What a granular spacetime could actually do to a photon, what the data already rules out, and what is still broken
Loop quantum gravity has made no confirmed prediction. No experiment has tested it, and none currently planned would decisively confirm or refute it. That sentence should be read without softening, and it should also be read without triumph, because it is true of every approach to quantum gravity that anyone is working on. The reason is arithmetic rather than sociology: the Planck energy is about 9.0e+14 times the energy the LHC collides protons at, which is fifteen orders of magnitude, and no accelerator anyone can imagine closes that gap.
What can be done instead is amplification. Take an effect whose relative size is some power of E/EP, a number around 10-19 for the most energetic photons we ever see, and give it billions of years to accumulate. That is the entire strategy, and there are only two places it has been carried out with real data: the arrival times of gamma-ray burst photons that have crossed a cosmological distance, and the primordial power spectrum, where the largest angular scales carry modes that left the horizon closest to the beginning.
The first of those has produced a genuine, publishable constraint. Fermi saw a 31 GeV photon from GRB 090510, a short burst at redshift 0.903, arrive within about 0.86 seconds of the trigger after 7.5 billion years of travel. That pushes the energy scale of any linear-in-energy vacuum dispersion above 1.34 times the Planck energy. It is a beautiful measurement. It is also, and this is the point of the first half of this lesson, not a test of loop quantum gravity, because loop quantum gravity does not unambiguously predict the effect being constrained.
The second half of the lesson is the audit. This is the last page of the module, and the right way to end it is not with a flourish but with a list: what has been proved, what has been partly shown, what is genuinely open, and what critics have pressed hardest on. The programme has real results and real holes, and a reader who has come this far deserves to know exactly which is which.
A high-energy photon and a low-energy one leave the same burst together. If the vacuum has structure at the scale EQG, they do not arrive together. Choose a burst, choose the photon energy, choose the scale, and choose whether the effect is linear or quadratic in the energy. Then slide EQG downward and watch the model die.
Short burst seen by Fermi; the benchmark linear-order constraint.
The fractional lag in the last box is the whole strategy in one number: the effect is a part in 1018 or smaller, and it is only measurable because it has ten billion years to accumulate. Slide EQG down from the Planck energy and watch the model die — at linear order, GRB 090510 alone kills every scale below roughly the Planck energy itself. Then switch to n = 2 and slide all the way to 1011 GeV before anything goes red: a quadratic effect is suppressed so much harder that gamma-ray bursts say almost nothing about it. What is being excluded here is a dispersion relation someone wrote down, not loop quantum gravity, which does not require one.
Suppose granularity leaves a mark on how light propagates. The standard way to parametrise it is to let the vacuum dispersion relation pick up a correction suppressed by some power of the ratio of the photon energy to an unknown scale EQG. Differentiate and you get a group velocity that depends on energy: colours travel at slightly different speeds, and the vacuum behaves faintly like a dispersive medium. For a 31 GeV photon and EQG at the Planck energy the fractional speed difference is about 10-18, which is not a number any laboratory will measure.
Distance is what rescues it. Over a cosmological baseline the tiny velocity difference integrates into a time difference, and the integral is not merely the light-travel time: because the photon was blueshifted in the past, the effect accumulates preferentially at high redshift, and the weighting carries a factor of (1+z)n. Jacob and Piran wrote the correct expression down in 2008. For GRB 090510 at z = 0.903 the linear-order integral comes to 1.001 in units of the Hubble time, so the lever arm is roughly the age of the universe.
The catch is the same one that limits every timing measurement: you are comparing the arrival of a high-energy photon with the arrival of the low-energy emission, and the source itself may not have emitted them simultaneously. Nobody has a first-principles model of when a gamma-ray burst releases its hardest photons. That intrinsic emission lag, not the timing precision of the detector, is the dominant systematic, and the second demo lets you see how much the answer depends on it.
Invert the calculation. Each burst that has been seen puts a floor under EQG, and the four floors are plotted against the Planck energy for both orders. The linear bars cross the Planck line; the quadratic ones are eight decades short. Drag the arrival window to see how much of the result rests on an astrophysical assumption.
These are bounds on a phenomenological dispersion relation, not on loop quantum gravity. The theory does not unambiguously predict linear-in-energy Lorentz violation; some model calculations produce it, others argue that local Lorentz invariance survives quantum geometry, since a discrete operator spectrum is not the same thing as a preferred frame. A bound above the Planck energy closes off a class of ideas that quantum gravity might have produced. It does not test the theory in this module.
Two things to notice. First, the n = 1 bars cross the Planck line and the n = 2 bars do not come within eight decades of it, because a quadratic suppression throws away almost all of the lever arm that billions of light years bought. Second, drag the window slider: a factor of ten in the assumed arrival window moves the linear bound by a factor of ten and the quadratic bound by only its square root. The astrophysics of how a burst emits its hardest photons, which nobody has nailed down, is the limiting uncertainty in the whole programme.
The bound is real and it matters. Linear-in-energy Lorentz violation at the Planck scale was, for about fifteen years, the standard thing people said quantum gravity might do, and the standard reason for saying it was that a minimum length ought to behave like a lattice spacing and a lattice spacing picks a frame. The data have closed that off. From GRB 090510 alone the linear-order scale is above 1.64e+19 GeV, or 1.34 Planck energies. A null result that excludes a whole class of models is informative, and it is the only place where quantum gravity phenomenology has actually bitten.
What it does not do is test the theory in this module. Loop quantum gravity has no derivation of a linear vacuum dispersion. Some model calculations, particularly in the semiclassical weave states studied in the late 1990s, produced such terms; others argue that local Lorentz invariance survives quantum geometry entirely. The reason to take the second view seriously is the argument made three lessons ago: the discreteness in this theory is discreteness of the spectrum of an operator, not a fixed length built into a background. The area gap of 5.17 Planck areas is an eigenvalue, and a boosted observer measures a different operator rather than a contracted version of the same one, exactly as the eigenvalues of angular momentum are the same integers in every frame with the rotation group entirely unbroken. So what the gamma-ray bursts rule out is a hypothesis quantum gravity might have produced, and did not have to.
The quadratic case is a different story and is worth stating plainly. Suppression by two powers of the energy ratio throws away almost the entire lever arm: the same burst bounds EQG at only 3.34e+10 GeV, about 2.7e-9 of the Planck energy. Eight decades of room remain, and gamma-ray bursts will not close them, because the bound improves only as the square root of the observing time and the arrival windows cannot be tightened without understanding the sources.
The other handle is cosmological. Loop quantum cosmology replaces the initial singularity with a bounce, and modes that left the horizon nearest the bounce carry the imprint of the quantum-corrected background. Those are the largest scales in the sky today, so the predicted signature is a modification of the primordial power spectrum at low multipoles, together with the non-Gaussianity and tensor-to-scalar behaviour that come with it. Current data neither confirm nor exclude it. Two things make it hard. The largest angular scales are exactly where cosmic variance is worst, because there are only a handful of independent modes to measure, so the error bars cannot be shrunk by building a better telescope. And the size of the predicted effect depends on how much expansion happened between the bounce and the start of inflation, which the theory does not fix, so a null result constrains a corner of a parameter space rather than the framework.
Fourteen claims, each with a status, the state of the argument, and the strongest objection anyone has raised against it. Filter by status to see the shape of the programme, and open the comparison panel for the trade against string theory on the axes where the two genuinely differ.
There is real progress. Coherent states peaked on classical geometries have been constructed, and the large-spin asymptotics of the vertex amplitude reproduce the Regge action, which is exactly what a discrete approximation to general relativity should give. Barrett and collaborators established this for the EPRL vertex in 2009, extending the Ponzano-Regge asymptotics of 1968.
Large-spin asymptotics of a single vertex is not a continuum limit. Taking many vertices and refining the triangulation is uncontrolled, and there are indications the limit misbehaves: the flatness problem, raised by Bonzom and developed by Hellmann and Kaminski, suggests the amplitude may be dominated by flat configurations in the regime that matters. Until a controlled continuum limit exists, the claim that the theory contains general relativity at low energy is a hope with evidence, not a result.
Read the objections, not just the claims. Four items are settled mathematics, and one of them — the uniqueness theorem — has no counterpart in ordinary quantum field theory. Three of the biggest, the semiclassical limit, the Hamiltonian constraint and the equivalence of the two formulations, are open in the strong sense that nobody knows the answer rather than the weak sense that nobody has written it up. A reader who finishes this list should be able to say precisely which is which.
The most serious problem is the semiclassical limit. Nobody has shown, in a controlled way, that this theory contains general relativity at low energies. There is real progress to weigh against that: the large-spin asymptotics of the spin foam vertex reproduce the Regge action, which is the correct discrete form of the Einstein-Hilbert action, and the graviton two-point function computed from a boundary amplitude has the expected falloff. But asymptotics of one vertex is not a continuum limit, refining the triangulation is uncontrolled, and there are results suggesting the amplitude may be dominated by flat configurations in the regime that matters. Any claim that the theory reduces to general relativity is currently a well-supported hope.
Next to it sits the Hamiltonian constraint. Thiemann’s operator exists and is finite, which was a genuine achievement after thirty years of the Wheeler-DeWitt equation resisting definition. It is also not unique: the attachment of the new loop, the spin it carries, the operator ordering and the regularisation of the inverse triad are all choices that no principle fixes, and different choices give different theories. The criticism that the resulting dynamics may fail to propagate has been pressed since the early 2000s and has not been answered. The dynamics of loop quantum gravity is not pinned down.
Then the shorter list. Whether the canonical and covariant formulations are the same theory is unproven; their boundary states match, which is suggestive and no more. Matter can be coupled but is entirely unconstrained, so the theory explains nothing about the particle content. The Barbero-Immirzi parameter is fixed by the black hole entropy calculation rather than predicted, which makes that celebrated result a calibration in part. And Freidel and Speziale showed that generic states are twisted geometries, in which glued faces agree on area but not on shape, so the picture of space as a tidy packing of polyhedra is the special case rather than the rule.
Against that, what the programme delivers is not small. Background independence is maintained from the first line to the last, with no metric ever fixed. The kinematical theory is finite and its representation is essentially unique, which is a theorem with no counterpart in ordinary quantum field theory. Area and volume come out discrete because the operators say so, not because anyone put a lattice in. The Big Bang singularity is replaced by a bounce in every model that can actually be solved. Horizon entropy proportional to area falls out of an explicit state count. And it achieves all of this in four dimensions, with no supersymmetry, no extra dimensions and no landscape of vacua.
The comparison with string theory is a comparison of bets, not of evidence. String theory bet on unification and accepted a fixed background, and has extraordinary mathematical results and a landscape problem to show for it. Loop quantum gravity bet on background independence and gave up unification, and has a rigorous kinematics and an unsolved dynamics to show for it. Neither has been tested. Anyone declaring a winner is expressing a preference about which bet is more likely to pay, and that is worth saying out loud rather than dressing up as a result.