Representations of \bmit{\su(2)}

Section 4.2 Representations of \(\bmit{\su(2)}\)

We have the following basis elements for \(\sl(2,\RR)\cong\su(2,\CC')\cong\so(2,1)\text{,}\) a real form of \(\su(2)\text{:}\)

\begin{equation} \sigma_0=\frac12\sigma_z, \quad \sigma_\pm=\frac12(\sigma_x\mp s_y)=\frac12(\sigma_x\pm i\sigma_y),\tag{4.2.1} \end{equation}

with commutation relations

\begin{equation} [\sigma_0,\sigma_\pm]=\pm\sigma_\pm, \quad [\sigma_+,\sigma_-]=2\sigma_0.\tag{4.2.2} \end{equation}

These basis elements also form a basis of the complexified Lie algebra \(\su(2)\otimes\CC\text{.}\)

Figure 4.2.1. The root diagram of \(\su(2)\text{.}\)

We can thus represent \(\sl(2,\RR)\) graphically as the points \(0,\pm1\in\RR\text{,}\) representing \(\sigma_z\) acting on itself and \(\sigma_\pm\text{,}\) respectively, connected by oriented arrows representing the action of \(\sigma_\pm\text{,}\) as shown in Figure 4.2.1. This diagram fully captures the algebraic description \(\sl(2,\RR)\) acting on itself, the so-called adjoint representation of \(\sl(2,\RR)\text{.}\) Each of these statements can be reinterpreted as being about \(\su(2)\otimes\CC\text{;}\) Figure 4.2.1 is normally called the root diagram of \(\su(2)\).

We can now ask about more general representations of \(\su(2)\text{,}\) with \(\rho(\su(2))\) acting on some vector space \(V\text{.}\) The commutation relations (4.2.2) show that \(\sigma_0\) is diagonal in the given basis. It turns out that \(L_z=\rho(\sigma_0)\) is diagonalizable in any representation \(\rho\text{,}\) ¹ so we can choose a basis for \(V\) consisting entirely of eigenvectors of \(L_z\text{.}\) If \(w\ne0\) is one such eigenvector, we have

\begin{equation} L_z w = \lambda w\tag{4.2.3} \end{equation}

for some \(\lambda\in\CC\text{.}\) Letting \(L_\pm=\rho(\sigma_\pm)\text{,}\) we have

\begin{equation} L_z L_\pm w = [L_z,L_\pm] w + L_\pm L_z w = \pm L_\pm w + L_\pm \lambda w = (\lambda\pm1) L_\pm w\tag{4.2.4} \end{equation}

Thus, \(L_\pm w\) is also an eigenvector of \(L_z\text{,}\) with eigenvalue \(\lambda\pm1\text{.}\)

We want \(V\) to be an irreducible representation of \(\su(2)\text{,}\) by which we mean that there should be no (nonzero, proper) subrepresentations of \(\su(2)\) in \(V\text{.}\) Thus, acting repeatedly on \(w\) with \(L_\pm\) must generate a basis for \(V\text{,}\) as any vector not contained in the resulting span would itself generate a disjoint subrepresentation.

We also want \(V\) to be finite. Since we are changing the eigenvalue at each step, this can only happen if there is a “biggest” eigenvalue. That is, we can assume without loss of generality that

\begin{equation} L_+ w = 0\tag{4.2.5} \end{equation}

and that the remaining basis vectors are obtained by repeated action of \(L_-\text{.}\)

We now compute

\begin{equation} \begin{aligned} L_+L_- w \amp= [L_+,L_-] w + L_-L_+ w = 2L_z w = 2\lambda w \\ L_+L_- L_-w \amp= [L_+,L_-] L_-w + L_-L_+L_- w \nonumber\\ \amp= 2 L_zL_-w + 2\lambda L_-w = 2(2\lambda-1) L_w \\ \vdots\qquad \amp= \qquad\qquad\vdots \nonumber\\ L_+(L_-)^k w \amp= ... = \bigl(2k\lambda-k(k-1)\bigr) (L_-)^{k-1} w \end{aligned}\tag{4.2.6} \end{equation}

But for \(V\) to be finite, \((L_-)^k w\) must be zero for some positive integer \(k\text{.}\) Assume that \(k\) is the smallest such integer. Then \((L_-)^{k-1}w\) is not zero, and therefore

\begin{equation} 2k\lambda-k(k-1)=0\tag{4.2.7} \end{equation}

by (4.2.6). Since \(k\ne0\text{,}\) it must be true that

\begin{equation} \lambda = \frac{k-1}{2}\tag{4.2.8} \end{equation}

is an integer or half-integer, often called the \(spin\) of the representation. Thus, the spin-\(\lambda\) representation of \(\su(2)\) has \(k=2\lambda+1\) basis vectors, with eigenvalues

\begin{equation} \lambda, \lambda-1, ..., \lambda-2\lambda=-\lambda .\tag{4.2.9} \end{equation}

We conclude that there is exactly one (irreducible) representation of \(\su(2)\) for each dimension \(k\ge2\text{,}\) with eigenvalues \(\{-\frac{k-1}{2},...,\frac{k-1}{2}\}\text{.}\) This representation is called the spin-\(\lambda\) representation of \(\su(2)\text{,}\) with \(\lambda\) given by (4.2.8). Put differently, we can reproduce the commutation relations (4.2.2) using \(k\times k\) matrices for any \(k\ge2\text{,}\) and can do so in essentially just one way (up to change of basis).

This property holds for any semisimple Lie algebra, one for which the Killing form \(B\) is nondegenerate, but is not true in general.