Sing “Twistors killed the Feynman diagram” like “Video Killed the Radio Star,” by The Buggles.

A theoretical physicist is a human Feynman’s diagram calculator, nearly by definition. “Richard Feynman’s famous diagrams allow the calculation of how particles interact. However, new mathematical tools,” based on twistor theory, “are simplifying the results and suggesting improved underlying principles.” Let us summarize Neil Turok, “Particle physics: Beyond Feynman’s diagrams,” Nature 469: 165–166, 13 January 2011.

The mathematical framework used to describe elementary particles such as electrons and photons, and their interactions, is known as quantum field theory. In realistic quantum field theories, the particles collide, scatter off each other, and emit or absorb additional particles at rates that are governed by an overall parameter called the interaction coupling. The physicist Richard Feynman developed a beautiful pictorial shorthand, called Feynman diagrams, for describing all of these processes. Generations of physicists have spent large parts of their lives working out Feynman’s formulae for many kinds of scattering processes, and then testing those formulae in detailed experiments. The the number of contributing diagrams grows rapidly and calculations quickly become arduous as successive orders of the interaction coupling are added.

In 1985, two particle physicists at Fermilab in Batavia, Illinois, Parke and Taylor, decided to compute all of the Feynman contributions to one of the simplest processes involving the strong nuclear force, whose elementary particle — the gluon — binds quarks together into protons and neutrons. They considered two incoming gluons colliding and producing four outgoing gluons. This is one of the most common processes, for example, in the Large Hadron Collider (LHC), located at CERN, near Geneva, Switzerland. The leading contribution to this six-gluon process involves no less than 220 Feynman diagrams, encoding tens of thousands of mathematical integrals. Yet Parke and Taylor found that they could express the final result in just three simple terms. This was the first indication that Feynman diagrams were somehow complicating the story, and that there might be a simpler and more efficient description of these scattering processes. The technical paper is Stephen J. Parke and T. R. Taylor, “Amplitude for n-Gluon Scattering,” Phys. Rev. Lett. 56: 2459–2460, 1986. Further insight into this simplicity was gained by Bern, Dixon and Kosower, by Britto, Cachazo and Feng, and by Britto, Cachazo, Feng and Witten, who developed powerful new techniques — not involving Feynman diagrams — to infer higher-order scattering processes from lower-order ones.

Work done over the past year has shown why these new methods are simpler than Feynman’s. The formulation of quantum field theory used in Feynman’s rules emphasizes locality, the principle that particle interactions occur at specific points in space-time; and unitarity, the principle that quantum-mechanical probabilities must sum to unity. However, the price of making these features explicit is that a huge amount of redundancy (technically known as gauge freedom) is introduced at intermediate steps, only to eventually cancel out in the final, physical result.

The calculations of Luis F. Alday, Davide Gaiotto, Juan Maldacena, Amit Sever and Pedro Vieira, and Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Simon Caron-Huot and Jaroslav Trnka, work differently. They assert relations between quantities in a new way, so that the relations are free of these redundancies and they turn out to be sufficient to define the theory. The first big surprise is that such relations exist, and the second is that they are expressed in quantities that are explicitly non-local — that is, quantities that are spread out over space and time. Both sets of authors perform calculations within a particularly simple family of four-dimensional quantum field theories, with interactions, known as N = 4 supersymmetric theories. These theories are not realistic descriptions of real-world particle physics, but they do have elementary particles such as gluons and quarks (and even Higgs bosons), and they provide a valuable testing ground for new calculational techniques.

Arkani-Hamed and colleagues exploit a combination of twistor theory — a non-local description of space-time developed by Roger Penrose in the 1970s — and algebraic geometry to obtain a complete description of the scattering of all the elementary particles in these theories, in ascending powers of the interaction coupling, but only for small coupling. By contrast, Alday and colleagues derive relations between non-local quantities known as Wilson loops which represent the flux of the strong nuclear-force fields through various geometrical areas. Using the powerful mathematical machinery of quantum integrability, they are able to determine the behaviour of these fluxes in the limit at which the interaction coupling is large. The two sets of authors have therefore described the theory in its two opposite extreme limits — small and large coupling — and the hunt is now on for a complete description, one that is valid for any value of the interaction coupling.

Quantum field theory is the most powerful mathematical formalism known to physics. The recent discovery of mathematical structures that are now seen to control quantum field theory is likely to be of enormous significance, allowing us not only to calculate complex physical processes relevant to real experiments, but also to tackle fundamental questions such as the quantum structure of space-time itself. Maybe an improved set of founding principles may also be at hand.