# Mathematicians Prove a 2D Version of Quantum Gravity Works

It’s a chic concept that yields concrete solutions just for choose quantum fields. No recognized mathematical process can meaningfully common an infinite variety of objects masking an infinite expanse of area usually. The trail integral is extra of a physics philosophy than a precise mathematical recipe. Mathematicians query its very existence as a legitimate operation and are bothered by the best way physicists depend on it.

“I’m disturbed as a mathematician by one thing which isn’t outlined,” stated Eveliina Peltola, a mathematician on the College of Bonn in Germany.

Physicists can harness Feynman’s path integral to calculate actual correlation features for under probably the most boring of fields—free fields, which don’t work together with different fields and even with themselves. In any other case, they need to fudge it, pretending the fields are free and including in gentle interactions, or “perturbations.” This process, referred to as perturbation principle, will get them correlation features for many of the fields in the usual mannequin, as a result of nature’s forces occur to be fairly feeble.

But it surely didn’t work for Polyakov. Though he initially speculated that the Liouville area is likely to be amenable to the usual hack of including gentle perturbations, he discovered that it interacted with itself too strongly. In comparison with a free area, the Liouville area appeared mathematically inscrutable, and its correlation features appeared unattainable.

Up by the Bootstraps

Polyakov quickly started searching for a work-around. In 1984, he teamed up with Alexander Belavin and Alexander Zamolodchikov to develop a method referred to as the bootstrap—a mathematical ladder that progressively results in a area’s correlation features.

To start out climbing the ladder, you want a perform which expresses the correlations between measurements at a mere three factors within the area. This “three-point correlation perform,” plus some extra details about the energies a particle of the sphere can take, varieties the underside rung of the bootstrap ladder.

From there you climb one level at a time: Use the three-point perform to assemble the four-point perform, use the four-point perform to assemble the five-point perform, and so forth. However the process generates conflicting outcomes for those who begin with the mistaken three-point correlation perform within the first rung.

Polyakov, Belavin, and Zamolodchikov used the bootstrap to efficiently remedy a wide range of easy QFT theories, however simply as with the Feynman path integral, they couldn’t make it work for the Liouville area.

Then within the Nineties two pairs of physicists—Harald Dorn and Hans-Jörg Otto, and Zamolodchikov and his brother Alexei—managed to hit on the three-point correlation perform that made it attainable to scale the ladder, fully fixing the Liouville area (and its easy description of quantum gravity). Their end result, recognized by their initials because the DOZZ formulation, let physicists make any prediction involving the Liouville area. However even the authors knew they’d arrived at it partially by likelihood, not by sound arithmetic.

“They had been these type of geniuses who guessed formulation,” stated Vargas.

Educated guesses are helpful in physics, however they don’t fulfill mathematicians, who afterward needed to know the place the DOZZ formulation got here from. The equation that solved the Liouville area ought to have come from some description of the sphere itself, even when nobody had the faintest concept the best way to get it.

“It appeared to me like science fiction,” stated Kupiainen. “That is by no means going to be confirmed by anyone.”

Taming Wild Surfaces

Within the early 2010s, Vargas and Kupiainen joined forces with the chance theorist Rémi Rhodes and the physicist François David. Their objective was to tie up the mathematical unfastened ends of the Liouville area—to formalize the Feynman path integral that Polyakov had deserted and, simply possibly, demystify the DOZZ formulation.

As they started, they realized {that a} French mathematician named Jean-Pierre Kahane had found, many years earlier, what would develop into the important thing to Polyakov’s grasp principle.