Area and volume become operators with discrete spectra, and there is a smallest nonzero patch of surface
The previous lesson built a Hilbert space out of graphs. That was construction work. This lesson is where the construction pays: take the classical formula for the area of a surface, promote the field in it to an operator on that Hilbert space, and ask what the answer can be. The answer is not any positive number. It is a sum of square roots of j(j+1), one term for each point where a link of the graph crosses the surface, and the smallest nonzero value it can take is about 5.17 Planck areas.
Rovelli and Smolin published the calculation in 1995; Ashtekar and Lewandowski put the operator on rigorous footing shortly after and fixed its normalisation. It is the single most quoted result of the whole programme, and it is worth being precise about why. Discrete spectra are not exotic. Angular momentum has one, and the operator here is literally built from the same Casimir. What is unusual is that the quantity being quantised is a piece of geometry, that nobody put the discreteness in by hand, and above all that the spectrum has a gap. Between zero and 5.17 ℓP2 there is nothing. A surface can have no area at all, or it can have that much, and there is no state in between.
Volume is the same story told badly. The operator exists, it acts at the nodes rather than the links, its spectrum is discrete, and a node needs at least four links meeting at it to enclose any volume at all. Beyond that the honest summary is that the operator is a messy antisymmetric matrix on the intertwiner space, that there are two inequivalent well-studied versions of it which do not agree, and that its detailed spectrum is a research problem rather than a formula. This page states the area result exactly and the volume results only as far as they are actually established.
A surface floating in a spin network. It picks up area only where a link pierces it, and each piercing contributes an amount fixed entirely by that link’s spin. Add punctures, change their spins, delete them, and watch the total. The disc is drawn as a disc for convenience; the operator only ever sees the punctures.
Watch the total rather than the picture. Every click moves the area by a fixed amount that depends only on the spin you added, never by an amount you chose, and the surface grows in visible steps because there is nothing in between. Clearing the surface gives exactly zero; one j = 1/2 puncture gives 5.17 ℓP2; nothing in that interval is an eigenvalue. The disc is drawn as a disc for convenience only. The operator knows nothing about its shape, only about where the graph crosses it.
Classically, the area of a surface is an integral over that surface of the densitised triad, the variable that took the place of the metric when Ashtekar rewrote general relativity in 1986. The triad is a field, so the integral runs over a continuum of points, each contributing infinitesimally. Quantising changes the character of that expression completely. The triad becomes an operator that is not a smooth field at all: acting on a spin network state, it does nothing except where a link of the graph is, and there it returns the SU(2) generator carried by that link.
The consequence is that the integral collapses. Almost every point of the surface contributes exactly zero, because almost every point has no link through it. What survives is a finite sum over the punctures, the isolated points where links cross, and the square root in the classical formula turns into the square root of the Casimir operator of each link’s representation. Spin network states are therefore not merely convenient; they are the eigenstates of the area operator, and their eigenvalues are read off the labels.
The smallest label a link can carry is j = 1/2, whose Casimir is √3/2, so the smallest nonzero eigenvalue is 4√3 π γ ℓP2, or 5.169 Planck areas at the standard value γ = 0.2375. Notice where γ sits. The Barbero-Immirzi parameter drops out of the classical equations of motion entirely, but it multiplies every geometric eigenvalue, so the quantum theory has a one-parameter family of geometry spectra that the classical theory cannot distinguish. Fixing γ from the black hole entropy count, as Ashtekar, Baez, Corichi and Krasnov did in 1998, is the standard move, and it is a genuine weakness that this is the way the scale gets set.
Every allowed area, drawn as a rung. The left edge is the point: however you set the sliders, nothing appears between zero and the gap. The right end is the other point: the rungs crowd together as the area grows, because the number of ways to build a given area out of spins grows much faster than the area does.
Two things are worth watching for. First, the left edge does not move: however you set the sliders there is nothing between zero and the first rung, because the smallest Casimir is √(3)/2 and one puncture is the least you can have. Raising γ slides the whole ladder outwards without ever opening a level below the gap. Second, the right end fills in. Near 16 ℓP2 the mean spacing is 1.034; near 35 it is 0.450, and it keeps falling, because the number of spin multisets summing to a given area grows far faster than the area does. The plot truncates at 4 punctures and j = 2, so the true spectrum is denser still than what is drawn here, especially at the right-hand end.
Area is the easy case because the classical expression involves the triad quadratically and a surface meets a graph in isolated points. Volume involves a triple product of triads, and a region meets a graph in whole segments of link. The regularisation that makes sense of the product forces the operator to have support only where three link directions can be independent, which is to say at the nodes. Links contribute nothing. A node with three links meeting at it contributes nothing either, because the antisymmetric triple product of three coplanar-in-the-relevant-sense directions vanishes. You need valence four before a node encloses any volume, which is exactly the combinatorics of a tetrahedron, the reading that Barbieri made precise in 1998 and that the next lesson is built on.
What is solidly established: the volume operator is well defined, self-adjoint on the kinematical Hilbert space, supported at nodes, zero on nodes of valence three or less, and its spectrum is discrete. What is not: almost everything quantitative. There are two inequivalent operators in the literature, the Rovelli-Smolin version and the Ashtekar-Lewandowski version, differing in how the regularisation treats the orientation of the triples of links, and they give different eigenvalues in general. The matrix whose eigenvalues you need is antisymmetric on the intertwiner space and grows fast with valence, so most of what is known past the smallest cases is numerical. Whether the volume spectrum has a gap in the way the area spectrum does is not settled either: numerical work by Brunnemann and Rideout found eigenvalues of the Ashtekar-Lewandowski operator that approach zero as the spins grow, which would mean no uniform smallest nonzero volume.
It is worth resisting the temptation to say space comes in grains of a definite size. The area result licenses a precise statement and the volume result does not. What both support is the weaker and still striking claim that geometry is a spectrum: the quantities we call area and volume are observables of a quantum system, they take values in a discrete set, and the scale of that set is set by ℓP2 and γ rather than by anything we chose.
A discrete spectrum you could never resolve. Count the eigenvalues in a window and you find the mean gap between neighbours collapsing exponentially with the area. Slide out from the gap towards a proton, a virus and a square centimetre, and watch the vertical axis need a second logarithm to keep up.
The solid stretch is an exact count: enumerate every eigenvalue built from at most 7 punctures of spin at most 6, and the mean gap between neighbours falls by a factor of ten for every 20 Planck areas of surface. The dashed line continues that measured trend, and the vertical axis has to be logarithmic twice over to hold the result. At a proton’s cross-section the gaps are already smaller than any number physics has a use for. Two honest caveats: the truncation throws away eigenvalues, so the real spectrum is denser and the real spacing smaller than the dashed line says, and an extrapolated fit is not a theorem. Neither caveat changes the conclusion, because both push the same way.
Whose surface, and how do you specify it? The area operator is invariant under the internal SU(2) rotations, so it passes the Gauss constraint. Diffeomorphism invariance is more delicate. Asking for the area of the surface at fixed coordinates is not a meaningful question in a theory where moving the coordinates is gauge; the surface has to be picked out physically, for instance as the boundary of a region defined by matter, or as a black hole horizon defined by the geometry itself. Only then does the question the operator answers correspond to something an observer could set up.
It is a kinematical observable. The area operator does not commute with the Hamiltonian constraint, so it is not a Dirac observable of the full theory. It is an operator on the space of states before the dynamics has been imposed. Dittrich and Thiemann argued in 2007 that discreteness of such kinematical operators does not by itself imply that the corresponding physical, fully constrained observables have discrete spectra, and constructed examples where the two come apart. Rovelli replied that for the relational observables that matter physically the discreteness does survive. The exchange has not been settled to everyone’s satisfaction, and anyone telling you the granularity of space is a theorem is overstating the case.
What about Lorentz contraction? The standard objection is that a smallest area cannot be Lorentz invariant, because a boosted observer sees lengths contracted and would report a smaller one. The standard reply is that the gap is the smallest eigenvalue of an operator, not a fixed length built into a background, and operators are not required to have Lorentz-invariant spectra. Angular momentum is the ready example: its spectrum is discrete in every frame, the rotation group is unbroken, and no one concludes that space has a preferred axis, because a boost or a rotation changes which operator you are measuring rather than shifting the spectrum of the one you measured. Rovelli and Speziale made this concrete in 2003 for the area operator in particular. It is a real answer, and it is not universally accepted; the fully Lorentz-covariant formulation of the theory is still an active area rather than a closed one.
None of that undoes the demonstration above. Whatever the resolution, the spectrum crowds so fast that a surface the size of a proton has gaps between its levels smaller than any quantity in physics, and a square centimetre needs at least 7.41 × 10^64 punctures of the smallest spin. Granularity at the Planck scale and perfectly smooth geometry at every scale we can measure are not in tension. That is a feature of the theory, and also the reason it is so hard to test, which is the subject of the last lesson in this module.