Rewrite gravity in terms of what a vector does when you carry it around a loop, and the infinities go away
The previous lesson ended with a new set of variables. Ashtekar’s 1986 reformulation replaced the metric of space with an SU(2) connection A and its conjugate densitised triad E, and general relativity came out looking like a gauge theory. That is real progress, because we know how to quantise gauge theories. The obvious next move is to promote A and E to operators and impose the canonical commutation relations.
The obvious next move does not work, and the reason is not a technical inconvenience. A quantum field is a distribution, not a function. The object A(x) at a sharp point is not an operator on any Hilbert space; only smeared objects are. In ordinary field theory you smear against a test function over a three-dimensional region, which requires a volume element, which requires a metric. In a theory whose whole point is that the metric is the unknown, there is no metric available to do the smearing with.
Loop quantum gravity’s answer is a change of variables that solves both problems at once. Integrate the connection along a one-dimensional curve instead. A connection is a one-form, so a curve is exactly the right dimension to integrate it over, and no metric appears anywhere. The result is the holonomy: the group element telling you what happens to a vector carried along that curve. Its conjugate partner, the flux, is integrated over a two-dimensional surface, again with no metric. One plus two is three, and between them the pair covers a slice of space without ever assuming a background.
This lesson works through why that trade is a good one, what is lost (nothing, as long as you keep every loop) and what is gained (a well-defined algebra, and a uniqueness theorem about its representation that is one of the programme’s genuine successes). It ends with the reason the original loop states had to be abandoned, which is what the next lesson repairs.
A connection is a rule for carrying a vector along a curve without turning it. Follow the rule around a closed loop on a flat plane and the vector comes back exactly as it left. Do it on a sphere and it comes back rotated. Change the loop and watch two numbers that have no obvious reason to agree agree anyway.
The enclosed area and the holonomy angle are the same number, to every digit shown, for any loop you can build here. That is Girard’s theorem: on the unit sphere the rotation a vector picks up going round a loop equals the area the loop encloses. Shrink the loop and both go to zero together, but their ratio stays pinned at the Gaussian curvature, 1. That ratio is what survives the limit, which is why a theory that keeps only loops has not thrown the curvature away.
The equality in the demo is Girard’s theorem, first proved for spherical triangles in the seventeenth century and later recognised as the two-dimensional case of Gauss-Bonnet. On the unit sphere the angle a vector picks up going round a closed loop equals the area the loop encloses. On a sphere of radius R it equals the area divided by R squared, which is the Gaussian curvature times the area. That is the statement in its useful form: the holonomy of a small loop is the curvature multiplied by how much area the loop caught.
Shrink the loop and both numbers vanish, but their ratio does not. What survives the limit is the curvature, so a holonomy is not a coarse summary of the connection with the fine detail thrown away. It is the integrated form of the curvature, and running the limit backwards recovers the curvature tensor component by component: pick a small square loop in the plane spanned by two directions and the leading term of its holonomy is the component of F in that plane. Keep the holonomies of all loops and you have kept everything.
There is a second reason to prefer the group element to the tensor, and it matters more than the first. The curvature F is an unbounded field: it can be arbitrarily large, and in a quantum theory an unbounded operator-valued distribution is exactly where the divergences live. A holonomy is an element of SU(2), which is compact. Its matrix entries can never exceed one in absolute value, no matter what the connection does along the way. The change of variables trades an unbounded object for a bounded one without losing any information, which is not a trade you often get.
Three ways to make sense of a field in a region, side by side, as the region shrinks. Two of them fail, for two different reasons. Then a control that shrinks a loop and checks that the holonomy really does reduce to the curvature, with a residual that behaves the way the expansion says it should.
The three columns are one argument. A field at a sharp point is not an operator; smearing it over a ball repairs that but needs a volume element, and in a theory whose unknown is the metric there is no volume element to use. A one-form does not care: it integrates along a curve using nothing but the curve. That is the whole reason the variables of this theory are holonomies and fluxes rather than A and E themselves. In the plot, the residual between the measured holonomy angle and the linear prediction tracks the dashed reference over the small-loop end of the range: slope 4 in the loop radius, which is slope 2 in the area, exactly what h = 1 + (area)F + O(area²) claims. The last box holds that ratio steady as you shrink the loop. Shrinking loops do not lose the curvature; they are how you recover it.
The dimension counting is the whole argument. The connection A is a one-form, so its natural integration domain is a curve. The densitised triad E is a vector density of weight one, which is the same thing as a two-form with an internal index, so its natural integration domain is a surface. Neither integral needs a metric, because in both cases the object being integrated already carries exactly the right density weight. Curves are one-dimensional, surfaces are two-dimensional, and a spatial slice is three-dimensional, so the pair between them reaches everywhere.
Their Poisson bracket closes. A flux through a surface and a holonomy along a curve commute unless the curve pierces the surface, and when it does the bracket inserts an SU(2) generator at the puncture. That algebra, called the holonomy-flux algebra, is the background-independent replacement for the canonical commutation relations. It is well defined where the naive relations are not, and every subsequent construction in this module is built on a representation of it.
Then comes the result that makes this more than a convenient choice. Lewandowski, Okolow, Sahlmann and Thiemann proved in 2006, with Fleischhack arriving independently by a different route, that the representation of the holonomy-flux algebra carrying a state invariant under spatial diffeomorphisms is essentially unique. Up to unitary equivalence there is one, the Ashtekar-Lewandowski representation, and it is the one the theory uses. This is genuine rigidity and it is rare. In ordinary quantum field theory the choice of vacuum is notoriously not unique once the background stops being flat, and much of the ambiguity of quantum field theory in curved spacetime comes from exactly that. Here the demand for diffeomorphism invariance removes the freedom.
Two honest qualifications. First, the theorem is only as strong as its hypotheses: it assumes this particular algebra, generated by this particular pair of smeared objects, and a critic who thinks the choice of algebra is the thing that should have been derived is not answered by a uniqueness proof about it. Second, the resulting representation is not weakly continuous in the holonomy. Shrink an edge to nothing and the holonomy operators do not converge to the identity operator, which means there is no operator corresponding to the connection A itself, and the Stone-von Neumann theorem does not apply. That discontinuity is what allows the discrete geometry of the following lessons to exist at all, and it is also why the representation is unitarily inequivalent to the Fock representation of a graviton field. Recovering ordinary low-energy physics from it is not a solved problem. It is one of the two or three open problems the programme is judged on.
A lattice with a random SU(2) element on every edge. Walk a path node by node, then hit the gauge transform button. Every individual edge holonomy changes. Whether anything else does depends entirely on whether your path came back to where it started.
Press the gauge transform button. Every node gets its own random SU(2) element, so every individual edge holonomy changes, and the top row of the table moves. The bottom two rows do not: for a closed loop the gauge elements at the interior nodes cancel in pairs and the two ends are the same node, so the holonomy is merely conjugated, and neither its rotation angle nor its character notices. The Wilson trace agrees to every digit the arithmetic has. That number, and not the connection, is what the theory is allowed to call an observable. Turn the curvature slider to zero and every trace collapses to 2j + 1, the dimension of the representation, because a flat connection has trivial holonomy.
A holonomy is gauge covariant, not gauge invariant. Under a gauge transformation with a group element at every point, the holonomy of an edge is hit by that element at one end and by its inverse at the other. Compose a path out of edges and the factors cancel at every interior node, because the end of one edge is the start of the next. Only the two extremities survive. So an open path still transforms, and its holonomy is still not something the theory is allowed to call an observable.
Close the path and the two extremities are the same point. What is left is conjugation, and a trace does not notice conjugation. That trace, taken in the spin-j representation, is the Wilson loop, and for SU(2) it is the character χj, a function of nothing but the rotation angle of the holonomy. It is the entire gauge-invariant content of a single loop: two loops with the same rotation angle are indistinguishable to every Wilson loop you can write down. Wilson introduced these in 1974 for lattice gauge theory, where they diagnose confinement, and they arrived in gravity for the same reason they were useful there. They are the gauge-invariant functions you can actually get your hands on.
Rovelli and Smolin built the first version of loop quantum gravity out of exactly these objects in 1990. States were functionals of loops, and the Gauss constraint, the one demanding invariance under internal SU(2) rotations, was solved by construction rather than imposed afterwards. The spatial diffeomorphism constraint became almost trivial as well, since moving a loop smoothly around does not change its knot class. For a formalism that had been stuck for fifteen years, two of the three constraints falling out for free was a startling result.
It also did not work, and the reason is worth stating precisely because it is what the next lesson fixes. Wilson loops are not independent. For SU(2) in the fundamental representation, the trace of a product is determined by the traces of the factors, through an identity that goes back to Mandelstam. Given loops that share edges, the loop states satisfy relations, so the set of all loop states is overcomplete: you cannot expand a general state in them uniquely, and you cannot compute an inner product by counting. The fix, found by Rovelli and Smolin in 1995 and reaching back to a combinatorial construction Penrose had written down in 1971 for entirely different reasons, is to stop using loops and use graphs whose edges carry spins and whose nodes carry intertwiners. Those give an orthonormal basis. That is the subject of the next lesson.
One more thing the demo is quietly showing. The group is SU(2), not SO(3). A rotation by 2π is the identity in SO(3) but not in SU(2), where it is minus the identity, and the half-integer characters change sign under it. This is not a technicality that could be tidied away by choosing the other group. The connection couples to fermions, and fermions see the sign, so SU(2) is the physically correct choice. It is also the reason the labels on spin network edges run over half-integers rather than integers, and therefore the reason the smallest nonzero quantum of area in the next lessons carries j = 1/2 rather than j = 1.