This is an elegant idea that only produces specific answers for selected quantum fields. There is no known mathematical program that can meaningfully cover an infinite number of objects in an infinite vast space on average. Path integral is more like a philosophy of physics than a precise mathematical formula. Mathematicians questioned its existence as an effective operation and were troubled by the way physicists depend on it.
“As a mathematician, I feel uneasy about some undefined things,” said Evelina Pertola, A mathematician at the University of Bonn, Germany.
Physicists can use Feynman’s path integral to calculate the exact correlation function of the most boring field-the free field, which does not interact with other fields, or even with themselves. Otherwise, they must fabricate it, pretend that these fields are free, and add gentle interactions or “disturbances.” This procedure is called perturbation theory, and it obtains correlation functions for most of the fields in the standard model, because natural forces happen to be very weak.
But it has no effect on Polyakov. Although he initially speculated that the Liouville field might be suitable for standard hackers with mild disturbances, he found that it interacted too strongly with himself. Compared with the free field, the Liuweier field is mathematically unpredictable, and its correlation function is also unattainable.
Polyakov soon began to find a solution. In 1984, he collaborated with Alexander Belavin and Alexander Zamolodchikov to develop a system called Bootloader-A mathematical ladder that gradually leads to the correlation function of a field.
To start climbing the ladder, you need a function to express the correlation between the measured values of only three points in the field. This “three-point correlation function”, plus some additional information about the energy that the particles of the field can use, forms the bottom step of the guiding ladder.
From there you climb one point at a time: construct a four-point function with a three-point function, construct a five-point function with a four-point function, and so on. However, if you start with the wrong three-point correlation function in the first rung, the process will produce conflicting results.
Polyakov, Belavin and Zamolodchikov used bootstrap to successfully solve various simple QFT theories, but like Feynman path integrals, they could not make it suitable for Liouville field.
Then in the 1990s, two pairs of physicists–Harald Dorn and Hans-Jörg Otto, with Zamorochkov and his brother Alexei-Managed to find a three-point correlation function that can scale the ladder and completely solve the Liuville field (and its simple description of quantum gravity). Their results, in their acronym DOZZ formula, allow physicists to make any prediction involving the Liuville field. But even the authors know that they were discovered partially by accident, not through reliable mathematics.
“They are the kind of geniuses who guess formulas,” Vargas said.
Educated guesses are useful in physics, but they do not satisfy mathematicians, who later wanted to know the source of the DOZZ formula. The equation for solving the Liuville field should come from some description of the field itself, even if no one knows how to get it.
“In my opinion, it’s like science fiction,” Kupiainin said. “This will never be proven by anyone.”
Tame the wild surface
In the early 2010s, Vargas and Kupianin joined forces with probability theorist Remirod and physicist François David. Their goal is to connect the mathematical loose ends of the Liouville field—formalize the Feynman path integral that Polyakov abandoned and, perhaps, uncover the mystery of the DOZZ formula.
When they started, they realized that a French mathematician named Jean-Pierre Kahane had discovered the key to Master Polyakov’s theory decades ago.