It is an elegant idea that gives concrete answers only for selected quantum fields. No known mathematical procedure can meaningfully average an infinite number of objects that cover the infinite space of space in general. The integral of times is more a philosophy of physics than an exact mathematical recipe. Mathematicians doubt its existence as a valid operation and are bothered by how physicists rely on it.
“As a mathematician, I am disturbed by something that is not defined,” he said Eveliina Peltola, a mathematician from the University of Bonn in Germany.
Physicists can use the Feynman path integral to compute exact correlation functions only for the most annoying fields — free fields, which do not interact with other fields, or even with themselves. Otherwise, they have to deceive it by pretending that the fields are free and adding mild interactions or “perturbations”. This procedure, known as perturbation theory, obtains correlation functions for most fields in the standard model because the natural forces are quite weak.
But the Poles did not succeed. Although he initially speculated that the Liouville field might be subject to the standard hack of adding mild disturbances, he found that it communicates too strongly with itself. Compared to the free field, Liouville’s field seemed mathematically inconceivable, and its correlation functions seemed unattainable.
Up by Bootstraps
Poljakov soon started looking for a job. In 1984, he teamed up with Alexander Belavin and Alexander Zamolodchikov to develop a technique called bootstrap—Mathematical scale that gradually leads to field correlation functions.
To start climbing ladders, you need a function that expresses the correlations between measurements at just three points on the ground. This “three-point correlation function”, plus some additional information about the energies that the particle of the field can receive, forms the lower rung of the trunk ladder.
From there, you climb one point at a time: Use the three-point function to construct a four-point function, construct a four-point function, and so on. But the procedure generates conflicting results if you start with the wrong three-point correlation function in the first step.
Polyakov, Belavin, and Zamolodchikov used bootstrap to successfully solve various simple QFT theories, but just as with the Feynman path integral, they could not make it work for the Liouville field.
Then in the nineties two pairs of physicists –Harald Dorn and Hans-Jörg Otto, i Zamolodchikov and his brother Alexei—He managed to guess the three-point correlation function that allowed the scale to be scaled, completely solving the Liouville field (and its simple description of quantum gravity). Their result, known by the initials as the DOZZ formula, allows physicists any prognosis that includes the Liouville field. But even the authors knew that they came to this in part by chance rather than by sound mathematics.
“They were the geniuses who guessed the formulas,” Vargas said.
Educated conjectures are useful in physics, but do not satisfy mathematicians, who afterwards wanted to know where the DOZZ formula came from. The equation that solved the Liouville field should have come from some description of the field itself, even if no one had the slightest idea how to get to it.
“It sounded like science fiction to me,” Kupiainen said. “No one will ever prove this.”
Taming of wild surfaces
In the early 2010s, Vargas and Kupiainen joined forces with probability theorist Rémi Rhodes and physicist François David. Their goal was to connect the mathematically loose ends of the Liouville field — to formalize the Feynman path integral that Poljakov abandoned and, perhaps only, to demystify the DOZZ formula.
When they started, they realized that a French mathematician named Jean-Pierre Kahane had discovered, decades earlier, what would prove crucial to Polyakov’s main theory.