Physicists Nima Arkani-Hamed and Jaroslav Trnka recently published a significant advance in the study of Scattering Amplitudes. These are formulas that physicists use to calculate everything from the chance an unstable particle will decay to the probability of new discoveries at the Large Hadron Collider. The two reformulated scattering amplitudes within a popular framework called N=4 super Yang-Mills, treating them as properties of abstract geometrical objects. In doing so, Arkani-Hamed and Trnka hope to gain a deeper understanding of the nature of quantum field theory.

For over half a century, scattering amplitudes in quantum field theories (the class of theories that physicists use to describe subatomic particles) have been calculated using what are known as Feynman Diagrams. Created by Nobel laureate Richard Feynman, these diagrams depict events in particle physics as combinations of all possible paths that particles could travel. While powerful, the method gets increasingly involved for more complicated processes, and many important calculations can’t be done using it, even with today’s most advanced computers.

This difficulty spurred physicists like Zvi Bern, Lance Dixon, and David Kosower to develop new methods for calculating scattering amplitudes. Starting in the early '90s, they reinvigorated the field, inventing a technique called generalized unitarity, work which was recently honored with the prestigious J. J. Sakurai Prize for Theoretical Physics. Generalized unitarity let researchers skip laborious calculations of Feynman diagrams and go more directly to results that are often surprisingly simple.

At a recent conference on the geometry and physics of scattering amplitudes, several presenters joked about “revolutions” in quantum field theory, depicting protesters demanding the overthrow of Feynman diagrams and other traditional elements of particle physics. Expressing scattering amplitudes without relying on the traditional principles of the field serves two roles: it helps physicists see which principles are truly essential, and it can radically streamline complex calculations.

Arkani-Hamed and Trnka sought to express scattering amplitudes in a part of N=4 super Yang-Mills known as the planar limit without relying on two key principles: locality and unitarity. These normally enforce light’s speed limit and the rules of probability, but they can cause paradoxes in theories of quantum gravity. To get there, the two scientists found a way to express each scattering amplitude in terms of a corresponding mathematical object called an Amplituhedron, a many-dimensional polyhedral object in an abstract mathematical space.

By describing amplitudes in terms of geometrical objects without relying on physical principles, Arkani-Hamed and Trnka hope to spur more thorough mathematical investigation. Their work issues a challenge to mathematicians: if they can understand the properties of the mathematical object that is the Amplituhedron, they may be able to gain dramatic mastery of scattering amplitudes in the planar limit of N=4 super Yang-Mills, potentially calculating any such amplitude in just a few lines.

Arkani-Hamed and Trnka also hope to generalize their work away from this specific theory and toward models closer to the physics of the real world. Recently, they unveiled the first steps toward translating the more complicated non-planar part of N=4 super Yang-Mills into an Amplituhedron-like form.

The author has collaborated with two of the physicists mentioned in this article, Zvi Bern and Lance Dixon. His current revolutionary slogan is “down with integrands!” His blog is here.