6.3 A summary

6.3 A summary

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · adjoint representation · irreducible representation · Lie group · Lie bracket · group homomorphism

To summarize, we have shown that in the three dimensional case we have two distinct Lie groups:

$Spin(3)$ , which geometrically is the space $S^3$ . Its Lie algebra is ${\mathbf{R}}^3$ with Lie bracket the cross-product. We have seen two diferent explicit constructions of $Spin(3)$ , in terms of unit quaternions $(Sp(1))$ , and in terms of 2 by 2 unitary matrices of determinant 1 $\left(SU(2)\right)$ ).

$SO(3)$ , which has a Lie algebra isomorphic to that of $Spin(3)$

There is a group homomorphism $\Phi$ that takes the first group to the second, which is a two-fold covering map. Its derivative ${\Phi }^{\prime }$ is an isomorphism of the Lie algebras of the two groups.

We can see from these constructions two interesting irreducible representations of these groups:

• A representation on ${\mathbf{R}}^3$ which can be constructed in two diferent ways: as the adjoint representation of either of the two groups, or as the defining representation of $SO(3)$ This is known to physicists as the “spin $1^{\mathfrak{s}}$ representation.

• A representation of the first group on ${\mathbf{C}}^2$ , which is most easily seen as the defining representation of $SU(2)$ . It is not a representation of $SO(3)$ since going once around a non-contractible loop starting at the identity takes one to minus the identity, not back to the identity as required. This is called the “spin $\frac{1}{2}d{s}^{,}$ or “spinor” representation and will be studied in more detail in chapter 7.

Previous section: 6.2 Spin groups in three and four dimensions · Next section: 6.4 For further reading · 中文版本

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 62、63、64、65、66、67、68、69、70、71、72、73、74

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：9523b4fdcb950eebc22282c613f6a0d19c5f46af4d841268eb20e4448e54cea8

OCR 产物 SHA-256：9523b4fdcb950eebc22282c613f6a0d19c5f46af4d841268eb20e4448e54cea8