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It is an elegant idea that provides specific answers only for selected quantum fields. An unknown mathematical procedure cannot, in general, give an average of an infinite number of objects that cover an infinite expanse of space. The path to the integral is a philosophy of physics, rather than an exact mathematical recipe. Mathematicians question its very existence as a valid operation and are disturbed by the ways in which physicists base it.
“Something that doesn’t define me bothers me as a mathematician,” he said Eveliina Peltola, Mathematician at the University of Bonn in Germany.
Physicists can use Feynman’s integral path to calculate exact correlation functions only for the most boring fields (free fields that do not interact with or with other fields). Otherwise, they must fail, pretending that the fields are free and adding slight interactions or “disturbances”. This procedure, known as perturbation theory, achieves correlation functions in most areas of the standard model because the forces of nature are relatively weak.
But it didn’t work out for Polyakov. Although he initially speculated that the Liouville area might be under the influence of the standard for adding slight disturbances, he found that it interacted very strongly with itself. Compared with a free field, the Liouville field seemed mathematically effortless, and its correlation functions seemed inaccessible.
Up with Bootstraps
Poliakov soon began looking for work. In 1984, he teamed up with Alexander Belavin and Alexander Zamolodchikov to develop a technique called bootstrap—The mathematical ladder that gradually leads to the correlation functions of a field.
To start climbing the ladder, you need a function that indicates the correlations between the measurements at three simple points in the field. This “three-point correlation function,” which adds additional information about the energies that a particle in the field can take up, is at the bottom of the bootstrap ladder.
From there you raise one point at a time: Use the three-point function to build a four-point function, use the four-point function to build a five-point function, and so on. But the procedure produces conflicting results if you start with a incorrect three-point correlation function at the first level.
Polyakov, Belavin, and Zamolodchikov used bootstrap to successfully solve many simple QFT theories, but as was the case with the Feynman comprehensive path, they were unable to function for the Liouville field.
Then, in the 1990s, two pairs of physicists—Harald Dorn and Hans-Jörg Otto, and Zamolodchikov and his brother Alexei—The ladder was scaled to a three-point correlation function that allowed it to be scaled, completely fixing the Liouville field (and a simple description of quantum gravity). Their results, known by the initials DOZZ formula, allow physicists to make any predictions about the Liouville field. But the authors also knew that part of it came about by chance, not through the sound of mathematics.
“They were the kind of geniuses who invented formulas,” Vargas said.
Educational inventions are useful in physics, but they do not satisfy mathematicians, then they wanted to know where the DOZZ formula comes from. The equation that solved the Liouville field would have to come from a description of the field itself, even though no one knew how to achieve it.
“It sounded like science fiction,” Kupiainen said. “No one will ever prove that.”
Taming wild surfaces
In the early 2010s, Vargas and Kupiain met Rémi Rhodes with theoretical probability and physicist François David. Their goal was to link the mathematical loose ends of the Liouville field – to formalize the integral Feynman path left by Polyakov and, perhaps, to demystify the DOZZ formula.
When they started, they realized that a French mathematician named Jean-Pierre Kahane had discovered, a few decades earlier, what would have been the key to Polyakov’s master theory.
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