Algebra -> Feynman Diagrams (2D) -> Amplituhedron (multi-dimensional)

Source: Nautilus, May 2014

even the next-simplest case—trading two photons instead of one—is ferociously difficult to calculate. One of Heisenberg’s intrepid students attempted such a calculation in 1936; his published equation spilled over several pages.

Feynman identified each element of his drawings with a corresponding mathematical expression. With those simple translation rules in hand, he could dispatch in half an hour the kind of calculations that had stymied the world’s best theoretical physicists for decades.

On Dec. 6, 2013, Arkani-Hamed and Trnka uploaded a paper describing the amplituhedron onto the preprint server, arXiv.4 Their claim is bold: Here, they write, is a replacement for the venerable Feynman diagram, at least for treating the types of interactions—like the nuclear forces—in which force-carrying particles can scatter directly off each other.

When comparing apples to apples (restricting themselves to models that incorporate supersymmetry), their new method can reproduce in just a few lines of algebra the scattering amplitudes that others have painstakingly calculated from hundreds (even thousands) of Feynman diagrams.

To perform their new kind of calculation, Arkani-Hamed and Trnka use their exquisitely simple-looking geometrical constructions, the amplituhedra. Unlike Feynman’s doodles, the new objects are not depicted in space and time; rather, they live in an imaginary, multidimensional mathematical space (see The Amplituhedron)

Almost like magic, Arkani-Hamed and Trnka have demonstrated—at least in several tough test cases—that they arrive at the same value for the amplitude of various particle scatterings by calculating the volume of the corresponding amplituhedron, thereby doing an end-run around all those closed-loop, ghost-filled Feynman diagrams.

The result is a breathtaking economy. Physicists like Andrew Hodges at Oxford University and Jacob Bourjaily at Harvard University have marveled at the extraordinary compression and simplification that the amplituhedron approach promises. “The degree of efficiency is mind-boggling,” Bourjaily exclaimed to a journalist recently—an uncanny echo of physicists’ responses to witnessing Richard Feynman first calculate with his diagrams 65 years ago.5

… just as a single amplituhedron stands in for hundreds of Feynman diagrams—and as each Feynman diagram, in turn, represents dozens of lines of algebra. Each new generation tends to reassess the symmetries that had previously been taken for granted. One physicist’s sacred spherical cow becomes another’s inefficient accounting scheme.

For now, though, the amplituhedron and the supersymmetry on which it relies remain unproven ideas. What we do know is that by avoiding local symmetries and enforcing global ones, the amplituhedron allows physicists to calculate complicated interactions with astonishing ease. Even if supersymmetry proves not to be an accurate description of our universe, the success of the amplituhedron suggests that the most basic forces of nature might be governed by a deeper, simpler mathematical structure than what Feynman diagrams have been able to reveal.

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