← quantum · contents

§ Phase xv · Open Questions · Chapter 03

Quantum gravity

Two cathedrals of 20th-century physics stand a few miles apart and refuse to speak. General relativity describes gravity as the smooth curvature of spacetime, and it predicts the orbits of planets and the bending of starlight with breathtaking accuracy. Quantum mechanics describes everything else, electrons and photons and quarks, in the language of probability and operators. A century after both theories were finished, we still do not know how to write them down in the same sentence. This last chapter is about why the marriage is so hard, what we have learned by trying, and where the cracks in the picture are loudest.

On November 25, 1915, Albert Einstein walked into the lecture hall of the Prussian Academy of Sciences in Berlin and presented the final form of the field equations of general relativity. He had spent eight years on the problem, the last three in a state he later described as bordering on madness. The equations he wrote on the board that afternoon said something almost too simple to be revolutionary. Mass and energy bend spacetime. Bent spacetime tells mass and energy how to move. There was no force of gravity at all. There was only geometry. A falling apple, a planet circling the Sun, a beam of light grazing the limb of an eclipsed star, all of them were following the straightest possible lines through a curved four-dimensional landscape. The mathematics was the language of Riemann and Ricci, the picture was the language of cartography. It was the most beautiful theory in physics, and a hundred and ten years later, no observation has ever contradicted it.

Ten years after Einstein finished general relativity, a separate revolution finished in Göttingen and Copenhagen. In the summer of 1925, Werner Heisenberg, recovering from hay fever on the island of Helgoland, invented matrix mechanics. The following winter, Erwin Schrödinger wrote down the wave equation. Within five years, Paul Dirac, Max Born, Pascual Jordan, and Wolfgang Pauli had pulled the new mechanics into a single coherent framework, and a generation of physicists had stopped asking what an electron is and started asking what its state vector is. Quantum mechanics described every other corner of physics. Atoms, light, chemistry, eventually nuclei and the particles inside them, all of it bent to the new rules. The two theories grew up side by side in the 1920s and 1930s, and their builders all knew each other, and yet they never quite touched.

For a long time the avoidance was practical. Gravity is feeble. The electrical attraction between a proton and an electron in a hydrogen atom is forty orders of magnitude stronger than their gravitational attraction. Anywhere you can do a quantum experiment, gravity is utterly negligible, so you can use ordinary flat spacetime and Newtonian gravity as a backdrop and forget about Einstein. Anywhere gravity matters (planets, stars, galaxies), the number of quanta involved is so astronomical that the quantum graininess averages out into smoothness, and Einstein’s classical geometry is all you need. The two regimes were so cleanly separated that physicists could spend entire careers in one and never visit the other. John Wheeler, who later became the great evangelist for the problem, used to say that for fifty years the two theories occupied the same physical universe like two friendly tenants in adjacent apartments, never quite needing to share a kitchen.

The trouble is that the universe does not respect the partition. There are situations where both gravity and the quantum matter at once. The interior of a black hole. The first instant of the Big Bang. The vacuum on scales so small that the geometry itself must fluctuate. In those places, you cannot use one theory as the backdrop for the other. You need a single law that knows about both. Finding that law is the open problem we are about to walk through.

The first thing to understand is that, naively, the merger should not be that hard. We already know how to take a classical field theory and quantize it. Electromagnetism began as Maxwell’s classical equations for the electric and magnetic fields. Quantize them, and you get photons, the particle excitations of the field. The same recipe turns the strong and weak nuclear forces into gluons and W and Z bosons. General relativity is also a field theory. Its field is the metric of spacetime, the object that tells you the distance between two nearby points. Run the same recipe on the metric and you should get a particle, the quantum excitation of the gravitational field. That particle has a name. It is called the graviton, it is massless, and it has spin two. Two photons of gravity beat against each other inside a wave, and the wave is what LIGO detects coming from merging black holes.

So why is this not the end of the story? Why is there a graduate course called Quantum Gravity rather than a tidy chapter at the back of the quantum field theory textbook? The answer is buried in a technical word that we will translate slowly. The word is renormalization. Every quantum field theory generates infinities when you compute. An electron passing near another electron exchanges a photon, then exchanges two photons, then three, and as you sum the infinite ladder of possibilities, the answer wants to diverge. For electromagnetism, the strong force, and the weak force, those divergences can be absorbed into a small number of redefinitions of the masses and couplings, and finite predictions come out the other side. The theory is then called renormalizable. Computations match experiment to twelve decimal places. The world works.

When you try the same procedure on gravity, the procedure fails. Each new order of computation introduces new types of infinity that cannot be absorbed by any finite redefinition. The theory needs an infinite number of independent parameters to be made finite, which is the same as saying it makes no predictions at all. Quantum general relativity, treated as just another field theory, is not renormalizable. This is not a small technical wart. It is a flashing red sign on the door of the building that says the building is not the building you thought it was. Either you need a fundamentally new theory at short distances, or you need a radically different way of organizing the math, or both.

Quantum gravity (QG) is a field of theoretical physics that seeks unification of the theory of gravity with the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vicinity of black holes or similar compact astrophysical objects, as well as in the early stages of the universe moments after the Big Bang.

So how do you build a sensible theory? Over the past fifty years, several serious programs have grown up around the question, each pulling on a different thread. The most developed is string theory. Its starting move is to replace point particles with one-dimensional extended objects, tiny loops or open strings, vibrating in a higher-dimensional spacetime. Different vibration modes of the same string correspond to different particles. The lowest vibration mode of the closed string turns out to be exactly the massless spin-2 particle we wanted, the graviton. The catch is that string theory only seems mathematically consistent in nine or ten spatial dimensions, which means the extra dimensions have to be curled up too small to see. The framework has produced beautiful mathematics, deep relations to gauge theory, and the strongest piece of evidence we have that a quantum theory of gravity can be written down at all. It has not produced a unique prediction about the world we live in, and many physicists now treat it more as a laboratory for ideas than as a finished theory.

A different program is loop quantum gravity. Rather than replacing particles with strings, it tries to quantize the geometry of spacetime itself, directly, in four dimensions, without adding new ingredients. The math reorganizes Einstein’s equations so that the variables are not the metric components but holonomies, integrals of the gravitational connection around little loops. When you quantize these loops, area and volume turn out to take discrete values, with a smallest possible area on the order of the Planck length squared. Spacetime, in this picture, is not a smooth manifold at all. It is a kind of woven fabric with a finest weave. Loop quantum gravity has not yet produced the everyday limit of ordinary general relativity in a clean way, and it does not unify gravity with the other forces. But it offers a vivid alternative philosophy. Maybe geometry itself is what should be quantized, not the matter on top of it.

There are other programs. Asymptotic safety is the conjecture that the apparent failure of renormalization is an illusion of low-order computation, and that gravity is well-defined at all energies because its coupling constants run to a finite fixed point. Causal dynamical triangulations builds spacetime out of tiny tetrahedra and lets the geometry emerge statistically, the way a fluid emerges from molecules. Each program has lovely features and stubborn problems, and the community is honest that none of them has yet earned the right to be called the answer.

The other reason the problem refuses to go away is a discovery that nobody saw coming. In 1974, a young Stephen Hawking calculated what happens when you do quantum field theory near a black hole’s event horizon, treating the gravity as a fixed classical background. He expected nothing dramatic. What he found, after several months of computation, was that black holes are not perfectly black. They radiate. The vacuum around the horizon is filled with virtual pairs of particles popping into and out of existence (we met those pairs in the chapter on vacuum fluctuations) and now and then one member of a pair falls inside the horizon while its partner escapes to infinity. From far away, the black hole looks like a hot thermal body, glowing with a spectrum of every particle in the standard model, slowly losing mass to its surroundings. The temperature is tiny for a stellar-mass black hole, around a hundred-millionth of a kelvin, but it is not zero. Over astronomical time scales, the black hole evaporates.

Hawking’s result was a quiet bomb. It said that gravity, the quantum, and thermodynamics are linked at a level much deeper than anyone had suspected. A black hole has a temperature. A temperature implies an entropy. An entropy implies that the black hole is, in some sense, a thermodynamic object made of many microscopic states, even though classical general relativity describes it as having only a handful of parameters (mass, charge, spin) and nothing else. There is a counting problem here. Whatever a black hole is, it has, by Hawking’s formula, a number of internal states equal to the exponential of one-quarter of its horizon area measured in Planck units. The states cannot be labeled within classical general relativity. The classical description is missing them.

↘ Why does naive quantum gravity blow up?

The technical heart of the non-renormalization problem is dimensional analysis on the gravitational coupling. Newton’s constant G has units of length squared over mass times time squared, or equivalently, in natural units where ħ and c are set to one, units of (energy)^(-2). The gravitational coupling at energy E is therefore the dimensionless combination G · E^2. As you push to higher energy, the effective coupling grows. By the time you reach the Planck energy E_P ≈ 10^19 GeV, the coupling becomes order one. Beyond that, perturbation theory (the technique that made QED and the standard model work) simply fails. Every new diagram is comparable in size to the previous one, and there is no good answer to any question.

Compare this to electromagnetism. The fine-structure constant α ≈ 1/137 is a dimensionless number. Higher-order diagrams are suppressed by powers of α, and the series converges. Compare to QCD. The QCD coupling actually runs the other way: it gets weaker at high energy (asymptotic freedom) and stronger at low energy. Either way, the dimensionless coupling stays bounded somewhere and lets you compute. Gravity has no such luck. Its coupling has the wrong dimensions, and that single fact dooms the naive recipe.

The number of new independent infinities you have to absorb grows with each order of perturbation theory. A renormalizable theory has a finite number, and once you fix them with experimental measurements, you can predict everything else. A non-renormalizable theory has an infinite number, and you have to measure all of them to predict anything. That is not a theory, it is a parametrization. The breakdown happens precisely at the Planck scale, where new physics, whatever it is, must take over. String theory’s claim is that the new physics is the finite size of the string. Loop quantum gravity’s claim is that the spectrum of areas is discrete. Asymptotic safety’s claim is that the perturbative breakdown is a perturbative artifact, and the true coupling runs to a fixed point. Each claim is a different guess at what the Planck-scale physics actually is, and the experimental data that could distinguish them lies far beyond any conceivable particle accelerator.

Hawking’s discovery led to a still-unresolved puzzle. If a black hole forms from a star (a definite configuration of quantum information) and then evaporates entirely into thermal Hawking radiation (which, naively, has lost track of the information), what happened to that information? Quantum mechanics says information is conserved. If you know the state of an isolated system at one time, you know it for all time. Pure states evolve into pure states. Thermal radiation looks like a mixed state, an average over many microstates, with the details washed out. So the calculation seems to say that quantum mechanics has been violated by a process involving gravity. This is the black hole information paradox, and it has been an industry for half a century. The current consensus, painstakingly assembled by people like Don Page, Andrew Strominger, and most recently a generation of younger workers building on holographic ideas, is that the information does come out, encoded in extremely subtle correlations between the late Hawking photons and the early ones. The proof passes through Maldacena’s holographic duality, which lets you translate the black hole interior into the boundary CFT, where unitarity is manifest. Whether the calculation extends to our universe, which is not anti-de Sitter, is one of the most active questions in theoretical physics.

If you step back, the picture you get is humbling. We have a finished theory of gravity and a finished theory of the quantum. We have, separately, tested each of them to absurd precision: general relativity to roughly a part in 10^15 in the orbital decay of binary pulsars, quantum mechanics to about a part in 10^12 in the electron’s magnetic moment. We have indirect, beautiful, partial evidence that they fit together. We have Hawking’s formula, the holographic principle, fifty years of failed and half-successful programs. We do not have the synthesis. The places where the missing theory matters (the singularity inside a black hole, the first instant of the Big Bang, the structure of spacetime at the Planck scale) are precisely the places we cannot reach with experiments. We are like nineteenth-century thermodynamicists trying to learn the atomic theory of matter without ever being able to see an atom.

And yet, here we are, closing a book that began with a glowing iron poker in 1900. Planck’s tiny constant, the number h, ran through every chapter. The wave-particle duality, the orbital, the periodic table, the antimatter, the standard model, the Higgs, all of it grew out of the consequence of that one number being small but not zero. The last open problem is what happens when h and Newton’s G meet on equal footing. Wheeler used to write his thoughts on yellow legal pads, and one of his most famous notes, scribbled in the late 1960s, read simply, “It from bit.” Maybe the universe is information, and gravity is the way information talks to itself. Maybe spacetime emerges from entanglement, the way a fluid emerges from molecules. Maybe none of these guesses will turn out to be right, and the answer will come from a direction we are not even looking. The story is not over. You have read the first hundred and twenty-five years of it. The next chapter, somebody is going to write.

⁂

A century ago, two cathedrals were finished and have stood ever since. The walk between them is not yet a road, only a footpath worn by curious people who keep trying. Maybe the next person to widen it is reading this sentence. Either way, the universe is still patient, the constants are still small, and the questions, thankfully, are still open.