Edinburgh PhD Student's Tweak to the Born Rule Pulls the Rug from Under String Theory

Source: Theories of Everything | Published: 2026-06-22T14:04:19Z

After losing his funding, Bateman found that a minimal modification to the Born rule—letting quantum states inhabit Krein space—eliminates a core assumption string theory depends on. Quantum gravity may not need extra dimensions after all.

The Simplest Answer, Forgotten for Fifty Years

Around 1977, a physicist named Kellogg Stelle published a paper proposing the addition of quadratic curvature terms to Einstein's gravitational action — what is now called "higher-derivative gravity" or "quadratic gravity." It has a very attractive mathematical property: it is renormalizable.

What does renormalizable mean? When you do quantum field theory calculations, various infinities appear — but they can be absorbed into redefinitions of the coupling constants, leaving the theory fully self-consistent at short distances. Standard general relativity cannot do this, which is one of the core reasons quantum gravity is so hard. Quadratic gravity solves this problem naturally. By the 1980s it had also been shown to be "asymptotically free" — just like QCD, the coupling weakens at higher energies, and at short distances the theory becomes nearly free.

The theory was always there. The problem was that nobody knew how to use it safely.

Two Ticking Bombs

Quadratic gravity has been repeatedly picked up and set aside, for two reasons.

The first is the Ostrogradsky instability. In 1850, Ostrogradsky proved a theorem in Saint Petersburg: if the equations of motion contain derivatives higher than second order, the system's energy is unbounded below — you can extract arbitrarily large amounts of negative energy from it. A physical system with no energy floor becomes a bottomless energy sink, triggering runaway decay. Our universe is obviously not like that.

The second problem appears in the quantum version: the space of quantum states in higher-derivative theories contains "negative-norm states," which physicists call ghosts. The standard interpretation is that negative norm implies negative probability — a catastrophe in any quantum theory.

These two ticking bombs drove people away from quadratic gravity, time and again, toward more elaborate alternatives: string theory, extra dimensions, M-theory. The structures grew increasingly sophisticated, while the real questions — what happened at the Big Bang? what happens inside a black hole? — remained unanswered.

A PhD Student and an "Impossible Problem"

The turning point came in Edinburgh, and its protagonist is Sam Bateman.

Bateman is from Ireland, with a background in mathematical physics from Trinity College Dublin. He came to Edinburgh for a one-year master's degree, unsure whether he wanted to continue. At the end of the year, he decided to pursue a PhD and began working with Turok. Turok gave him a problem that seemed nearly impossible at the time: quantize a higher-derivative theory.

"Anyone would tell you this can't be done," Turok said. "You're basically ruining this student's career."

Four years passed. Bateman published nothing. His funding ran out. With no results and no job prospects in sight, he worked his way through the algebraic methods of quantum field theory from scratch — a rigorous 1980s mathematical physics textbook by Bogoliubov that Turok estimates 99.9999% of practicing theoretical physicists have never cracked open.

Last September, after the funding was gone, Bateman walked in and said: "I think I know how to do it."

All it required was a small modification to the Born rule.

Ghosts Aren't a Problem Because They're Unobservable

Turok and Bateman's core argument begins with negative norm.

The standard claim is: negative norm = negative probability = unphysical. But they point out that this inference is itself wrong. A quantum state is just a label for a system's configuration. Its norm is not an observable — you cannot measure the norm of a quantum state in a laboratory.

What is actually observable is transition probability: the probability that a system evolves from an initial state to a final state. And the calculation of that probability does not require that norms be positive.

Mathematically, this corresponds to a structure called a Krein space — a generalization of Hilbert space that allows positive-norm, negative-norm, and zero-norm states to coexist, just as Minkowski spacetime accommodates timelike, spacelike, and lightlike directions. Physicists have actually been using something like this all along — BRST formalism, Faddeev-Popov ghosts — but have always projected onto a "physical subspace" at the end.

Turok and Bateman's approach is more direct: don't project. Construct probabilities directly in the larger space.

A Small Adjustment to the Born Rule

In standard quantum mechanics, probability = |⟨f|S|i⟩|², where S is the scattering matrix and i and f are normalized initial and final states.

In a Krein space, states cannot be normalized — because some have negative norm. Bateman's insight was to rewrite the formula in terms of a trace over projection operators.

Specifically, probability = Tr(A†A), where A is the combination of "initial-state projector × S matrix × final-state projector." The key is that the trace is taken over all states, including ghosts. No projection. Nothing discarded.

They proved that as long as the theory possesses a discrete symmetry called "ghost parity" — an operator that acts as +1 on positive-norm states and −1 on negative-norm states — all computed probabilities are necessarily positive and sum to one.

This is not a patch. It is a minimal extension of the quantum-mechanical framework that preserves causality, unitarity, and everything else you'd want.

One Assumption Behind String Theory Is Removed

Why does string theory require ten dimensions? There is a chain of assumptions behind this. One of them — which Turok considers the strongest — is that any sensible quantum theory must live in a Hilbert space, meaning all quantum state norms must be positive.

Under that assumption, higher-derivative theories are excluded. With those theories excluded, the viable paths to a renormalizable theory of gravity narrow sharply. String theory becomes part of the argument for the "only way out."

That assumption has now been removed. Turok is direct: "We have an example of a theory that doesn't need that assumption, and it is UV-complete. With just a small extension of the canonical principles, you no longer need string theory, you no longer need extra dimensions to describe gravity."

This doesn't falsify string theory. But one of its foundations has shifted.

Where Did the Ostrogradsky Instability Go?

The first bomb also has an answer, and it's surprisingly simple.

Turok points out that gravity itself has a peculiar property: gravitational potential energy is negative. The energy of a gravitationally bound system is naturally negative — physicists have long accepted this as a feature of gravity, not a bug.

When they analyzed the expanding-universe solutions of quadratic gravity, they found that the so-called Ostrogradsky instability is actually just ordinary cosmological expansion — not a genuine catastrophe, but normal behavior for a gravitational system. Analyzed with the same tools used in standard general relativity, the expanding solutions are perfectly stable.

"The Ostrogradsky instability disappears because it gets reinterpreted as a gravitational theory."

The Cosmic Microwave Background May Contain a Quantum Gravity Signal

There is an unexpected connection between this framework and Turok's earlier work.

The temperature fluctuations in the cosmic microwave background — the seeds of galaxies and everything else that exists today — have a specific power spectrum: more power on large scales, a so-called "red-tilted" spectrum. If you simply ask what kind of quantum field would produce this spectrum, the answer is: a higher-derivative field, and nothing else.

Turok's CPT-symmetric cosmology model does not rely on inflation to generate these fluctuations. Instead, it asserts directly: what we see in the sky is a signal from quantum gravity. Observations on the largest scales and theory on the smallest scales converge at the same place — higher derivatives.

A Way Out of the Hierarchy Problem

There is a long-standing puzzle in physics: the Planck mass is 10¹⁹ GeV, the Higgs boson mass is 125 GeV — a gap of nearly 17 orders of magnitude. In the Standard Model, this gap can only be maintained through fine-tuning: manually adjusting two numbers to differ by this much, with no underlying reason.

Quadratic gravity is asymptotically free, meaning its coupling constants run extremely slowly with energy scale — logarithmically. Starting from the Planck scale, a coupling of order 1/30 at that scale grows slowly as energy decreases, becoming strongly coupled around 1 GeV — just like QCD. This enormous mass ratio is not tuned by hand; it is the natural result of logarithmic running, just as QCD's hadron masses are vastly smaller than the Planck mass — and nobody considers that a problem.

If the Higgs boson is some kind of composite state of this scalar field, its mass would naturally be exponentially smaller than the Planck mass. The hierarchy problem may not require supersymmetry or extra dimensions — just asymptotic freedom.

The Axioms of Quantum Mechanics Should Be Questioned

Sam Bateman ultimately earned a postdoctoral position at the Simons Center with zero published papers. Raju Venugopalan — a leading expert in nonlinear quantum field theory, senior theoretical physicist at Brookhaven National Laboratory, director of the EIC Theory Institute, and Royal Society Wolfson Visiting Professor at the University of Edinburgh beginning October 2025 — happened to catch him giving an informal talk in Edinburgh, immediately sensed "there's something here," and extended his visit through the end of the year.

Turok's conclusion is not that this is simply the achievement of one clever student. He points toward a deeper structural problem: an assumption tacitly accepted by the entire field for decades — that quantum theories must live in Hilbert space — was never written down explicitly, never seriously questioned, yet has been quietly determining what theories are allowed and what theories are excluded.

"In theoretical physics, if you make one wrong step, you're completely lost. That's the danger. I don't think people worry about this enough. We should very, very carefully check every assumption: is it actually necessary? Or do we do it simply because we're used to it?"

If an assumption this fundamental can be relaxed, how many similar assumptions are shaping the mainstream directions of today — and going unchallenged?