The Bootstrap Program for Quantum Field Theory

Project Information

QFTinAdS

Grant agreement ID: 101087025

DOI

10.3030/101087025

EC signature date 23 May 2023

Start date 1 September 2023

End date 31 August 2028

Funded under

• European Research Council (ERC)

Total cost

€ 1 930 051,25

EU contribution

€ 1 930 051,00

Coordinated by

ECOLE POLYTECHNIQUE
France

Periodic Reporting for period 1 - QFTinAdS (The Bootstrap Program for Quantum Field Theory)

Reporting period: 2023-09-01 to 2026-02-28

Summary of the context and overall objectives of the project

This project studies the general mathematical properties of certain observables in quantum field theories. The general philosophy is that of the "bootstrap" and proceeds in two steps. First, rigorous properties are established that should hold universally for these observables, irrespective of the specific quantum field theory under consideration. This is a formal and mathematical exercise, where one tries to understand the structure by starting from some completely general axioms. One derives for example analyticity, crossing symmetry, and other properties. Second, these rigorous and sometimes rather abstract properties are translated into concrete predictions for experiment, or more precisely constraints for the range of possible experimental values for these observables. These predictions can be completely general, so valid for every quantum field theory under the sun, or more specific by supplementing the universal properties with specific information about the quantum field theory under consideration. This second stage often uses advanced numerical methods.

What this project specifically focuses on is the derivation of mathematical properties of so-called scattering amplitudes, and the observable consequences that follow from them. Its approach to doing so is distinguished by a new method, called QFT in AdS, which uses curved-space quantum field theory. This change of perspective allows one to start from well-understood mathematical structures and leverage existing numerical methods to derive new properties about the relatively poorly understood scattering amplitudes.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

In the first two years of the grant we made progress on various fronts. First, we rigorously proved a property of conformal correlation functions which provides an important first step in rigorously deriving results on scattering amplitudes using QFT in AdS. In fact, our result applies much more generally and the new methodology can also be used to study the asymptotic structure of other observables like partition functions in two-dimensional conformal quantum field theories.

We also made some progress in the numerical component of the research. One important exploration studies the specific quantum field theory that governs the strong interactions in the standard model of particle physics. We found constraints on some observables that might be difficult to measure experimentally but should be computable in simulations in the near future. This study also gave us an insight into the capabilities of the current numerical methods and promising avenues for future research.

In the next three years we plan to combine various ingredients, strengthen the mathematical rigor of existing derivations, and arrive at a novel way to describe the fundamental structure of scattering amplitudes.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

This is theoretical work and as such the impact will mostly be restricted to the theoretical physics community. That said, it is part of an important line of research to rigorously define (aspects of) quantum field theory. Indeed, even though quantum field theory is essential to describe physics ranging from elementary particles to boiling water at high temperatures and pressures, its status as a mathematical structure remains deeply unsatisfactory. Making some progress in this direction is an important subject in the mathematical physics field.

Our numerical problems can often be formulated as semidefinite programs or as an infinite-dimensional variation of linear programs. These kind of programs are used more widely, both in academia and in the professional world, so the algorithms we develop may also be useful to others.