2023年12月29日 · There exists a map 2 → 1 that associates a couple of matrices U, − U ∈ SU(2) to a M ∈ SO(3). This mapping is a homomorphism between the two groups because the composition rule is preserved, i.e. U1 ∗IU2 = M1 ∗IIM2, where ∗I is the composition rule in SU(2) and ∗II is the composition rule in SO(3).

给定一个 U\in SU(2) ，我们总是可以得到一个唯一的 R\in SO(3). 值得指出的是， U(\theta) = e^{i\frac{\theta}{2} \boldsymbol{\sigma} \cdot\boldsymbol{n}} 和 U(\theta+2\pi) 并不相等，但是它们对应了同一个正交矩阵 R. 因此，从SU(2)到SO(3)的映射并不是一一对应，而是二对一。

┣ 利用林檎定理 (同态核定理): \[\text{SO}\left( 3 \right)={\text{SU}\left( 2 \right)}/{{{\mathbb{Z}}_{2}}}\]. ┗ 碰到这种情况我们就称 \[\text{SU}\left( 2 \right)\] 是 \[\text{SO}\left( 3 \right)\] 的二重覆盖群. 确定同态映射下的具体对应关系:

2 SU(2) is locally isomorphic to SO(3) One of the most interesting properties of SU(2) is its aforementioned relationship to SO(3), namely that the two are locally isomorphic. Essentially, this means that, in a neighborhood of any U 2 SU(2), SU(2) and SO(3) are isomorphic. This manifests itself as an isomorphicm of their Lie algebras.

SU(2) is isomorphic to unit quaternions. In this this article you can find a way to represent every quaternion with a rotation of R3, namely an element of SO(3), and for every rotation of R3 there's a quaternion that represents it.

I have two maps from $SU(2) \to SO(3)$. For the first map, think of $SU(2)$ as the group of unit quaternions. Under this identification, we can give a map $f: SU(2) \to SO(3)$ given by $SU(2)$ acting via conjugation on the 3-dimensional vector subspace of …

我们将了解 \text{SO}(3) 的通用覆盖群是 \text{SU}(2) ，这样以后便可以方便地讨论 \text{SO}(3) 的射影表示问题。 四元数 给出了立体旋转的 \text{SU}(2) 表示和四元数表示。二维 特殊酉群 \text{SU}(2) \lhd \text U(2) ，它的定义中对行列式所要求的条件，相当于单位四元数(unit ...

To show that φ is a homomorphism with the properties as mentioned in Section 1, we need to show that φ preserves the group law and the identity in SU(2) is mapped to the identity in SO(3). By definition, φU(x) = A−1(TU[A(x)]) = A−1[UA(x)U∗] or UA(x)U∗ = A[φU(x)]. meaning that φUV = φUφV . Also, we have IA(x)I∗ = A(x) so.

They are the analogy of vectors in three dimensional rotation group (SO(3)): = 1, 2. Theorem (Cornwell 1984): “There exists a two-to-one homomorphic mapping of the group SU (2) onto the group SO(3). If A 2 SU (2) maps onto R(A) 2 SO(3), then R(A) = …

SU(2), SO(3), and SU(3) These three groups play a large role in physics, particle physics in particular SO(3) is used to describe and calculate external rotations SU(2) and SU(3) are used to describe and calculate internal rotations, while SU(2) deals with systems with two states, and SU(3) deals with systems with three states