Spin Foams & State Sum Models
A covariant path-integral approach to quantum gravity through histories of spin networks
1. From Spin Networks to Spin Foams
In canonical LQG, spatial geometry is described by spin network states. The covariant (path integral) counterpart seeks transition amplitudes between spin network states. A spin foam is a 2-complex $\mathcal{F}$ whose boundary is a spin network: it represents a “history” of the spatial geometry. The transition amplitude between an initial spin network $s_i$ and a final one $s_f$ is:
$$\langle s_f | s_i \rangle = \sum_{\mathcal{F}:\,\partial\mathcal{F} = s_f \cup s_i} \prod_{f \in \mathcal{F}} A_f(j_f)\,\prod_{e \in \mathcal{F}} A_e(j_f, \iota_e)\,\prod_{v \in \mathcal{F}} A_v(j_f, \iota_e)$$
The sum is over all spin foams $\mathcal{F}$ bounded by the given spin networks. The amplitude factorizes into face amplitudes $A_f$ (one per 2-face), edge amplitudes $A_e$ (one per 1-edge), and vertex amplitudes $A_v$ (one per vertex, the most important factor). Each face carries a spin $j_f$ and each edge an intertwiner $\iota_e$.
2. The Ponzano-Regge Model (3D)
The simplest spin foam model is Ponzano-Regge for 3D Euclidean gravity. Given a triangulation $\Delta$ of a 3-manifold, the partition function is:
$$Z_{\rm PR}(\Delta) = \sum_{\{j_f\}} \prod_f (2j_f + 1) \prod_t \begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix}$$
where the product over $t$ runs over tetrahedra, and the curly bracket is the Wigner $6j$-symbol. The remarkable result of Ponzano and Regge is that in the large-spin limit $j \to \infty$:
$$\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix} \sim \frac{1}{\sqrt{12\pi V_t}}\,\cos\!\left(S_{\rm Regge}(t) + \frac{\pi}{4}\right)$$
where $V_t$ is the volume of the tetrahedron and $S_{\rm Regge} = \sum_f j_f\,\theta_f$ is the Regge action with dihedral angles $\theta_f$. This shows the spin foam amplitude recovers discrete gravity in the semiclassical limit.
3. The Barrett-Crane Model (4D)
Extending to 4D required new ideas. Barrett and Crane proposed a model based on the geometry of 4-simplices. The key variables are $\text{Spin}(4) \cong SU(2)_+ \times SU(2)_-$ representations labeled by $(j^+, j^-)$. The simplicity constraints, which reduce $BF$ theory to gravity, impose:
$$j^+ = j^- = j \quad \text{(Barrett-Crane simplicity constraint)}$$
The vertex amplitude for a 4-simplex is built from the $\{10j\}$-symbol of $SU(2)$. Its large-spin asymptotics gives:
$$A_v^{\rm BC} \sim \sum_{\pm} \frac{N_\pm}{\sqrt{j^{12}}}\,e^{\pm i\, S_{\rm Regge}^{(4)}(j_f)}$$
However, the Barrett-Crane model has a known deficiency: its simplicity constraints are too strong, killing some intertwiner degrees of freedom and leading to incorrect graviton propagator behavior.
4. The EPRL-FK Vertex Amplitude
The EPRL (Engle-Pereira-Rovelli-Livine) and FK (Freidel-Krasnov) models resolve the Barrett-Crane issues by imposing simplicity constraints weakly. For Lorentzian signature with Immirzi parameter $\gamma$, the map from $SU(2)$ spin $j$ to $SL(2,\mathbb{C})$ representations is:
$$\rho = \gamma\, j, \qquad k = j \quad \text{(for } \gamma > 1\text{)}$$
where $(\rho, k)$ label the unitary irreps of $SL(2,\mathbb{C})$ in the principal series. The EPRL vertex amplitude for a 4-simplex with boundary spin network is:
$$A_v^{\rm EPRL} = \int_{SL(2,\mathbb{C})^5/SL(2,\mathbb{C})} \prod_{a<b} K_{j_{ab}}^{(\gamma)}(g_a^{-1} g_b)\,\prod_a dg_a$$
where $K_j^{(\gamma)}$ is the EPRL propagator kernel. The large-spin asymptotics now correctly reproduces the Regge action and yields the correct graviton propagator in the semiclassical limit.
5. Semiclassical Limit and Regge Gravity
The central consistency check of any spin foam model is the recovery of Regge calculus (and hence general relativity) in the semiclassical limit. For the EPRL model, the stationary phase analysis of the vertex amplitude gives:
$$A_v^{\rm EPRL}(j_f) \underset{j \to \infty}{\longrightarrow} \frac{N}{\sqrt{\det H}}\,\cos\!\left(\sum_f j_f\,\Theta_f + \frac{\sigma\pi}{4}\right)$$
where $\Theta_f$ are the 4D dihedral angles and $H$ is the Hessian of the Regge action at the critical point. The full partition function in the large-$j$ limit thus gives:
$$Z \sim \sum_{\text{geom}} e^{i\,S_{\rm Regge}[\Delta, j_f]} + \text{degenerate configurations}$$
The degenerate (non-geometric) configurations are a known issue: they correspond to orientations where the 4-simplex is “inside out.” Various proposals exist for suppressing them, including orientation-dependent face amplitudes.
6. Finiteness and the Cosmological Constant
The Ponzano-Regge model diverges (the sum over spins is unbounded). Turaev and Viro showed that introducing a cosmological constant $\Lambda > 0$ regularizes the model by replacing $SU(2)$ with the quantum group $SU(2)_q$ at root of unity $q = e^{2\pi i / (k+2)}$:
$$Z_{\rm TV} = \sum_{j_f = 0}^{k/2} \prod_f [2j_f + 1]_q \prod_t \begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix}_q$$
where $[n]_q = (q^{n/2} - q^{-n/2})/(q^{1/2} - q^{-1/2})$ is the quantum dimension and the $6j$-symbol is now the quantum one. The level $k$ is related to the cosmological constant by $\Lambda = 12\pi^2 / (k\,\ell_P^2)$. The Turaev-Viro amplitude is finite and is a topological invariant.
Simulation: Spin Foam Visualization
We visualize the structure of a simple spin foam as a 2-complex interpolating between spin networks, show the vertex amplitude structure, large-spin asymptotics, and partition function convergence:
Spin foam structure and vertex amplitudes
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