Spin-Foam Amplitudes

A spin foam assigns a quantum amplitude to a labeled two-complex whose boundary is a spin network. Four-dimensional models begin with topological BF theory, whose bivector data contain no local gravitational degrees of freedom, and impose simplicity constraints intended to make each bivector arise from a tetrad. The resulting vertex amplitude is a precise finite-complex object. Establishing its composition, refinement, and continuum interpretation is a separate problem.

Required background. Variational and Symplectic Geometry supplies actions, constraints, and boundary symplectic data; Frames, Tetrads, and Spinors supplies coframes and Lorentz representations.

Helpful background. Flat-Spacetime Quantum Field Theory supplies path integrals and amplitudes; Einstein Equations supplies the classical dynamics to recover; Quantum Information supplies boundary-state and tensor-contraction language; and Quantum Gravity supplies the distinction between a regulated amplitude and a continuum theory.

From BF theory to gravitational bivectors

For a Lorentz connection $\omega^{IJ}$ and Lie-algebra-valued two-form $B^{IJ}$, four-dimensional BF theory has action

$$S_{\mathrm{BF}}[B,\omega] =\frac{1}{16\pi G}\int_M B_{IJ}\wedge F^{IJ}[\omega].$$

Varying $B$ sets $F=0$, while varying $\omega$ sets $D_\omega B=0$; the theory is topological. General relativity is obtained only after algebraic constraints restrict a nondegenerate $B$ to a tetrad sector. With one common Holst convention,

$$B^{IJ}=\left({}^\star+\frac{1}{\gamma}\right) (e^I\wedge e^J),$$

up to sector and orientation signs. Here the internal Hodge dual is $({}^\star X)^{IJ}=\tfrac12\epsilon^{IJ}{}_{KL}X^{KL}$; on Lorentzian internal bivectors ${}^\star{}^\star=-1$. Simplicity is most transparently stated for the untwisted geometric bivectors $\Sigma_f^{IJ}=\int_f e^I\wedge e^J$, rather than imposing the same equation on the Holst combination $B_f=({}^\star+\gamma^{-1})\Sigma_f$. For two triangles of one tetrahedron, diagonal and cross-simplicity give

$$\epsilon_{IJKL}\Sigma_f^{IJ}\Sigma_{f'}^{KL}=0, \qquad f,f'\subset e.$$

The zero has a geometric meaning: both bivectors lie in the same three-dimensional tetrahedral hyperplane, so their four-volume vanishes. Closure is $\sum_{f\subset e}\Sigma_f=0$ with outward orientations; it also holds for $B_f$ after the linear Holst map. A nonzero oriented four-volume instead comes from appropriate bivectors spanning four independent directions in the four-simplex. Cross-simplicity, closure, nondegeneracy, and gluing consistency together select geometric sectors. Degenerate and sign-related sectors remain possible and must be selected or controlled. Plebanski’s constrained action and its sectors are the classical starting point Plebanski 1977, pp. 2512–2515.

Discretization and representation constraints

Choose an oriented two-complex $\mathcal C$ dual to a cellular decomposition. A face $f$ is dual to a triangle, an edge $e$ to a tetrahedron, and a vertex $v$ to a four-cell. BF quantization assigns a group representation to each face and an invariant tensor to each edge. A local state sum has the general form

$$Z_{\mathcal C}(\Psi_{\partial\mathcal C}) =\sum_{\{j_f,\iota_e\}_{\mathrm{internal}}} \left(\prod_f A_f\right) \left(\prod_e A_e\right) \left(\prod_v A_v\right) \Psi_{\partial\mathcal C}.$$

Face and edge amplitudes, gauge-volume factors, boundary normalization, and the treatment of internal sums are part of the definition, not harmless decoration.

In the Lorentzian EPRL model, the linear simplicity constraint maps a boundary $SU(2)$ representation $j$ into a unitary principal-series representation of $SL(2,\mathbb C)$ with labels

$$(\rho,k)=(\gamma j,j)$$

in the displayed convention. The injection $Y_\gamma:\mathcal H_j\to\mathcal H_{(\gamma j,j)}$ lets Lorentz-group parallel transport act while retaining the canonical $SU(2)$ boundary Hilbert space. Group averaging at each tetrahedron enforces gauge invariance and couples the four incident faces. The EPRL construction, including the distinction between strongly and weakly imposed simplicity, is developed in Engle, Livine, Pereira, and Rovelli 2008, §§2–4.

For Euclidean signature the gauge group splits as $Spin(4)\cong SU(2)_+\times SU(2)_-$. In the sector $0\lt\gamma\lt1$, a common convention correlates the two BF spins by

$$j^\pm=\frac{1\pm\gamma}{2}\,j.$$

Only spins for which both $j^\pm$ are half-integers are admitted. The Lorentzian and Euclidean label maps are not interchangeable; neither formula should be applied without declaring signature and sector.

Vertex amplitudes and composition

For a four-simplex, five tetrahedra provide five group variables and ten triangles provide ten boundary representations. Define the EPRL-mapped four-valent intertwiner schematically by

$$\mathcal I_\gamma(\iota_e)= \int_{SL(2,\mathbb C)}dg_e\, \left[\bigotimes_{f\supset e}g_eY_\gamma\right]\iota_e , \qquad A_v(j_f,\iota_e)= \operatorname{Contr}_v\! \left[\bigotimes_{e\ni v}\mathcal I_\gamma(\iota_e)\right],$$

with one redundant group average removed or gauge-fixed. Here each $\iota_e$ is a four-valent tensor in the product of the four incident $SU(2)$ representation spaces; $Y_\gamma$ acts separately on each leg, and $\operatorname{Contr}_v$ pairs magnetic indices belonging to the same triangle across neighboring tetrahedra. Thus no tetrahedral intertwiner is incorrectly treated as a state in a single face representation. This expression displays the essential ingredients: boundary spin-network data, Lorentz transport, the simplicity map, and contraction according to four-simplex adjacency. Exact normalization depends on the declared model.

If two complexes are glued along a common boundary graph, composition requires summing over an orthonormal basis of shared boundary states:

$$Z_{\mathcal C_2\circ\mathcal C_1} =\sum_\alpha Z_{\mathcal C_2}(\Psi_{\mathrm{out}},\alpha)\, Z_{\mathcal C_1}(\alpha,\Psi_{\mathrm{in}}).$$

This identity is a check on boundary normalization and gauge fixing. It does not imply invariance under refinement: a gravity state sum generally changes when internal faces and vertices are added unless amplitudes and coarse-graining maps sit at a suitable fixed point.

Regge asymptotics and their limit

Scale all spins of a nondegenerate Lorentzian Regge boundary geometry as $j_f\mapsto\lambda j_f$. Stationary-phase analysis of the EPRL four-simplex amplitude gives paired contributions of the form

$$A_v(\lambda j_f,n_{ef}) \sim \lambda^{-12} \left(N_+e^{\,i\lambda S_{\mathrm R}} +N_-e^{-\,i\lambda S_{\mathrm R}}\right),$$

where $S_{\mathrm R}$ is the Lorentzian Regge action of that single simplex, with triangle areas and orientation fixed as in the stated convention Barrett et al. 2010, §6.4, Eq. (53). Closure and shape matching are discriminating checks: boundary data lacking a consistent nondegenerate four-simplex do not acquire the same Regge saddle. Reversing orientation exchanges the conjugate phases.

This result is stronger than a formal analogy but weaker than a continuum derivation. It controls one vertex at large quantum numbers. It does not sum arbitrary foams, remove infrared or ultraviolet regulators, establish the Hamiltonian constraint, or show Einstein propagation over many scales. Perez surveys these distinctions, radiative corrections, and refinement questions in Perez 2013, Parts III–IV. String/holographic constructions, semiclassical black-hole thermodynamics, classical alternative gravity, and generic QFT remain outside this chapter’s scope.

Exercises

In the Euclidean $0\lt\gamma\lt1$ convention, take $\gamma=\tfrac13$ and boundary spin $j=3$. Compute $(j^+,j^-)$, compare the number of independent BF labels before simplicity with the constrained labels, and state one reason this check does not establish a continuum limit.

Solution

The simplicity map gives

$$j^+=\frac{1+1/3}{2}\,3=2, \qquad j^-=\frac{1-1/3}{2}\,3=1.$$

Both are allowed integer spins. Unconstrained Euclidean BF theory permits $j^+$ and $j^-$ to be chosen independently; simplicity restricts them to a one-parameter subset labeled by $j$ (and subject to half-integrality). That verifies the intended reduction of representation data for this face. It does not prove that many constrained vertices reproduce general relativity: composition, degenerate sectors, radiative corrections, refinement, and continuum observables have not been controlled by this calculation.

References

• Barrett, John W., Richard J. Dowdall, Winston J. Fairbairn, Frank Hellmann, and Roberto Pereira. “Lorentzian Spin Foam Amplitudes: Graphical Calculus and Asymptotics.” Classical and Quantum Gravity 27 (2010): 165009. Open manuscript.
• Engle, Jonathan, Etera Livine, Roberto Pereira, and Carlo Rovelli. “LQG Vertex with Finite Immirzi Parameter.” Nuclear Physics B 799 (2008): 136–149. Open manuscript.
• Perez, Alejandro. “The Spin-Foam Approach to Quantum Gravity.” Living Reviews in Relativity 16 (2013): 3. Open manuscript.
• Plebanski, Jerzy F. “On the Separation of Einsteinian Substructures.” Journal of Mathematical Physics 18 (1977): 2511–2520. DOI.