Apr 7, 2023 03:32 PM
In the math of particle physics, every calculation should result in infinity. Physicists get around this by just ignoring certain parts of the equations — an approach that provides approximate answers. But by using the techniques known as “resurgence,” researchers hope to end the infinities and end up with perfectly precise predictions.
https://www.quantamagazine.org/alien-cal...-20230406/
INTRO: The secret to fixing a fatal flaw at the heart of quantum theory may lie in three obscure textbooks from the 1980s. But physicists can be forgiven for overlooking the potentially transformative ideas within, as the volumes appear simultaneously amateurish and intimidating.
The few physical copies that exist of Jean Écalle’s magnum opus look like little more than glorified photocopies. Oversize mathematical symbols scrawled in thick black ink frequently interrupt the neatly typed sentences. The text is also written in French, an inconvenience for researchers in the English-speaking world.
The mathematics itself poses another barrier. The trilogy’s 1,110 pages brim with original mathematical objects and bizarre coinages. Odd-sounding terms like “trans-series,” “analyzable germs,” “alien derivations” and “accelero-summation” abound.
“If you have a look at this for the first time and you don’t read it very carefully, you could think it’s a crackpot writing some crazy things,” said Marcos Mariño, a mathematical physicist at the University of Geneva who keeps what he calls the “historical documents” on his bookshelf and uses tools developed by Écalle daily. “Of course, he is not. He’s one of these visionary mathematicians.”
His visionary mathematics might be just what’s needed to overcome a profound conceptual embarrassment — one that physicists have been more or less ignoring for the past 70 years. In that time, physicists have learned to make breathtakingly accurate predictions about the subatomic world. But these predictions, precise though they may be, are approximations. If one seeks absolute precision, textbook quantum theory breaks down and yields infinite answers — nonsensical results many physicists consider to be mathematical trash.
By studying Écalle’s vintage textbooks, physicists are coming to suspect that these infinite answers contain countless treasures, and that, with sufficient effort, the mathematical tools he developed should let them take any infinity and dig out a finite and faultless answer to any quantum question.
“Indeed, it works very beautifully” in many cases, said Marco Serone, a physicist who studies this strategy, which goes by the name of “resurgence.” “At some point this process ends, and what you have in front of your eyes is the exact solution to your original problem.”
The resurgence community is small but has made steady progress over the years. A proto-version of the technique obtained exact results in quantum mechanics, which limits itself to the behavior of particles. And more sophisticated incarnations have allowed some physicists to venture further into the murky waters of quantum field theory, and recently string theory. But that’s just the beginning of the big dreams harbored by resurgence practitioners. They aim at nothing less than a new way of thinking about infinities in physical theories — one that better matches our finite world in theory and, just maybe, in practice too... (MORE - details)
https://www.quantamagazine.org/alien-cal...-20230406/
INTRO: The secret to fixing a fatal flaw at the heart of quantum theory may lie in three obscure textbooks from the 1980s. But physicists can be forgiven for overlooking the potentially transformative ideas within, as the volumes appear simultaneously amateurish and intimidating.
The few physical copies that exist of Jean Écalle’s magnum opus look like little more than glorified photocopies. Oversize mathematical symbols scrawled in thick black ink frequently interrupt the neatly typed sentences. The text is also written in French, an inconvenience for researchers in the English-speaking world.
The mathematics itself poses another barrier. The trilogy’s 1,110 pages brim with original mathematical objects and bizarre coinages. Odd-sounding terms like “trans-series,” “analyzable germs,” “alien derivations” and “accelero-summation” abound.
“If you have a look at this for the first time and you don’t read it very carefully, you could think it’s a crackpot writing some crazy things,” said Marcos Mariño, a mathematical physicist at the University of Geneva who keeps what he calls the “historical documents” on his bookshelf and uses tools developed by Écalle daily. “Of course, he is not. He’s one of these visionary mathematicians.”
His visionary mathematics might be just what’s needed to overcome a profound conceptual embarrassment — one that physicists have been more or less ignoring for the past 70 years. In that time, physicists have learned to make breathtakingly accurate predictions about the subatomic world. But these predictions, precise though they may be, are approximations. If one seeks absolute precision, textbook quantum theory breaks down and yields infinite answers — nonsensical results many physicists consider to be mathematical trash.
By studying Écalle’s vintage textbooks, physicists are coming to suspect that these infinite answers contain countless treasures, and that, with sufficient effort, the mathematical tools he developed should let them take any infinity and dig out a finite and faultless answer to any quantum question.
“Indeed, it works very beautifully” in many cases, said Marco Serone, a physicist who studies this strategy, which goes by the name of “resurgence.” “At some point this process ends, and what you have in front of your eyes is the exact solution to your original problem.”
The resurgence community is small but has made steady progress over the years. A proto-version of the technique obtained exact results in quantum mechanics, which limits itself to the behavior of particles. And more sophisticated incarnations have allowed some physicists to venture further into the murky waters of quantum field theory, and recently string theory. But that’s just the beginning of the big dreams harbored by resurgence practitioners. They aim at nothing less than a new way of thinking about infinities in physical theories — one that better matches our finite world in theory and, just maybe, in practice too... (MORE - details)