Neil Turok presents a new approach to quantum gravity that challenges the dominant paradigm requiring strings and extra dimensions, instead reviving a simpler framework from the 1970s called quadratic gravity — and his student Sam Bateman has found a way to resolve its long-standing fatal flaw.
The standard approaches to quantum gravity (string theory, loop quantum gravity) have grown increasingly complex without solving core problems like what happened at the Big Bang, what occurs inside black holes, or whether information is lost.
Turok’s key claim: the assumptions that forced physicists into strings and extra dimensions were never actually necessary — and one in particular (that quantum states must live in a Hilbert space with all positive norms) can be relaxed, opening a path to a complete theory of quantum gravity in four dimensions without any extra structure.
Quadratic Gravity: The Simplest Renormalizable Theory of Gravity
In 1977, Kellogg Stelle showed that if you add curvature-squared terms to Einstein’s action, the resulting theory is renormalizable — meaning quantum calculations that normally produce infinite answers can be systematically controlled.
Einstein’s action contains the Ricci scalar (two derivatives of the metric). Adding terms quadratic in curvature (four derivatives) makes gravity structurally similar to gauge theories like QCD, where the action is the integral of field strength squared.
In the 1980s, Avramidi and Barvinsky further showed this theory is asymptotically free: at short distances (high energies), the coupling constant goes to zero, and the theory becomes simple and non-interacting — exactly like QCD.
This theory has been known for decades but was largely abandoned due to two problems identified through the Ostrogradsky theorem (1850): higher-derivative theories appear to have energies unbounded below (instabilities) and negative-norm quantum states (ghosts).
The Ostrogradsky Instability Is Not What It Seems
The classical Ostrogradsky theorem says that equations of motion with more than two derivatives produce Hamiltonians unbounded below — suggesting runaway instabilities.
Turok and collaborators argue this “instability” is actually just normal gravitational expansion. When analyzed as a gravitational theory (not a generic mechanical system), the expanding solution is stable.
Gravity already has unusual energy properties: gravitational potential energy is negative, and the universe’s accelerated expansion (driven by the cosmological constant) already looks formally like an instability — yet it is observed to be stable.
So the first “horn” of the Ostrogradsky problem dissolves upon proper interpretation within gravity.
Ghosts and the Krein Space Resolution
The second problem is negative-norm states (ghosts) in the quantum theory. The standard argument is that negative norms imply negative probabilities, which would be unphysical.
Turok and Sam Bateman argue this reasoning is wrong: the norm of a quantum state is not an observable. What matters are transition probabilities, not the norms of individual states.
The mathematical framework needed is a Krein space — a generalization of Hilbert space that allows positive, negative, and null norm states, much like Minkowski spacetime allows timelike, spacelike, and null intervals.
Krein spaces were studied by functional analysts but were essentially unknown to physicists. Bateman found this literature and recognized its relevance.
Physicists already work in such spaces routinely: Faddeev-Popov ghosts in gauge theory and BRST quantization use a larger unphysical space and then project onto a physical subspace. Turok and Bateman’s approach is a more economical generalization of this.
A Modified Born Rule
In standard quantum mechanics, the Born rule gives the probability of a transition as |⟨f|i⟩|², which requires normalized states. In a Krein space, states with negative norm cannot be normalized in the usual way.
The key innovation: replace the Born rule with a projection-based formula. Instead of squaring an amplitude, construct the probability directly as:
Project onto the initial state, evolve with the S-matrix, project onto the final state, evolve with S-dagger, and trace over all states (including ghosts).
This is mathematically equivalent to the standard Born rule in ordinary quantum mechanics, but it remains well-defined in a Krein space.
Crucially, if the S-matrix (or Hamiltonian) possesses a discrete symmetry called ghost parity — which assigns +1 to positive-norm states and −1 to negative-norm states — then all computed probabilities are guaranteed to be positive and sum to one.
This means the theory is fully consistent: causal, unitary in the generalized sense, and physically interpretable, even though it does not live in a Hilbert space.
What Has Been Solved and What Remains
The full quadratic gravity action has two independent curvature-squared terms: Ricci scalar squared and Weyl curvature squared, with two coupling constants.
Turok and Bateman have fully solved a limit of the theory where the Weyl-squared coupling is set to zero. In this limit, the tensor (graviton) and vector modes decouple, leaving only a scalar mode — the local scale of the metric.
This scalar theory is renormalizable, asymptotically free, ghost-parity symmetric, and yields positive probabilities. It is a consistent, UV-complete quantum field theory.
It is not the full theory of gravity (no gravitons, no gravitational waves), but it is a toy model relevant to cosmology and potentially to understanding the hierarchy problem.
The full theory, including the Weyl-squared term (which introduces spin-2 ghosts), has not yet been solved. The critical open question is whether the full theory also possesses the ghost-parity symmetry needed for consistency.
If it does, this would be a complete, four-dimensional, string-free theory of quantum gravity.
Implications for the Hierarchy Problem and Higgs Physics
The hierarchy problem — why the Higgs mass (~100 GeV) is so far below the Planck mass (~10¹⁹ GeV) — has no explanation in the standard model; it requires fine-tuning.
In an asymptotically free theory, couplings run only logarithmically with energy. Starting from a coupling of order 0.1 at the Planck scale, the scale at which the coupling becomes strong is exponentially smaller — exactly as happens in QCD, where the confinement scale (~1 GeV) is exponentially below the Planck scale.
If the Higgs boson is a composite of the scalar mode in this theory (analogous to how the Higgs-like mode in a superconductor is made of Cooper pairs), then the hierarchy between the Planck scale and the Higgs mass is natural, not fine-tuned.
Turok notes that Peter Higgs himself would likely not have considered the Higgs fundamental — his original inspiration was the composite Cooper pair condensate in superconductivity.
Connection to the CPT-Symmetric Universe and Cosmology
Turok’s earlier work with Latham Boyle on the CPT-symmetric universe was motivated by extreme minimalism: explain all observations with the fewest assumptions.
That framework successfully explains dark matter, the smoothness and flatness of the universe, and the horizon problem without inflation.
The one thing it could not explain was the primordial fluctuations seen in the CMB. These fluctuations have a spectrum that does not match a normal two-derivative scalar field — but matches exactly what a four-derivative field would produce.
This is what led Turok to study four-derivative theories, and now the connection closes: the same four-derivative structure needed to explain the CMB fluctuations is precisely what is needed for quantum gravity.
Turok’s striking interpretation: the CMB fluctuations are a direct signal of quantum gravity — we are seeing the birth of the universe imprinted with the quantum fluctuations of gravity itself.
The 36 Fields Puzzle
In earlier work, Turok and collaborators found that vacuum energy divergences from standard model fields are canceled if there are exactly 36 additional four-derivative scalar fields — and this only works if there are exactly three generations of elementary particles.
This provides a potential explanation for why there are three generations, a fact otherwise unexplained in the standard model.
The current quadratic gravity work involves only one such scalar field. Reconciling “36” with “1” is an open problem.
Turok speculates that the full tensor sector of gravity (the Weyl-squared term) might naturally produce 36 fields — noting that BF theory (related to loop quantum gravity) naturally involves 36 objects.
Contrast with Other Approaches (Bender, Mannheim, Conformal Gravity)
Bender and Mannheim have also studied higher-derivative gravity (particularly Weyl/conformal gravity) and attempted to resolve the negative-norm problem by redefining the inner product with a minus sign for ghost states.
Turok argues this procedure is not covariant — it breaks Lorentz and translation symmetry — and will not yield a consistent quantum field theory, though it may work in quantum mechanics.
Turok claims his and Bateman’s resolution is the only covariant one, fully respecting spacetime symmetries.
Mannheim has claimed Weyl-squared gravity can explain galaxy rotation curves without dark matter — Turok is skeptical but collegial.
Gravitational Entropy and Why the Universe Is Simple
Turok addresses why the universe is so simple on large scales (smooth, homogeneous, spatially flat) without invoking inflation.
Building on Hawking’s concept of gravitational entropy, Turok and collaborators generalized Hawking’s black hole entropy formula to cosmologies.
They find that the universe with the greatest number of microstates (highest entropy) is smooth, homogeneous, isotropic, spatially flat, and has a small positive cosmological constant — exactly matching observations.
This is analogous to why gas in a room is uniform: it is the typical state, not a special one. No dynamics (like inflation) is needed to smooth the universe out — you just need a way to count states and pick the most probable one.
This contrasts with the inflationary view, which assumes special initial conditions and then uses dynamics to smooth them. Turok’s view is more aligned with statistical mechanics: the universe is typical because typical states dominate.
On the Health of Theoretical Physics
Turok is concerned that theoretical physics has become too orthodox, dominated by string theory despite its lack of testable predictions.
He sees this orthodoxy reflected in referee reports, grant decisions, and hiring: young researchers are channeled toward popular paradigms (string theory, supersymmetry) that have not worked, rather than toward foundational questioning.
He worries that mathematical spin-offs of string theory have diluted the field, with math departments hiring string theorists who are not concerned with observational predictions.
He advocates for young people to question axioms — as Bateman did — rather than follow fashion. He wishes he had spent more of his own career on foundations.
He notes that the current global instability (political, technological) may actually stimulate unorthodox thinking, as people question established orders.
Sam Bateman’s Story
Sam Bateman, Turok’s PhD student, is central to this work. He came to Edinburgh uncertain about pursuing a PhD and was given what seemed like an impossible problem: quantize four-derivative field theories.
After four years of modest progress, with his funding run out and no papers, Bateman had a breakthrough in September — realizing that a slight modification of the Born rule resolves the ghost problem.
Despite having zero published papers, he was offered a postdoc at the Simons Center at Stony Brook after Raju Venugopalan (a leading expert in non-linear QFT) recognized the significance of his work during a visit to Edinburgh.
Bateman had taught himself rigorous algebraic quantum field theory from Bogoliubov’s textbook — a level of mathematical sophistication most practicing theorists never acquire.
Simplicity as a Guiding Principle
Turok’s core philosophy: simplicity leads to understanding and predictivity.
The universe is astonishingly simple on both the smallest scales (particles, atoms) and the largest scales (black holes, cosmology). All the complexity is at intermediate (human) scales.
He is skeptical of frameworks that start simple but lead to enormous complexity (the multiverse, many-worlds), arguing that these often rest on unexamined assumptions — like the Hilbert space axiom — and suffer from unsolved measure problems that make them unpredictive.
He believes the physics we already know may be 99.9% of the story, and that resolving its internal contradictions with minimal modifications is more fruitful than pursuing increasingly elaborate new structures.
