Elements of the Representation Theory of SU(2)

This appendix collects the results from the representation theory of ${\mathrm{SU}}(2)$ that are used in the main text: irreducible representations, the Peter–Weyl theorem, intertwiners, and the $3j$ and $6j$ symbols. The conventions are those of the Notation and Conventions page.

A.1 The group SU(2) and Its Lie Algebra

The group ${\mathrm{SU}}(2)$ consists of the unitary complex $2\times2$ matrices of unit determinant; it is diffeomorphic to the sphere $S^3$ and is the double cover of ${\mathrm{SO}}(3)$. Its Lie algebra ${\mathfrak{su}}(2)$ is generated by the anti-Hermitian matrices $\tau_i = -\frac{\mathrm{i}}2\sigma_i$, with commutation relations $[\tau_i,\tau_j] = \epsilon_{ij}{}^k\tau_k$. Every element can be written as $h = \exp(\theta\,\hat n^i\tau_i)$, where $\theta\in[0,2\pi]$ is the rotation angle and $\hat n$ is a unit vector. In terms of the Euler angles $(\phi,\vartheta,\psi)\in[0,2\pi)\times[0,\pi]\times[0,4\pi)$, the normalized Haar measure, which is both left- and right-invariant, reads

A.2 Irreducible Representations

The irreducible unitary representations of ${\mathrm{SU}}(2)$ are labeled by a spin $j\in\mathbb N/2$. The carrier space $V_j$ of the spin-$j$ representation has dimension $d_j = 2j+1$ and admits the orthonormal basis $\{{\left|{j,m}\right\rangle}\}_{m=-j,\dots,j}$ of eigenvectors of $J_3 = {\mathrm{i}}\tau_3$:

with $J_\pm = J_1\pm{\mathrm{i}} J_2$. The Wigner matrices $D^{(j)}_{mn}(h) = {\left\langle{j,m}\middle|{h}\middle|{j,n}\right\rangle}$ represent the group elements, and their trace defines the character

which depends only on the conjugacy class of $h$, that is, on the angle $\theta$. For $j = \tfrac12$, we recover $\chi^{1/2} = {\operatorname{Tr}} h = 2\cos(\theta/2)$, the trace that appears in Wilson loops.

A.3 The Peter–Weyl Theorem

The functions $\sqrt{2j+1}\,D^{(j)}_{mn}$ form an orthonormal basis of $L^2({\mathrm{SU}}(2),{\mathrm{d}} h)$; the orthogonality relations are given by (3.20). This yields the decomposition

and, for class functions, the orthonormality of the characters, $\int{\mathrm{d}} h\;\overline{\chi^j(h)}\,\chi^{j'}(h) = \delta_{jj'}$. The delta distribution on the group, defined with respect to the Haar measure, admits the expansion

which is used in Chapter 5. Finally, the integral of a tensor product of representations is the projector onto the invariant subspace:

where the sum runs over an orthonormal basis of $\mathrm{Inv}(V_{j_1}\otimes\cdots\otimes V_{j_n})$.

A.4 Coupling of Angular Momenta and Intertwiners

The tensor product of two irreducible representations decomposes according to the Clebsch–Gordan rule

where $j$ varies in integer steps. From this rule one obtains the dimension of the intertwiner spaces, which appear at the nodes of spin networks.

Trivalent nodes. The space $\mathrm{Inv}(V_{j_1}\otimes V_{j_2}\otimes V_{j_3})$ is one-dimensional if ${\left|{j_1-j_2}\right|}\le j_3\le j_1+j_2$ and $j_1+j_2+j_3\in\mathbb N$, and trivial otherwise. The normalized intertwiner is given by the Wigner $3j$ symbol, which is related to the Clebsch–Gordan coefficients by

It is nonzero only if $m_1+m_2+m_3 = 0$, and it satisfies $\sum_{m_i}\big|\begin{smallmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3\end{smallmatrix}\big|^2 = 1$.

Four-valent nodes. A basis of $\mathrm{Inv}(V_{j_1}\otimes\cdots\otimes V_{j_4})$ is obtained by first coupling $j_1$ and $j_2$ into a “virtual” spin $k$, and then coupling $j_3$ and $j_4$ into the same spin: the intertwiner $\iota_k$ is the contraction of two $3j$ symbols along a virtual link of spin $k$, normalized by the factor $\sqrt{2k+1}$. The virtual spin ranges over

in integer steps, and the number of allowed values is the dimension of the intertwiner space. For four spins $\tfrac12$, we find $k\in\{0,1\}$: the space is two-dimensional, a fact used in the volume computation (3.36). The change of coupling channel, for instance from $(j_1j_2)(j_3j_4)$ to $(j_1j_3)(j_2j_4)$, is given by the $6j$ symbols.

A.5 The 6j Symbol

The $6j$ symbol is the contraction of four $3j$ symbols following the structure of a tetrahedron:

Each of the four triples $(j_1,j_2,j_3)$, $(j_1,j_5,j_6)$, $(j_4,j_2,j_6)$ and $(j_4,j_5,j_3)$ must satisfy the triangle conditions; they correspond to the four faces of the tetrahedron in Fig. 5.5. The $6j$ symbol is invariant under the $24$ symmetry operations of the tetrahedron, namely permutations of its columns and the interchange of the upper and lower entries in any two columns. It gives the recoupling coefficients of three angular momenta,

and satisfies the orthogonality relation

For example, ${\begin{Bmatrix}{1}&{1}&{1}\\{1}&{1}&{1}\end{Bmatrix}} = {\begin{Bmatrix}{\frac12}&{\frac12}&{1}\\{\frac12}&{\frac12}&{1}\end{Bmatrix}} = \frac16$ and ${\begin{Bmatrix}{2}&{2}&{2}\\{2}&{2}&{2}\end{Bmatrix}} = -\frac{3}{70}$. We have verified the asymptotic formula (5.22) numerically for the regular tetrahedron: for $j = 20$, the exact value $\{6j\}\simeq-5.029\times10^{-3}$ is reproduced to within $0.1\,\%$ by the Ponzano–Regge formula, which gives $-5.034\times10^{-3}$.