A graph with spins on its edges is not a picture of space — it is a quantum state of space itself
The previous lesson ended on a defect. Wilson loops solve the Gauss constraint automatically, which is why they were worth building a theory out of, but they are not independent of one another. For SU(2) the trace of a product of holonomies can be rewritten as a sum of traces of other loops, so the loop states overlap and overcount. A basis has to be independent, and the loop states are not.
Rovelli and Smolin fixed this in 1995 by throwing away the insistence on single loops. Take a graph instead. Put an irreducible representation of SU(2) on each link, and at each node put an invariant tensor in the product of the representations meeting there. The result is an orthonormal basis of the kinematical Hilbert space, with the overcounting gone: the Mandelstam identities are exactly the statement that the loop basis was writing the same graph state many times over.
The combinatorics were not new. Roger Penrose had written spin networks down in 1971 for a completely different purpose: a discrete model in which angles between directions emerge from representation theory alone, with no space assumed anywhere. He was not doing quantum gravity, and there was no reason to expect his objects to be the answer to a question about the quantised gravitational field. They were. This lesson builds them, takes a node apart to see what an intertwiner actually is, and then makes the point that the subject gets wrong more often than any other: a spin network is not a picture of space. It is a quantum state of space, and it is nowhere.
Drop nodes, join them with links, and put a spin on each link. Every node is checked against the Gauss constraint live: the panel underneath reports the dimension of the intertwiner space there, and tells you which node has failed and why when the labelling is illegal. Try the theta graph with three spins of one half and watch it refuse.
Drag any node. Nothing physical changes when you do. Click a link to select it, then pick a spin.
Two nodes joined by three links. The smallest graph with a genuinely three-valent node, and the smallest one whose labelling can fail.
The states of the theory are functions of the connection, and the well-behaved ones depend on it only through the holonomies along the links of some graph. Such a function is called cylindrical: it takes a finite number of group elements and returns a number. The Peter-Weyl theorem then does all the work. It says that any square integrable function on SU(2) decomposes into matrix elements of the irreducible representations, so expanding a cylindrical function link by link automatically produces a sum of terms, each carrying one spin per link.
Imposing the Gauss constraint contracts the free indices at each node with an invariant tensor. What is left is labelled by a graph, a spin on every link, and an intertwiner at every node, and these states are orthonormal in the measure Ashtekar and Lewandowski constructed in 1994 and 1995 on the space of generalised connections. Orthonormal is the word that matters. It is what the loop basis could not deliver, and it is what makes the rest of the theory calculable: operators become matrices in this basis, and the geometric ones turn out to be diagonal.
One piece of bookkeeping is worth stating plainly, because the demonstration above enforces it. A spin network with a link carrying spin zero is the same state as the network with that link deleted, so the labels are taken to be nonzero and the graph is taken to be the smallest one supporting the state. Once that convention is fixed, two different labelled graphs give orthogonal states, and there is no double counting left anywhere.
Four spins meet at a node. Dial them and watch the dimension of the intertwiner space change, then switch which pair you fuse first. The two schemes give different lists of virtual spins and the same dimension, because they are two bases of one space, and the matrix between them is built from 6j symbols computed live.
| (12)(34) \ (13)(24) | k’ = 1 | k’ = 2 |
|---|---|---|
| k = 1/2 | -0.4082 | 0.9129 |
| k = 3/2 | 0.9129 | 0.4082 |
The selected vector k = 1/2 of the (j1 j2)(j3 j4) scheme is a superposition of 2 vectors of the (j1 j3)(j2 j4) scheme: -0.408 |k' = 1> + 0.913 |k' = 2>. Its amplitudes squared sum to 1.000000, computed live, which is the 6j orthogonality relation doing its job.
The two pairings give completely disjoint lists of virtual spins, k in {1/2, 3/2} against k in {1, 2}, and yet both are bases of the same two-dimensional space. The virtual spin is a property of the basis, not of the node.
Gauge transformations in this formulation act at the nodes: an SU(2) rotation there twists every holonomy that ends on it. A state survives that twisting only if the tensor sitting at the node is invariant, which is why the Gauss constraint reduces to a question in representation theory. Does the tensor product of the representations on the incident links contain the trivial representation, and if so, how many times?
At a three-valent node the answer is the Clebsch-Gordan rule you already know. The invariant exists when the three spins obey the triangle inequalities and sum to a whole number, and when it exists it is unique up to scale, so nothing is left to choose. From four links upward the space of invariants is more than one-dimensional, and the node carries a quantum number that the spins on its links do not fix. Naming a vector in that space means choosing a recoupling scheme: fuse two of the legs, read the spin k of the resulting virtual link, and pair it against the rest. Choose a different pair to fuse first and you get a different basis of the same space. The change of basis is the Wigner 6j symbol, up to a phase which is nothing more than a sign convention for each basis vector.
There is a geometric reading of all this, due to Barbieri in 1998, and the next two lessons live on it. A four-valent intertwiner is the quantum state of a tetrahedron whose four face areas are set by the four spins; the virtual spin k is the area of an internal diagonal. The two recoupling schemes correspond to the two ways of cutting the tetrahedron in half, and the operators measuring those two internal areas do not commute. A state with a sharp answer for one is a superposition of answers for the other. That is not a defect of the description. It is the first place where quantum geometry says something a classical polyhedron cannot.
Drag the network into any shape you like. One panel tracks the coordinates, which change constantly and mean nothing; the other tracks the labelled adjacency and the area of a marked surface, which do not move at all. Then try the one deformation that is not a deformation: passing a link through another link.
Left is your arrangement, right is one nobody would recognise as the same picture. The verdict is computed by comparing the labelled adjacency of the two: identical state. Every geometric quantity the theory predicts takes the same value in both.
The two configurations have the same nodes, the same links and the same spins. Only the crossing has changed. The linking number, computed here from the signs of the crossings actually drawn, reads 0 in this configuration and 1 in the other. Linking number cannot change under a smooth deformation, so the two are genuinely different physical states with identical combinatorics. That is why physical states are knot classes of labelled graphs, called s-knots, and not merely graphs.
Drag until the picture is unrecognisable and watch the right-hand panel refuse to move. The node positions are gauge, the labelled graph is the state, and the area of the marked surface is fixed at 35.66 Planck areas by the spins of the links that cross it and by nothing else.
A spatial diffeomorphism acts on these states by dragging the graph: it sends a state on a graph to the same labelled state on the deformed graph. The diffeomorphism constraint says that states related this way are physically identical, so the theory works not with embedded graphs but with their orbits under smooth deformation. Technically the orbits are constructed by group averaging, since the averaged state does not live in the kinematical Hilbert space itself but in its dual. The upshot is simple to state and hard to internalise: the nodes have no positions. There is no fact of the matter about where a node is, how far apart two nodes are in the coordinate sense, or what shape the graph has. Ask any of those questions and the theory has no answer, because the question was not about anything.
What survives is the combinatorics, and one thing more. A diffeomorphism is smooth and invertible, so it carries the graph along continuously and can never pass one strand through another. In three dimensions, knotting and linking survive the quotient alongside the adjacency, and the physical states are knot classes of labelled graphs, which Rovelli and Smolin called s-knots. That is genuinely surprising: the states of quantum geometry are objects from knot theory, and the connection was noticed early enough that the loop representation and the Jones polynomial were being discussed in the same breath around 1990. It is not all tidy. Grot and Rovelli showed in 1996 that at nodes of valence six and above the knot classes carry continuous parameters, so the space of states is not simply discrete unless the diffeomorphism group is extended to a larger group that washes those moduli out. Which extension is the right one is still argued about.
Now the caveat that the whole lesson has been leaning towards. Everything built here lives in the kinematical Hilbert space, and the word is doing real work. Two of the three constraints are solved: Gauss by the intertwiners, diffeomorphisms by passing to knot classes. The third, the Hamiltonian constraint, is not. Thiemann gave a well-defined operator for it in 1996, which was a genuine achievement given that the corresponding expression in the older metric variables never made sense, but it comes with large regularisation ambiguities, its action creates new nodes in a way many people find suspicious, and whether the resulting dynamics has general relativity as its classical limit is not established. So the basis is solid ground and the dynamics is not. What we have at the end of this lesson is a complete, orthonormal, background independent set of states. The next question is what geometry any one of them describes, and that is where the theory produces its sharpest result.