
\end{pmatrix}\)
容易验证下面的等式:
①\(\gamma^{1} \cdot \gamma^{2} = \gamma^{3}{,\gamma}^{2} \cdot \gamma^{3} = \gamma^{1},\gamma^{3} \cdot \gamma^{1} = \gamma^{2},\gamma^{2} \cdot \gamma^{1} = - \gamma^{3}{,\gamma}^{3} \cdot \gamma^{2} = - \gamma^{1},\gamma^{1} \cdot \gamma^{3} = - \gamma^{2}\)
②\(\gamma^{1} \cdot \gamma^{1} = \gamma^{2} \cdot \gamma^{2} = \gamma^{3} \cdot \gamma^{3} = 1\)
(上面的两组等式可以合并为\(\gamma^{i} \cdot \gamma^{j} = \delta^{{ij}}E + i\varepsilon^{{ijk}}\gamma^{k}\)
此处采用哑指标求和规则,\(\delta^{{ij}}\)为Dirac记号,定义为\(i = j,\delta^{{ij}} = 1;i \neq j,\delta^{{ij}} = 0,\)
另一个符号\(\delta^{{ijk}}\)为Levi-Civita记号,定义为\(\varepsilon^{123} = 1,\varepsilon^{{ijk}} = - \varepsilon^{{jik}} = \varepsilon^{{jki}}\))
显然任何二阶无迹矩阵可以表示为伽马矩阵的线性组合\begin{equation}
M = m^{1}\gamma^{1} + m^{2}\gamma^{2} + m^{3}\gamma^{3}\label2\end{equation}
反过来,任何一个二阶无迹矩阵$M$都对应着一个三维矢量\(\overrightarrow{T}\)使得$[\overrightarrow{T}]=M$
式\eqref{2}和三维矢量的点积\(\overrightarrow{A} \cdot \overrightarrow{B} = A^{1}B^{1} + A^{2}B^{2} + A^{3}B^{3}\)是类似的,这启发我们对任一个三维矢量\(\overrightarrow{T} = \left( T^{1},T^{2},T^{3} \right)\)定义它对应的二阶无迹矩阵\(\lbrack T\rbrack = T^{1}\gamma^{1} + T^{2}\gamma^{2} + T^{3}\gamma^{3} = \begin{pmatrix}
T^{3} & T^{1} - iT^{2} \\
T^{1} + iT^{2} & - T^{3} \\
\end{pmatrix}\).
于是形式上可以写出\(\overrightarrow{T} \cdot \overrightarrow{\gamma} = \lbrack T\rbrack\),其中\(\overrightarrow{\gamma} = (\gamma^{1},\gamma^{2},\gamma^{3})\)
对于平面上一个点$p=x+iy$,定义它对应的三维矢量\(\overrightarrow{p} = (bx, - iab,ay)\).
那么点$p$对应的矩阵为\((p) = \begin{pmatrix}
ay& bx - ab \\
bx + ab & - ay \\
\end{pmatrix}\)
由于分式线性函数在复合运算下和二阶矩阵的一维线性空间在矩阵乘法运算下是同态的,即\(F_{p}(\omega) = \lbrack p\rbrack(\omega)\),因此映射\(F_{p}\)在复合运算下和一维矩阵空间\(\left\{ \lambda\lbrack p\rbrack \mid \lambda \in \mathbb{R}^{*} \right\}\)在矩阵乘法运算下是同态的.所以可以用\(\lbrack p\rbrack\)来描述\(F_{p}\),进而描述点$p$本身.
无穷远点的引入:\(\overrightarrow{p}\)和三维矢量并非一一对应,因为\(p\)的第二个分量\(-iab≠0\),称\(\overrightarrow{k}=(bu,0,av)\)所描述的点\(k\)为斜率为\(\frac{v}{u}\)的平行直线的无穷远点.
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