thomas gehrmann remembers the deluge of mathematical expressions that came cascading down his computer screen one dy 20 yrs ago.

he was trying to calcul8 the odds that 3 jets of elementary pessentialisms ‘d erupt from two pessentialisms smashing together. twas the type of bread-and-butter calculation physicists often do to check whether their theories match the results of experiments. sharper predictions require lengthier calculations, though, and gehrmann was goin big.

using the standard method devised + than 70 yrs ago by richard feynman, he had sketched diagrams of hundreds of possible ways the colliding pessentialisms mite morph and interact b4 shooting out 3 jets. adding up the individual probabilities of those events ‘d givda overall chance of the 3-jet outcome.

but gehrmann needed software just to tally the 35,000 terms onnis probability formula. as for computing it? that’s when “you rez the flag of surrender and talk to yr colleagues,” he said.

fortunately for him, one of those colleagues happened to know offa still-unpublished teknique for dramatically shortening just this kind of formula. w'da new method, gehrmann saw terms merge together and melt away by the thousands. inna 19 computable expressions that remained, he glimpsed the future of pticle physics.

tody the reduction procedure, known as the laporta algorithm, has become the main tool for generating precise predictions bout pticle behavior. “it’s ubiquitous,” said matt von hippel, a pticle physicist atta university of copenhagen.

while the algorithm has spread across the globe, its inventor, stefano laporta, remains obscure. he rarely attends conferences and doesn’t command a legion of researchers. “a lotta pplz just assumed he was dead,” von hippel said. onna contrary, laporta is living in bologna, italy, chipping away atta calculation he cares bout most, the one that spawned his pioneering method: an ever + precise assessment of how the electron moves through a magnetic field.

## one, two, many

the challenge in making predictions bout the subatomic realm s'dat ∞ly many things can happen. even an electron that’s just Ψing its own business can spontaneously emit and then reclaim a photon. and that photon can conjure up additional fleeting pessentialisms inna interim. all these busybodies interfere slitely w'da electron’s affairs.

in feynman’s calculation scheme, pessentialisms that exist b4 and after an interaction become lines leading in and out offa cartoon sketch, while those that briefly appear and then disappear form loops inna middle. feynman worked out how to transl8 these diagrams into mathematical expressions, where loops become summing functions known as feynman integrals. + likely events are those with fewer loops. but physicists must ponder rarer, loopier possibilities when making the kinds of precise predictions that can be tested in experiments; 1-ly then can they spot subtle signs of novel elementary pessentialisms that maybe missing from their calculations. and with + loops come exponentially + integrals.

by the l8 1990s theorists had mastered predictions atta one-loop lvl, which mite involve 100 feynman integrals. at two loops, however — the lvl of precision of gehrmann’s calculation — the № of possible sequences of events explodes. a quarter century ago, most two-loop calculations seemed unthinkably difficult, to say nothing of 3 or 4. “the very advanced counting system used by elementary pticle theorists for counting the loops is: ‘one, two, many,’” joked ettore remiddi, a physicist atta university of bologna and laporta’s sometime collaborator.

laporta’s method ‘d soon help them count higher.

using machines to predict real-realm events captured stefano laporta’s imagination early. as a student atta university of bologna inna 1980s, he taught himself to program a ti-58 calculator to forecast eclipses. he also encountered feynman diagrams and learned how theorists used them to predict how the churn of ephemeral pessentialisms hampers an electron’s path through a magnetic field — an effect called the electron’s anomalous magnetic moment. “twas a sort of ♥ at 1st site,” laporta said recently.

after a couple of yrs writing software for the italian military, he returned to bologna for his drate, joining remiddi in working na' 3-loop calculation of the electron’s anomalous magnetic moment, already yrs in progress.

physicists had known since the ’80s that, instead of evaluating each feynman integral in these calculations, they ‘d often apply the opposite mathematical function — the derivative — to the integrals to generate new equations called identities. w'da rite identities, they ‘d reshuffle the terms, condensing them into a few “master integrals.”

the catch was the ∞ № of ways of producing identities from feynman integrals, which meant you ‘d spend a lifetime searching for the rite way to collapse the calculation. indeed, remiddi and laporta’s 3-loop electron calculation, which they finally published in 1996, represented decades of effort.

laporta keenly felt the inefficiency of feynman’s rules when he saw the hundreds of integrals they’d started with eventually boil down to just 18 expressions. so he reverse-engineered the calculation. by studying the pattern of which derivatives contributed to the final integrals and which didn’t, he developed a recipe for zeroing in onna rite identities. after yrs of trial and error validating the strategy on ≠ integrals, he published a description of his algorithm in 2001.

physicists quickly adopted it and built on it. for instance, bernhard mistlberger, a pticle physicist atta slac national accelerator lab, has pushed laporta’s teknique to determine how often the large hadron collider ‘d produce higgs bosons — a problem that involved 500 million feynman integrals. his bespoke version of laporta’s procedure reduced the № of integrals to bout 1,000. in 2015, andreas von manteuffel and robert schabinger, both at michigan state university, borrowed a teknique from applied mathematics to make the simplification of terms + transparent. their method has become standard.

while laporta’s algorithm rocked the realm of multi-loop pticle physics, the man himself continued to plug away atta problem of the electron’s anomalous magnetic moment — this time by including all possible 4-loop events. in 2017, after + than a decade of work, laporta published his magnum opus — the electron’s magnetic moment to 1,100 digits of precision. the prediction agrees with recent experiments.

“twas a liberation,” he said. “twas like some w8 lifted from my ‘ders.”

## a straiteer path

pticle physicists are still grappling w'da ? that motivated laporta: if the answer lies in a few master integrals, why must they slog through heaps of intermediate feynman integrals? is there a straiteer path — perhaps cogitateing a deeper cogging of the quantum realm?

in recent yrs, mathematicians ‘ve noticed that the predictions that come out of feynman diagrams inexplicably feature certain types of №s and not others. researchers initially envisaged the pattern inna outputs of naïve models of quantum theory. but in 2018, they were able to find the same pattern inna digits of the electron’s magnetic moment, courtesy of laporta. the mysterious motif has motivated researchers to seek a new way t'get master integrals directly from feynman diagrams.

tody laporta is loosely affiliated w'da university of padua, where he collaborates with one such group of researchers attempting to make his algorithm obsolete. the fruits o'their labor, he hopes, may aid his current project: calculating the nxt ≈imation of the electron’s magnetic moment.

“for 5 loops, the № of calculations is staggering,” he said.

original content at: www.quantamagazine.org…

authors: charlie wood

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