In mathematics, the special unitary group of degree n, denoted SU (n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication.

When an element of SU(2) is written as a complex 2 × 2 matrix, it is simply a multiplication of column 2-vectors. It is known in physics as the spin-1/2 and, historically, as the multiplication of quaternions (more precisely, multiplication by a unit quaternion).

2024年1月1日 · 我们定义U (2)=\ {U\in C^ {2\times2}|U^+U=I\},可以证明这是一个群，称为二维酉群。 从定义可以知道，det (U)=\pm1，一种特殊的情况是det (U)=1，称为二维特殊酉群，记做SU (2)。

SU (2) 群是一个矩阵群，群元素为 2\times 2 的幺正矩阵，而且矩阵的行列式为1. 如果将 SU (2) 群的一个元素写做 e^ {iA} ，那么因为 e^ {iA} e^ {-iA^ {\dagger}}= I ，所以 e^ {-iA^ {\dagger}} = e^ {-iA} . 因此，就有 A = A^ {\dagger} ，也就是 A 是Hermite矩阵。 又因为 \text {det} (e^ {iA}) = e^ {i\text {Tr} A} = 1 ，所以 \text {Tr} A = 0 . 定义 泡利矩阵 为.

The group SU(2) is the group of unitary 2 2 complex matrices with determinant 1. Every such matrix can be uniquely written as. for (z;w) 2 C2, with the condition that jzj2 + jwj2 = 1. In other words, SU(2) is topologically equivalent to the unit sphere in C2, which is the same as the real 3 …

2017年4月25日 · We will now work out in detail the properties of SU(2) and its representations. We have already seen that the generators may be chosen to be. with σi = the Pauli matrices. βij = −ǫaibǫbja = 2δij. Thus our generators are not quite canonically normalized, but are all nor-malized equally, and β is positive definite.

SU(2) is isomorphic to the description of angular momentum – SO(3). SU(2) also describes isospin – for nucleons, light quarks and the weak interaction. We see how to describe hadrons in terms of several quark wavefunctions. SU(2) corresponds to special unitary transformations on complex 2D vectors.

I am having hard time showing that every matrix in $SU(2)$ is of the form $\begin{pmatrix}z & -\overline w \\ w & \overline z\end{pmatrix}$ for some $z, w \in \mathbb{C}$ such that $|z|^2 + |w|^2 =1$. In most of the sources (including answers here) it is stated as something well-known.

2020年11月22日 · The special unitary group SU(n) is a real matrix Lie group of dimension n 2 − 1. Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below). The center of SU(n) is isomorphic to the cyclic group Z n.

顾名思义，它是由行列式为 1 的二维 幺正矩阵 构成的。 这个群与我们之前提到的SO (3)是局域等价的，它拥有与后者一样的 李代数。 从宏观性质来说，它拥有紧致性以及单连通性。 因此所有李代数的不可约表示都是SU (2)群的单值表示。 在Chapter7中，我们已经展示了，所有SO (3)群的元素都可以映射到行列式为 1 的 2\times 2 的幺正矩阵 D^ {1/2} (\alpha,\beta,\gamma) 上。 不难证明，所有的SU (2)群的元素也可以用同样的方式参数化，简单的，我们创造一个任意二维矩阵 …