复群的实形式 / Real form of a complex group

Woit · §40.4 Spin and the Lorentz group

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英文 OCR 草稿

Three explicit Spin real forms and their shared complexification; not a full abstract definition of a real form.

机器草稿中的相关讨论摘录

These three diferent constructions for the cases

$$Spin(4)=SU(2)\times SU(2),Spin(3,1)=SL(2,\mathbf{C})$$

and

$$Spin(2,2)=SL(2,\mathbf{R})\times SL(2,\mathbf{R})$$

correspond to diferent so-called “real forms” of a fact about complex groups that one can get by complexifying any of the examples (considering elements $(x_0,x_1,x_2,x_3)\in {\mathbf{C}}^4$ , not just in ${\mathbf{R}}^4)$ . For instance, in the $Spin(4)$ case, taking the $x_0,x_1,x_2,x_3$ in the matrix

$$\left(\begin{array}{cc}{x}_0-i{x}_3& -x_2-i{x}_1\\ x_2-i{x}_1& x_0+i{x}_3\end{array}\right)$$

to have arbitrary complex values $z_0,z_1,z_2$ , $z_3$ one gets arbitrary 2 by 2 complex matrices, and the transformation

$$\left(\begin{array}{cc}{z}_0-i{z}_3& -z_2-i{z}_1\\ z_2-i{z}_1& z_0+i{z}_3\end{array}\right)\to {\Omega }_1\left(\begin{array}{cc}{z}_0-i{z}_3& -z_2-i{z}_1\\ z_2-i{z}_1& z_0+i{z}_3\end{array}\right){\Omega }_2$$

preserves this space as well as the determinant $(z_0^2+z_1^2+z_2^2+z_3^2)$ for ${\Omega }_1$ and ${\Omega }_2$ not just in $SU(2)$ , but in the larger group $SL(2,\mathbf{C})$ $So$ we find that the group $SO(4,\mathbf{C})$ of complex orthogonal transformations of ${\mathbf{C}}^4$ has spin double cover

$$Spin(4,\mathbf{C})=SL(2,\mathbf{C})\times SL(2,\mathbf{C})$$

Since spin $(4,\mathbf{C})=\mathfrak{so}(3,1)\otimes \mathbf{C}.$ , this relation between complex Lie groups corresponds to the Lie algebra relation

$$\mathfrak{so}(3,1)\otimes \mathbf{C}=\mathfrak{sl}(2,\mathbf{C})\oplus \mathfrak{sl}(2,\mathbf{C})$$

we found explicitly earlier when we showed that by taking complex coeficients of generators $l_j$ and $k_{j}of\mathfrak{so}(3,1)$ we could find generators $A_j$ and $B_j$ of two diferent $\mathfrak{sl}(2,\mathbf{C})$ sub-algebras.

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正文：英文 OCR 草稿

来源版本：2025-10-20

原文位置：§40.4；PDF 页 450、451；印刷页 433、434

处理记录：MinerU OCR

核对状态：英文 OCR 草稿 · 待校对

中文机译草稿

Three explicit Spin real forms and their shared complexification; not a full abstract definition of a real form.

机器草稿中的相关讨论摘录

这三种不同构造针对这些情形

$$Spin(4)=SU(2)\times SU(2),Spin(3,1)=SL(2,\mathbf{C})$$

和

$$Spin(2,2)=SL(2,\mathbf{R})\times SL(2,\mathbf{R})$$

对应于复群的某个事实的不同所谓“实形式”，该事实可通过将上述任一例子复化而得到（考虑元素 $(x_0,x_1,x_2,x_3)\in {\mathbf{C}}^4$，而不仅仅在 ${\mathbf{R}}^4)$ 中。例如，在 $Spin(4)$ 的情形下，取矩阵中的 $x_0,x_1,x_2,x_3$

$$\left(\begin{array}{cc}{x}_0-i{x}_3& -x_2-i{x}_1\\ x_2-i{x}_1& x_0+i{x}_3\end{array}\right)$$

使 $z_0,z_1,z_2$ 取任意复数值，$z_3$ 便得到任意 2 乘 2 复矩阵，而变换

$$\left(\begin{array}{cc}{z}_0-i{z}_3& -z_2-i{z}_1\\ z_2-i{z}_1& z_0+i{z}_3\end{array}\right)\to {\Omega }_1\left(\begin{array}{cc}{z}_0-i{z}_3& -z_2-i{z}_1\\ z_2-i{z}_1& z_0+i{z}_3\end{array}\right){\Omega }_2$$

保持该空间不变，以及保持行列式$(z_0^2+z_1^2+z_2^2+z_3^2)$不变，对于${\Omega }_1$和${\Omega }_2$而言，不仅在$SU(2)$中如此，在更大的群$SL(2,\mathbf{C})$ $So$中我们发现，${\mathbf{C}}^4$的复正交变换群$SO(4,\mathbf{C})$具有自旋双重覆盖。

$$Spin(4,\mathbf{C})=SL(2,\mathbf{C})\times SL(2,\mathbf{C})$$

由于自旋 $(4,\mathbf{C})=\mathfrak{so}(3,1)\otimes \mathbf{C}.$ ，复李群之间的这个关系对应于李代数关系

$$\mathfrak{so}(3,1)\otimes \mathbf{C}=\mathfrak{sl}(2,\mathbf{C})\oplus \mathfrak{sl}(2,\mathbf{C})$$

我们此前在证明通过取生成元$l_j$和$k_{j}of\mathfrak{so}(3,1)$的复系数可以找到两个不同$\mathfrak{sl}(2,\mathbf{C})$子代数的生成元$A_j$和$B_j$时，已明确地得出了这一点。

阅读所在节 · 完整章节

来源与版本

查看 PDF

正文：中文机译草稿

来源版本：2025-10-20

原文位置：§40.4；PDF 页 450、451；印刷页 433、434

处理记录：MinerU OCR；machine-translation/deepseek-v4.1-flash

核对状态：中文机译草稿 · 待校对