8.1 Representations of SU(2) : classification

8.1 Representations of SU(2) : classification

8.1.1 Weight decomposition

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If we make a choice of a $U(1)\subset SU(2)$ , then given any representation $(\pi ,V)$ of $SU(2)$ of dimension $m_{;}$ , we get a representation $({\pi }_{|U(1)},V)$ of $U(1)$ by restriction to the $U(1)$ subgroup. Since we know the classification of irreducibles of $U(1)$ we know that

$$(\pi {|}_{U(1)},V)={\mathbf{C}}_{q_1}\oplus {\mathbf{C}}_{q_2}\oplus \cdots \oplus {\mathbf{C}}_{q_m}$$

for some $q_1,q_2,\cdot \cdot \cdot ,q_m\in \mathbf{Z}$ , where ${\mathbf{C}}_q$ denotes the one dimensional representation of $U(1)$ corresponding to the integer $q$ (theorem 2.3). These $q_j$ are called the ${}^{66}{\mathrm{weights}}^3$ of the representation $V$ . They are exactly the same thing discussed in chapter 2 as “charges”, but here we’ll favor the mathematician’s terminology since the $U(1)$ here occurs in a context far removed from that of electromagnetism and its electric charges.

Since our standard choice of coordinates (the Pauli matrices) picks out the z-direction and diagonalizes the action of the $U(1)$ subgroup corresponding to rotation about this axis, this is the $U(1)$ subgroup we will choose to define the weights of the $SU(2)$ representation $V$ . This is the subgroup of elements of $SU(2)$ of the form

$$\left(\begin{array}{cc}{e}^{i\theta }& 0\\ 0& e^{-i\theta }\end{array}\right)$$

Our decomposition of an $SU(2)$ representation $(\pi ,V)$ into irreducible representations of this $U(1)$ subgroup equivalently means that we can choose a basis of $V$ so that

$$\pi \left(\begin{array}{cc}{e}^{i\theta }& 0\\ 0& e^{-i\theta }\end{array}\right)=\left(\begin{array}{cccc}{e}^{i\theta q_1}& 0& \dots & 0\\ 0& e^{i\theta q_2}& \dots & 0\\ \dots & & & \dots \\ 0& 0& \dots & e^{i\theta q_m}\end{array}\right)$$

An important property of the set of integers $q_j$ is the following:

Theorem

$Ifq$ is in the set $\{q_j\}$ , so $is-q$

Proof

Recall that if we diagonalize a unitary matrix, the diagonal entries are the eigenvalues, but their order is undetermined: acting by permutations on these eigenvalues we get diferent diagonalizations of the same matrix. In the case of $SU(2)$ the matrix

$$P=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$

has the property that conjugation by it permutes the diagonal elements, in particular

$$P\left(\begin{array}{cc}{e}^{i\theta }& 0\\ 0& e^{-i\theta }\end{array}\right)P^{-1}=\left(\begin{array}{cc}{e}^{-i\theta }& 0\\ 0& e^{i\theta }\end{array}\right)$$

So

$$\pi (P)\pi (\left(\begin{array}{cc}{e}^{i\theta }& 0\\ 0& e^{-i\theta }\end{array}\right))\pi (P{)}^{-1}=\pi (\left(\begin{array}{cc}{e}^{-i\theta }& 0\\ 0& e^{i\theta }\end{array}\right))$$

and we see that $\pi (P)$ gives a change of basis of $V$ such that the representation matrices on the $U(1)$ subgroup are as before, with $\theta -\theta$ . Changing $\theta -\theta$ in the representation matrices is equivalent to changing the sign of the weights $q_j$ . The elements of the set $\{q_j\}$ are independent of the basis, so the additional symmetry under sign change implies that for each non-zero element in the set there is another one with the opposite sign. □

Looking at our three examples so far, we see that, restricted to $U(1)$ , the scalar or spin 0 representation of course is one dimensional and of weight 0

$$({\pi }_0,\mathbf{C})={\mathbf{C}}_0$$

and the spin $\frac{1}{2}$ representation decomposes into $U(1)$ irreducibles of weights $-1,+1$ :

$$({\pi }_1,{\mathbf{C}}^2)={\mathbf{C}}_{-1}\oplus {\mathbf{C}}_{+1}$$

For the spin 1 representation, recall (theorem 6.1) that the double cover homomorphism $\Phi$ takes

$$\left(\begin{array}{cc}{e}^{i\theta }& 0\\ 0& e^{-i\theta }\end{array}\right)\in SU(2)\to \left(\begin{array}{ccc}\cos 2\theta & \sin 2\theta & 0\\ -\sin 2\theta & \cos 2\theta & 0\\ 0& 0& 1\end{array}\right)\in SO(3)$$

Acting with the $SO(3)$ matrix above on ${\mathbf{C}}^3$ will give a unitary transformation of ${\mathbf{C}}^3$ , which therefore is in the group $U(3)$ . One can show that the upper left diagonal 2 by 2 block acts on ${\mathbf{C}}^2$ with weights $-2,+2.$ , whereas the bottom right element acts trivially on the remaining part of ${\mathbf{C}}^3$ , which is a one dimensional representation of weight 0. So, restricted to $U(1)$ , the spin 1 representation decomposes as

$$({\pi }_2,{\mathbf{C}}^3)={\mathbf{C}}_{-2}\oplus {\mathbf{C}}_0\oplus {\mathbf{C}}_{+2}$$

Recall that the spin 1 representation of $SU(2)$ is often called the “vector” representation, since it factors in this way through the representation of $SO(3)$ by rotations on three dimensional vectors.

8.1.2 Lie algebra representations: raising and lowering operators

Concept links · terms present in this machine draft; source roles are unverified: vector space · linear map · eigenvalue · Lie algebra · Lie algebra representation · group representation · unitary representation · irreducible representation · Lie group · complexification

To proceed further in characterizing a representation $(\pi ,V)$ of $SU(2)$ we need to use not just the action of the chosen $U(1)$ subgroup, but the action of group elements in the other two directions away from the identity. The noncommutativity of the group keeps us from simultaneously diagonalizing those actions and assigning weights to them. We can however work instead with the corresponding Lie algebra representation $({\pi }^{\prime },V)$ of $\mathfrak{su}(2)$ . As in the $U(1)$ case, the group representation is determined by the Lie algebra representation. We will see that for the Lie algebra representation, we can exploit the complexification (recall section 5.5) $\mathfrak{sl}(2,\mathbf{C})$ of $\mathfrak{su}(2)$ to further analyze the possible patterns of weights.

Recall that the Lie algebra su(2) can be thought of as the tangent space ${\mathbf{R}}^3$ to $SU(2)$ at the identity element, with a basis given by the three skew-adjoint 2 by 2 matrices

$$X_j=-i\frac{1}{2}{\sigma }_j$$

which satisfy the commutation relations

$$[X_1,X_2]=X_3,[X_2,X_3]=X_1,[X_3,X_1]=X_2$$

We will often use the self-adjoint versions $S_j=i{X}_j$ that satisfy

$$[S_1,S_2]=i{S}_3,[S_2,S_3]=i{S}_1,[S_3,S_1]=i{S}_2$$

A unitary representation $(\pi ,V)$ of $SU(2)$ of dimension $m$ is given by a homomorphism

$$\pi :SU(2)\to U(m)$$

We can take the derivative of this to get a map between the tangent spaces of $SU(2)$ and of $U(m)$ , at the identity of both groups, and thus a Lie algebra representation

$${\pi }^{\prime }:\mathfrak{su}(2)\to \mathfrak{u}(m)$$

which takes skew-adjoint 2 by 2 matrices to skew-adjoint m by $m$ matrices, preserving the commutation relations.

We have seen in section 8.1.1 that restricting the representation $(\pi ,V)$ to the diagonal $U(1)$ subgroup of $SU(2)$ and decomposing into irreducibles tells us that we can choose a basis of V so that

$$(\pi ,V)=({\pi }_{q_1},\mathbf{C})\oplus ({\pi }_{q_2},\mathbf{C})\oplus \cdots \oplus ({\pi }_{q_m},\mathbf{C})$$

For our choice of $U(1)$ as matrices of the form

$$e^{i2\theta S_3}=\left(\begin{array}{cc}{e}^{i\theta }& 0\\ 0& e^{-i\theta }\end{array}\right)$$

with $e^{i\theta }$ going around $U(1)$ once as $\theta$ goes from $0$ to $2\pi$ , this means we can choose a basis of $V$ so that

$$\pi (e^{i2\theta S_3})=\left(\begin{array}{cccc}{e}^{i\theta q_1}& 0& \dots & 0\\ 0& e^{i\theta q_2}& \dots & 0\\ \dots & & & \dots \\ 0& 0& \dots & e^{i\theta q_m}\end{array}\right)$$

Taking the derivative of this representation to get a Lie algebra representation, using

$${\pi }^{\prime }(X)=\frac{d}{d\theta }\pi (e^{\theta X}){|}_{\theta =0}$$

we find for $X=i2S_3$

$${\pi }^{\prime }(i2S_3)=\frac{d}{d\theta }{\left(\begin{array}{cccc}{e}^{i\theta q_1}& 0& \dots & 0\\ 0& e^{i\theta q_2}& \dots & 0\\ \dots & & & \dots \\ 0& 0& \dots & e^{i\theta q_m}\end{array}\right)}_{|\theta =0}=\left(\begin{array}{cccc}i{q}_1& 0& \dots & 0\\ 0& i{q}_2& \dots & 0\\ \dots & & & \dots \\ 0& 0& \dots & i{q}_m\end{array}\right)$$

Recall that ${\pi }^{\prime }$ is a real-linear map from a real vector space $(\mathfrak{su}(2)={\mathbf{R}}^3)$ to another real vector space $(\mathfrak{u}(n)$ , the skew-Hermitian m by $m$ complex matrices). As discussed in section $5.5,$ , we can use complex linearity to extend any such map to a complex-linear map from $\mathfrak{su}(2{)}_{\mathrm{C}}$ (the complexification of $\mathfrak{su}(2))$ to $\mathfrak{u}(m{)}_{\mathbf{C}}$ (the complexification of $\mathfrak{u}(m),$ . Since $\mathfrak{su}(2)\cap i\mathfrak{su}(2)=0$ and any element of $\mathfrak{sl}(2,\mathbf{C})$ be written as a complex number times an element of $\mathfrak{su}(2)$ , we have

$$\mathfrak{su}(2{)}_{\mathbf{C}}=\mathfrak{su}(2)+i\mathfrak{su}(2)=\mathfrak{sl}(2,\mathbf{C})$$

Similarly

$$\mathfrak{u}(m{)}_{\mathbf{C}}=\mathfrak{u}(m)+i\mathfrak{u}(m)=M(m,\mathbf{C})=\mathfrak{gl}(m,\mathbf{C})$$

As an example, multiplying $X=i2S_3\in \mathfrak{su}(2)$ by $-\frac{i}{2}$ , we have $S_3\in \mathfrak{sl}(2,\mathbf{C})$ and the diagonal elements in the matrix ${\pi }^{\prime }(i2S_3)$ get also multiplied by $-\frac{i}{2}$ (since ${\pi }^{\prime }$ is now a complex-linear map), giving

$${\pi }^{\prime }(S_3)=\left(\begin{array}{cccc}\frac{q_1}{2}& 0& \dots & 0\\ 0& \frac{q_2}{2}& \dots & 0\\ \dots & & & \dots \\ 0& 0& \dots & \frac{q_m}{2}\end{array}\right)$$

We see that ${\pi }^{\prime }(S_3)$ will have half-integral eigenvalues, and make the following definitions:

Definition (Weights and weight spaces)

$If{\pi }^{\prime }(S_3)$ has an eigenvalue $\frac{k}{2}$ , we say that $k$ is a weight of the representation $(\pi ,V)$

The subspace $V_k\subset V$ of the representation $V$ satisfying

$$v\in V_k⟹{\pi }^{\prime }(S_3)v=\frac{k}{2}v$$

is called the k’th weight space of the representation. All vectors in it are eigenvectors of ${\pi }^{\prime }(S_3)$ with eigenvalue $\frac{k}{2}$ .

The dimension dim $V_k$ is called the multiplicity of the weight $k$ in the representation $(\pi ,V)$ .

$S_1$ and $S_2$ don’t commute with $S_3,$ so they won’t preserve the subspaces $V_k$ and we can’t diagonalize them simultaneously with $S_3$ . We can however exploit the fact that we are in the complexification $\mathfrak{sl}(2,\mathbf{C})$ to construct two complex linear combinations of $S_1$ and $S_2$ that do something interesting:

Definition (Raising and lowering operators)

Let

$$S_{+}=S_1+i{S}_2=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),S_{-}=S_1-i{S}_2=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$$

We have $S_{+},S_{-}\in \mathfrak{sl}(2,\mathbf{C})$ . These are neither self-adjoint nor skew-adjoint, but satisfy

$$(S_{\pm }{)}^{†}=S_{\mp }$$

and similarly we have

$${\pi }^{\prime }(S_{\pm }{)}^{†}={\pi }^{\prime }(S_{\mp })$$

We call ${\pi }^{\prime }(S_{+})a$ “raising operator” for the representation $(\pi ,V)$ , and ${\pi }^{\prime }(S_{-})$ a “lowering operator”.

The reason for this terminology is the following calculation:

$$[S_3,S_{+}]=[S_3,S_1+i{S}_2]=i{S}_2+i(-i{S}_1)=S_1+i{S}_2=S_{+}$$

which implies (since ${\pi }^{\prime }$ is a Lie algebra homomorphism)

$${\pi }^{\prime }(S_3){\pi }^{\prime }(S_{+})-{\pi }^{\prime }(S_{+}){\pi }^{\prime }(S_3)={\pi }^{\prime }([S_3,S_{+}])={\pi }^{\prime }(S_{+})$$

For any $v\in V_k$ , we have

$${\pi }^{\prime }(S_3){\pi }^{\prime }(S_{+})v={\pi }^{\prime }(S_{+}){\pi }^{\prime }(S_3)v+{\pi }^{\prime }(S_{+})v=(\frac{k}{2}+1){\pi }^{\prime }(S_{+})v$$

so

$$v\in V_k⟹{\pi }^{\prime }(S_{+})v\in V_{k+2}$$

The linear operator ${\pi }^{\prime }(S_{+})$ takes vectors with a well-defined weight to vectors with the same weight, plus 2 (thus the terminology “raising operator”). A similar calculation shows that ${\pi }^{\prime }(S_{-})$ takes $V_k$ to $V_{k-2}$ , lowering the weight by 2.

We’re now ready to classify all finite dimensional irreducible unitary representations $(\pi ,V)$ of $SU(2)$ . We define:

Definition (Highest weights and highest weight vectors)

A non-zero vector $v\in V_n\subset V$ such that

$${\pi }^{\prime }(S_{+})v=0$$

is called a highest weight vector, with highest weight $n$.

Irreducible representations will be characterized by a highest weight vector, as follows

Theorem (Highest weight theorem)

Finite dimensional irreducible representations of $SU(2)$ have weights of the form

$$-n,-n+2,\dots ,n-2,n$$

for n a non-negative integer, each with multiplicity 1, with $n$ a highest weight.

Proof. Finite dimensionality implies there is a highest weight $n$, and we can choose any highest weight vector $v_n\in V_n$ . Repeatedly applying ${\pi }^{\prime }(S_{-})$ to $v_n$ will give new vectors

$$v_{n-2j}={\pi }^{\prime }(S_{-}{)}^j{v}_n\in V_{n-2j}$$

with weights $n-2j$

Consider the span of the $v_{n-2j},j\ge 0$ . To show that this is a representation one needs to show that the ${\pi }^{\prime }(S_3)$ and ${\pi }^{\prime }(S_{+})$ leave it invariant. For ${\pi }^{\prime }(S_3)$ this is obvious, for ${\pi }^{\prime }(S_{+})$ one can show that

$${\pi }^{\prime }(S_{+})v_{n-2j}=j(n-j+1)v_{n-2(j-1)}\text{\hspace{1em}(8.1)}$$

by an induction argument. For $j=0$ this is the highest weight condition on $v_n$ Assuming validity for $j,$ validity for $j+1$ can be checked by

$$\begin{array}{rl}{\pi }^{\prime }(S_{+})v_{n-2(j+1)}& ={\pi }^{\prime }(S_{+}){\pi }^{\prime }(S_{-})v_{n-2j}\\ & =({\pi }^{\prime }([S_{+},S_{-}])+{\pi }^{\prime }(S_{-}){\pi }^{\prime }(S_{+}))v_{n-2j}\\ & =({\pi }^{\prime }(2S_3)+{\pi }^{\prime }(S_{-}){\pi }^{\prime }(S_{+}))v_{n-2j}\\ & =((n-2j)v_{n-2j}+{\pi }^{\prime }(S_{-})j(n-j+1)v_{n-2(j-1)}\\ & =((n-2j)+j(n-j+1))v_{n-2j}\\ & =(j+1)(n-(j+1)+1)v_{n-2((j+1)-1)}\end{array}$$

where we have used the commutation relation

$$[S_{+},S_{-}]=2S_3$$

By finite dimensionality, there must be some integer $k$ such that $v_{n-2j}\ne 0$ for $j\le k,$ and $v_{n-2j}=0$ for $j=k+1$ . But then, for $j=k+1$ , we must have

$${\pi }^{\prime }(S_{+})v_{n-2j}=0$$

By equation 8.1 this will happen only for $k+1=n+1$ (and we need to take $n$ positive). We thus see that $v_{-n}$ will be a “lowest weight vector”, annihilated by ${\pi }^{\prime }(S_{-})$ . As expected, the pattern of weights is invariant under change of sign, with non-zero weight spaces for

$$-n,-n+2,\dots ,n-2,n$$

Digression. Dropping the requirement of finite dimensionality, the same construction starting with a highest weight vector and repeatedly applying the lowering operator can be used to produce infinite dimensional irreducible representations of the Lie algebras su(2) or $\mathfrak{sl}(2,\mathbf{C})$ . These occur when the highest weight is not a non-negative integer, and they will be non-integrable representations (representations of the Lie algebra, but not of the Lie group).

Since we saw in section 8.1.1 that representations can be studied by looking at the set of their weights under the action of our chosen $U(1)\subset SU(2)$ , we can label irreducible representations of $SU(2)$ by a non-negative integer $n,$ the highest weight. Such a representation will be of dimension $n+1$ , with weights

$$-n,-n+2,\dots ,n-2,n$$

Each weight occurs with multiplicity one, and we have

$$({\pi }_{|U(1)},V)={\mathbf{C}}_{-n}\oplus {\mathbf{C}}_{-n+2}\oplus \dots {\mathbf{C}}_{n-2}\oplus {\mathbf{C}}_n$$

Starting with a highest weight or lowest weight vector, a basis for the representation can be generated by repeatedly applying lowering or raising operators. The picture to keep in mind is this

Figure 8.1: Basis for a representation of $SU(2)$ in terms of raising and lowering operators.

where all the vector spaces are copies of $\mathbf{C},$ and all the maps are isomorphisms (multiplications by various numbers).

In summary, we see that all irreducible finite dimensional unitary $SU(2)$ representations can be labeled by a non-negative integer, the highest weight $n$. These representations have dimension $n+1$ and we will denote them $({\pi }_n,V^n=$ ${\mathbf{C}}^{n+1})$ . Note that $V_n$ is the n’th weight space, $V^n$ is the representation with highest weight $n$. The physicist’s terminology for this uses not n, but $\frac{n}{2}$ and calls this number the ${}^6{\mathrm{spin}}^{\gamma }\mathrm{of}$ the representation. We have so far seen the lowest three examples $n=0,1,2$ , or spin $\begin{array}{r}s=\frac{n}{2}=0,\frac{1}{2},1\end{array}$ , but there is an infinite class of larger irreducibles, with dim $V_n=n\overline{+}1=\bar{2}s+1$

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