Let a spin network evolve and its history sweeps out a foam: the sum over spacetimes, one vertex at a time
Everything up to this point has been kinematics. We have a Hilbert space, an orthonormal basis of labelled graphs, and geometric operators with discrete spectra acting on it. What we do not have is any statement about what happens next. Two of the three constraints are solved. The third, the Hamiltonian constraint, is the dynamics, and it is where the subject stops being tidy.
The canonical route is to build that constraint as an operator. Thiemann managed it in 1996, and the result was finite, which was a genuine achievement given that finiteness is exactly what the perturbative approach could not deliver. But the construction makes regularisation choices that nothing fixes, so the operator is not unique, and the resulting equation is fearsomely hard to solve or even to interpret. There is no consensus that any candidate has the right classical limit.
So try the other route. Instead of asking how a state evolves, sum over histories. Push a spin network forward and each link drags out a two-dimensional face, each node drags out a one-dimensional edge, and where the graph itself changes there is a vertex. Colour the faces with the spins and the edges with the intertwiners and the result is a spin foam: a path integral for geometry in which no background metric appears at any point. That is the subject of this lesson, along with an honest account of how much of it works.
Scrub a spin network through the sweep direction and watch the two-complex assemble itself. Nothing happens while the graph merely persists; the foam is a cylinder and the geometry is frozen. All of the dynamics lives at the moment the graph changes, and that moment is a vertex.
Drag the sweep slowly. For most of its length the foam is a cylinder: nothing happens, because the graph is only being carried along and a spin foam with no vertices assigns the same geometry to the top slice as to the bottom one. Then the single node in the middle opens into a triangle. Three new faces appear, three new edges start, and the area of the slice jumps, because area is carried by the links and there are suddenly more of them. That one point is a vertex, and the amplitude of the entire history is a product over vertices. Switch to colour-by-role to see the dictionary directly: link to face, node to edge, change to vertex. The spoke spin slider shows the other half of the rule — the Gauss constraint has to hold on every slice, so a labelling that leaves any node with a zero-dimensional intertwiner space is not a history at all.
In ordinary quantum mechanics a path integral sums over trajectories through a fixed space, weighting each by the exponential of its action. Here there is no fixed space to move through, so the objects being summed over are not paths in a geometry. They are geometries, presented combinatorially: two-complexes with representation labels. The amplitude of a history factorises into a product over its faces, its edges and its vertices, and the transition amplitude between two boundary spin networks is the sum of that product over every labelling and every two-complex interpolating them.
Notice how naturally the boundary works. A spin foam bounded above and below has spin networks on those boundaries, because a face meeting the boundary is a link and an edge meeting the boundary is a node. The states of the canonical theory are exactly the boundary data of the covariant one, which is the strongest reason to believe the two are formulations of the same theory even though nobody has proved that they are.
The structure is also flatly local. The face and edge amplitudes are bookkeeping; everything interesting is in the vertex, which is the elementary event in which the geometry rearranges itself. Writing down a theory of quantum gravity in this language means writing down one function: the vertex amplitude.
In three dimensions the elementary rearrangements are clean: two tetrahedra sharing a face become three sharing an edge, or one tetrahedron subdivides into four. Every vertex of the two-complex is one of these. The reason three-dimensional gravity is exactly solvable is that the amplitude does not care which sequence of moves you take to get from one triangulation to another, and this demo checks that identity term by term.
The five vertices and nine edges of the bipyramid never move: the move is entirely internal. What changes is that the shaded face joining the two tetrahedra is replaced by the dashed edge joining the two apexes, and two tetrahedra become three. On the right, the dual graph does the same thing in the language of spin networks, one node per tetrahedron and one link per shared face. That change of graph is what a spin foam vertex is.
Gravity in three spacetime dimensions has no local degrees of freedom. There are no gravitational waves, no gravitons, and every solution is locally flat; all the physics is topological. That makes it a bad model of our world and an excellent laboratory, because the sum over histories can actually be carried out.
Ponzano and Regge wrote the answer down in 1968, decades before anyone said the words spin foam. Triangulate the three-manifold, label every edge with a spin, and assign to each tetrahedron the Wigner 6j symbol of its six edge labels. Sum over labels. That is the whole model. And it is independent of the triangulation, because the 6j symbols satisfy an identity, due to Biedenharn and Elliott, which says precisely that the 2-3 move does not change the amplitude. Triangulation independence is not imposed; it is a fact about representation theory.
Four dimensions has no such luck. The analogous invariance fails, so the sum over two-complexes is not a formality that can be quotiented away. It is a genuine sum that has to be defined and controlled, and nobody knows how to organise it. Group field theory is the leading attempt: reformulate the state sum as the Feynman expansion of an auxiliary field theory on a group manifold, so that the sum over two-complexes becomes a sum over Feynman diagrams. Whether that series converges, or what resumming it would mean, is open.
This is the evidence. The 6j symbol is pure representation theory: no metric, no coordinates, nothing continuous anywhere in its definition. Turn the spins up and watch it lock onto the cosine of the Regge action, the discrete Einstein-Hilbert action of the tetrahedron whose edges are those spins. Then push the spins into a configuration that cannot close into a tetrahedron and watch the oscillation give way to decay.
Start on the irregular preset and drag the scale. The green curve is the exact 6j symbol, computed from the Racah sum with no geometry anywhere in it; the dashed amber curve is the cosine of the Regge action of the tetrahedron whose edges are j + 1/2 long. At scale one they are already close and by scale twenty the discrepancy is a few parts in a thousand, falling roughly as one over the spin. The oscillation is not a coincidence of this configuration: change any of the six spins and both curves change together. Switch to log |6j| and pick “never closes”: those spins pass every triangle inequality but no Euclidean tetrahedron has those edges, so there is no action to oscillate with and the symbol falls off a cliff instead, straight down a log axis. The exponential decay of the non-geometric branch is real, but the library only evaluates the oscillating formula, so nothing is drawn in amber there.
The four-dimensional vertex took two decades to get right. The strategy comes from an observation of Plebanski: general relativity can be written as a topological theory, which has no local degrees of freedom and is easy to quantise, plus a set of constraints that break the topological invariance and put the local degrees of freedom back. Quantise the easy theory, then impose the constraints on the representation labels. The first serious attempt, the Barrett-Crane model of 1998, imposed them too strongly and ended up with a vertex whose boundary states were not the spin networks of the canonical theory and whose graviton propagator came out wrong.
The model that fixed this appeared in 2008, from Engle, Pereira, Rovelli and Livine, with Freidel and Krasnov arriving independently. Its constraints are imposed weakly, and the Barbero-Immirzi parameter appears exactly where it must: it is the ratio tying the labels of the covariant Lorentz group to the SU(2) labels on the boundary. The boundary states of this vertex are the spin networks we already had, intertwiners and all. And its large-spin asymptotics, worked out over the following years, again give the Regge action. That is the main reason to think the four-dimensional model is a theory of gravity rather than an interesting sum of symbols.
Now the debits, and there are several. The sum over two-complexes is not defined. There is a persistent difficulty known as the flatness problem, in which the large-spin limit of the model on a fixed triangulation appears to suppress everything except flat geometries, which would be fatal if it survives; the literature is actively divided on whether it does. The asymptotic formula produces a cosine rather than a single exponential, which means the amplitude contains both orientations of the geometry and it is not clear how to project onto one. And the equivalence with the canonical theory remains a matching of boundary data rather than a theorem. This is the least settled part of loop quantum gravity, and the final lesson returns to it.