It’s an elegant idea that only gives concrete answers for certain quantum fields. No known mathematical procedure can significantly average an infinite number of objects covering an infinite extent of space in general. The path integral is more a physical philosophy than an exact mathematical recipe. Mathematicians question its very existence as a valid operation and are embarrassed by how physicists trust it.
“I am disturbed as a mathematician by something that is not defined,” said Eveliina peltola, mathematician at the University of Bonn in Germany.
Physicists can exploit the Feynman’s path integral to calculate exact correlation functions only for the most boring fields, free fields, which do not interact with other fields or even with themselves. Otherwise, they have to fake it, pretending the fields are free and adding light interactions, or “disturbances.” This procedure, known as perturbation theory, gives them correlation functions for most of the fields of the Standard Model, because the forces of nature are quite weak.
But it didn’t work for Polyakov. Although he initially speculated that Liouville’s field might lend itself to the standard hack of adding slight disturbances, he found that he interacted too strongly with himself. Compared to a free field, Liouville’s field seemed mathematically impenetrable, and its correlation functions seemed inaccessible.
Until by the bootstraps
Polyakov quickly started looking for a workaround. In 1984, he joined forces with Alexander Belavin and Alexander Zamolodchikov to develop a technique called the to prime—A mathematical scale which progressively leads to the correlation functions of a field.
To start moving up the ranks, you need a function that expresses correlations between measurements at just three points in the field. This “three-point correlation function”, plus some additional information about the energies that a particle in the field can take, forms the bottom rung of the bootstrap scale.
From there, you go up one point at a time: use the three point function to build the four point function, use the four point function to build the five point function, and so on. But the procedure produces conflicting results if you start with the wrong three-point correlation function in the first rung.
Polyakov, Belavin, and Zamolodchikov used the bootstrap to successfully solve a variety of simple QFT theories, but just like with the Feynman’s path integral, they couldn’t make it work for the Liouville field.
Then in the 1990s two pairs of physicists—Harald Dorn and Hans-Jörg Otto, and Zamolodchikov and his brother Alexei– succeeded in touching the three-point correlation function which allowed scaling to scale, completely solving the Liouville field (and its simple description of quantum gravity). Their result, known by their initials as the DOZZ formula, allows physicists to make any prediction involving the Liouville field. But even the writers knew they got there in part by chance, and not through solid math.
“They were the kind of geniuses guessing formulas,” Vargas said.
Informed guesses are useful in physics, but they do not satisfy mathematicians, who then wanted to know where the DOZZ formula came from. The equation that solved Liouville’s field should have come from a description of the field itself, even though no one had a clue how to get it.
“It sounded like science fiction to me,” Kupiainen said. “It will never be proven by anyone. “
Taming wild surfaces
In the early 2010s, Vargas and Kupiainen joined forces with probability theorist Rémi Rhodes and physicist François David. Their objective was to complete the mathematical details of the Liouville field, to formalize the integral of the Feynman path that Polyakov had abandoned and, perhaps, to demystify the DOZZ formula.
At first, they realized that a French mathematician named Jean-Pierre Kahane had discovered, decades earlier, what would prove to be the key to Polyakov’s main theory.