Ⅲ: The Quantum Gravity Problem

Physics has four fundamental forces. Three of them have been successfully described within the framework of quantum mechanics. The fourth, gravitational force, has not and the failure to incorporate it into a quantum framework is not a gap waiting to be filled by better technology. It is a structural problem that has resisted every serious attempt for nearly a century, and whose resolution will require either a radical modification of quantum mechanics, a radical modification of general relativity, or both.

How Three Forces Were Quantised

In the early twentieth century, physics faced a specific challenge – how to make electromagnetic theory consistent with quantum mechanics. Classical electromagnetism, described by Maxwell’s equations, treated the electromagnetic field as a continuous entity filling space. Quantum mechanics described particles as probabilistic objects with wave properties. The two frameworks were in tension, and reconciling them required a new kind of theory.

The resolution was quantum field theory. Rather than describing particles as objects moving through a field, quantum field theory describes both particles and forces as excitations of underlying quantum fields. An electron is an excitation of the electron field. A photon is an excitation of the electromagnetic field. When two electrons repel each other, the repulsion is mediated by the exchange of virtual photons, temporary excitations of the electromagnetic field that carry the interaction from one electron to the other.

The quantum theory of electromagnetism, quantum electrodynamics (QED), was developed in the 1940s by Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga, who shared the Nobel Prize in 1965. QED became the model and template for all subsequent quantum field theories.

The strong nuclear force, which binds quarks into protons and neutrons, was quantised by the same method. Quantum chromodynamics (QCD), developed in the early 1970s, describes quarks interacting through the exchange of particles called gluons in precise analogy to electrons interacting through photons. The weak nuclear force, responsible for radioactive decay, was incorporated into a unified electroweak theory by Sheldon Glashow, Steven Weinberg, and Abdus Salam. Together, QED, QCD, and the electroweak theory form the Standard Model, whose predictions have been confirmed at the LHC and in every particle physics experiment conducted since.

The key to all three successes was a mathematical property called renormalisability. When physicists attempt to calculate the outcome of a particle interaction using quantum field theory, the calculation is performed as an infinite series of increasingly complex terms, each corresponding to a Feynman diagram with more virtual particles. At each order of this perturbation series, infinities appear. Renormalisability means these infinities can be systematically absorbed into a small number of measurable parameters, such as the electron’s mass and charge, leaving finite, testable predictions for everything else.

The procedure seems like sleight of hand, and physicists debated its legitimacy for decades. But renormalisation is now understood as reflecting the structure of effective field theories. The infinities arise because the calculation naively extends to arbitrarily high energies where the theory is not expected to apply, and the finite parameters absorb the ignorance about what happens at those energies. As long as the number of parameters that need to be fixed is finite, the theory retains predictive power.

Why Gravity Is Different

At first glance, gravity should be quantisable by the same method. General relativity describes the gravitational field as the curvature of spacetime. Quantising it should mean treating gravitational waves as excitations of the gravitational field, with a force-carrying particle – the graviton, mediating gravitational interactions. The graviton is predicted to be a massless particle with spin 2, and early attempts at quantisation confirmed that this is at least a consistent starting point.

On the surface, the incompatibility between general relativity and quantum theory might seem rather trivial. Why not just follow the model of QED and quantise the gravitational field, similar to the way in which the electromagnetic field was quantised? This is more or less the path that was taken, but it encounters extraordinary difficulties.

The problem is that quantum gravity is not renormalisable.

When physicists apply the perturbative methods of quantum field theory to gravity, infinities appear in the calculations just as they do for the other forces. But these infinities cannot be absorbed into a finite number of parameters. Each new order of the perturbation series introduces new, independent infinities that require new, independent parameters to absorb. There is no finite set of measurements that could determine all these parameters. The theory requires infinitely many inputs to make any prediction, which means it makes no predictions at all. As usually stated, the difficulty with non-renormalisable theories is that they are not predictive, since obtaining well-defined predictions potentially requires an infinite number of divergent renormalisations.

This failure was suspected from the 1960s onward, when Richard Feynman and Bryce DeWitt attempted to quantise gravity and found that the calculations behaved badly. But the definitive result came in 1985, when Marc Goroff and Augusto Sagnotti carried out the two-loop gravity calculation in four dimensions and showed that the theory is no longer finite at this order. They found that the S-matrix of pure Einstein gravity diverges at the two-loop order, confirming the absence of any miraculous cancellation that might have saved the theory. This established the perturbative non-renormalisability of quantum gravity as a proven mathematical result, not merely a suspicion.

Their result was confirmed by an independent calculation in 1992 and has not been challenged since.

Why Newton’s Constant Makes Everything Worse

The mathematical reason for gravity’s non-renormalisability is not accidental. It follows directly from a property of Newton’s gravitational constant G.

In quantum field theory, the coupling constant of a force determines how strongly particles interact. The coupling constant of electromagnetism is the fine-structure constant, approximately 1/137. It is dimensionless, meaning it is just a pure number with no units. This dimensionlessness is what makes QED renormalisable: when infinities appear in the calculations, they can be absorbed into a redefinition of the dimensionless coupling and the finite masses and charges of the particles. The number of independent infinities that appear is controlled by the number of coupling constants, which is finite.

Newton’s gravitational constant G is not dimensionless. It has dimensions of length squared divided by mass in units where the speed of light and Planck’s constant are set to one. This negative mass dimension means that the effective gravitational coupling grows with energy. At low energies, gravity is the weakest of the four forces by many orders of magnitude, its effects negligible at the scale of elementary particles. But as energy increases, gravity becomes stronger. At energies approaching the Planck scale, the gravitational coupling becomes of order one, gravity becomes as strong as the other forces, and the perturbation series breaks down completely.

More precisely: because G has a negative mass dimension, each additional loop in the perturbation series introduces a new type of infinity that cannot be absorbed by redefining the existing parameters. New parameters must be introduced at each order. An infinite tower of new coupling constants appears, each requiring a new measurement to fix, and the theory loses all predictive power. The quantum description of gravity requires virtual particles known as gravitons. Since all particles are affected by gravity, they would be surrounded by gravitons. This would include gravitons themselves, and their gravitons, and theirs, and so on, until an infinite gravitational field emerges.

This is the core of the mathematical problem. It is not a calculational difficulty that might yield to more powerful computers. It is a structural feature of how Newton’s constant interacts with the rules of quantum field theory.

The Deeper Conceptual Problem

Beyond the technical failure of renormalisability, gravity poses a deeper conceptual problem that has no parallel in the other forces.

Quantum electrodynamics describes electrons and photons interacting on a fixed background spacetime. The electromagnetic field exists in space. The photon moves through space. The metric of spacetime, which defines distances and times, is taken as given, external to the theory. Quantum field theory as developed for the Standard Model is formulated on flat Minkowski spacetime or, at most, on a fixed curved background.

General relativity does not treat spacetime as a background. Spacetime is itself a dynamical entity: it curves in response to mass and energy, and that curvature is what gravity is. There is no background spacetime in general relativity. The theory is background-independent: all of its content is about the relationships between spacetime events, not about positions in a pre-existing arena.

Quantising gravity therefore means quantising spacetime itself. A quantum spacetime cannot be described in terms of a smooth classical background, because the background is the thing being quantised. This creates a profound methodological problem: quantum field theory as standardly formulated requires a background spacetime to define what a particle is, what a vacuum is, what a time evolution is. All of these concepts break down when spacetime is itself quantum.

This is not merely a problem of formalism. It reflects a genuine physical question about what space and time are at the smallest scales. At the scales of atoms and elementary particles, spacetime can be treated as a smooth classical background to excellent approximation. At the Planck scale, this approximation fails. Quantum fluctuations of geometry are of order one. The very concept of a well-defined spacetime interval loses meaning. What replaces it is the question that drives the rest of this series.

What Effective Field Theory Tells Us

There is an important qualification to the non-renormalisability problem that is worth stating precisely.

Non-renormalisability does not mean that no quantum calculations involving gravity can be performed. At energies far below the Planck scale, gravity can be treated as an effective quantum field theory, a framework that makes no claim about the physics at high energies but correctly describes the low-energy behaviour. In this framework, the infinitely many coupling constants required by non-renormalisable gravity are all suppressed by powers of the Planck mass. At energies much lower than the Planck scale, only a finite number of these couplings contribute significantly, and the theory makes well-defined, finite predictions.

Non-renormalisability is not in itself an obstruction to performing predictive quantum calculations, provided the low-energy nature of these predictions is borne in mind. In this way, quantum corrections to Newton’s law of gravity between two slowly moving masses have been calculated using effective field theory methods, and these calculations are genuine predictions of quantum gravity, independent of whatever the correct high-energy theory turns out to be.

But this is a pragmatic workaround, not a solution. The effective field theory approach breaks down precisely where it is most needed: at the Planck scale, where quantum gravitational effects are of order one and where the structure of spacetime itself is expected to change character. Any attempt to compute Planck-scale physics using the effective field theory framework produces uncontrolled results. New physics must appear at or below the Planck scale to restore predictivity, but the effective field theory framework cannot tell us what that new physics is.

Why This Problem Has Resisted a Century of Effort

The quantum gravity problem has been recognised since the 1930s, when physicists first attempted to apply the methods of quantum mechanics to the gravitational field. The problem has not yielded in nearly a century of sustained effort by some of the most capable theoretical physicists in history. The field of quantum gravity is actively developing, but no theory has yet proven successful in describing the general situation where the dynamics of matter, modelled with quantum mechanics, affect the curvature of spacetime.

Long-time practitioners of quantum gravity have noted that any advance that improves one aspect of the problem tends to make other aspects worse. String theory, which succeeds in being perturbatively finite, requires extra dimensions and makes predictions in a space of possibilities so large it may be experimentally unconstrained. Loop quantum gravity, which succeeds in being background-independent, has not yet shown it correctly reproduces general relativity at large scales. Asymptotic safety proposes a non-perturbative fixed point for the renormalisation group that would make gravity predictive, but evidence for this fixed point remains contested.

coSym. | Exploring the Universe: Within & Beyond

Special Series

Topics