9 highlights
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But three recent papers from a group of physicists led by Pierpaolo Mastrolia of the University of Padua in Italy and Sebastian Mizera of the Institute for Advanced Study in Princeton, New Jersey, have revealed an underlying mathematical structure in the equations. The structure provides a new way of collapsing interminable terms into just dozens of essential components.
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Two quarks enter and two quarks leave, but a lot can happen in the middle. A full accounting, implying a perfect prediction, would demand an infinite number of diagrams. No one expects perfection, but the key to improving a calculation’s precision is getting further along in the infinite line of events.
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Feynman developed rules for turning this cartoon into an equation which calculates the probability that the event actually takes place
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The fleeting existence of the quark pair above, like many virtual events, is represented by a Feynman diagram with a closed “loop.” Loops confound physicists — they’re black boxes that introduce additional layers of infinite scenarios.
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Physicists have algorithms to compute the probabilities of no-loop and one-loop scenarios, but many two-loop collisions bring computers to their knees.
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But there is one small mercy: Physicists don’t need to calculate every last integral in a complicated Feynman diagram because the vast majority can be lumped together.
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Mastrolia and Mizera’s work is rooted in a branch of pure math called algebraic topology, which classifies shapes and spaces. Mathematicians pursue this classification with “cohomology” theories, which allow them to extract algebraic fingerprints from complicated geometric spaces.
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Their method takes a family of related physical scenarios, represents it as a geometric space, and calculates the twisted cohomology of that space.
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The cohomology theories that produce these intersection numbers may do more than just ease a computational burden — they could also point to the physical significance of the most important quantities in the calculation.