The origin of Feynman diagrams: A new viewpoint

At low energies, the structure of Feynman diagrams arises from a dispersion relation that is valid at all energy scales

Two studies published in Physical Review Letters (PRL) in May 2021 report powerful new techniques in quantum field theory (QFT).  The studies were carried out by Aninda Sinha, Professor at the Centre for High Energy Physics (CHEP) and his graduate student Ahmadullah Zahed (the second one is in collaboration with Rajesh Gopakumar, Professor at ICTS, Bangalore).

Physicists use QFT as a framework that is consistent with quantum theory and Einstein’s special relativity. The standard model of particle physics, which provides a microscopic understanding of nature, relies heavily on this framework. Condensed matter physicists use the machinery of field theory to explain universal properties and make quantitative predictions. Calculations that lead to experimental predictions are carried out using the well-known Feynman diagram techniques. While the standard model can be predictive, for instance, explain observations in the Large Hadron Collider in CERN, Geneva, several features are still ad hoc and lack a fundamental explanation. While there are other techniques relying on lattice methods on powerful computers, there are conceptual gaps in the relation between nonperturbative approaches, for example, depending on dispersion relations and the hugely successful diagrammatic expansion.

In the first study, Sinha and Zahed provide a novel explanation for the origin of Feynman diagrams, which also clarifies and constrains low energy physics. This makes use of an old and forgotten result found in 1972 by Auberson and Khuri. Sinha and Zahed’s work fills a significant conceptual gap by giving a fresh perspective on how Feynman diagrams emerge.

In a follow-up work, Sinha and Zahed, together with Rajesh Gopakumar, demonstrate that similar ideas promise to play a pivotal role in the context of conformal field theories, which are of enormous interest in statistical mechanics, for instance, in calculating the specific heat of water at the so-called critical point.

The two studies enable a profound new connection with an area of mathematics that is 100 years old and goes by the name of univalent functions in geometric function theory, with arguably the most famous example being the Bieberbach conjecture, named after Ludwig Bieberbach, which was put forth in 1916 and proven only in 1985 by Louis de Branges. Together with graduate students Parthiv Haldar and Ahmadullah Zahed, Sinha established this mysterious connection in a preprint study making heavy use of the formalism Zahed and Sinha developed in their PRL work.