Spin and Representations
Wonmyeong Cho
TL;DR
The paper derives the representation theory of $\,SU(2)$ from Lie group and Lie algebra theory and applies it to the non-relativistic quantum mechanics of spin, including multi-particle systems. It identifies the $\mathfrak{su}(2)$ generators $S_j$ (with Pauli-based realization) and their commutation relations, constructs the spin-$s$ irreducible representations of dimension $2s+1$, and connects group representations to Hilbert-space spin spaces via the complexification $\mathfrak{su}(2)_{\mathbb{C}}$. For two-particle systems, it shows the tensor-product decomposition $V_{s_1}\otimes V_{s_2}=\bigoplus_{s=|s_1-s_2|}^{s_1+s_2} V_s$, and derives Clebsch-Gordan coefficients $C_{m_1 m_2 m}^{s_1 s_2 s}$ that relate the product basis to total-spin basis. The paper provides explicit spin-$\tfrac{1}{2}$ ($\tfrac{1}{2}\otimes\tfrac{1}{2}$) Clebsch-Gordan coefficients and discusses entanglement and measurement within this representation-theoretic framework, illustrating how angular momentum coupling arises from $SU(2)$ representations.
Abstract
We derive the representation theory of $SU(2)$ from the expository theory of Lie groups and Lie algebras. Based on this, the mathematics of non-relativistic quantum mechanics of a spin $\frac{1}{2}$ particle are described from a representation-theoretic perspective, and are extended to many particle systems.