The Bootstrap Program for Quantum Field Theory

Introduction

The importance of quantum field theory (QFT) in modern theoretical physics is hard to overstate. Nevertheless, a proper mathematical framework to describe field theories is still lacking. This is felt most concretely in the world of scattering amplitudes, whose general properties are difficult to define from existing axioms. This hampers our ability to make progress in understanding strongly coupled field theories.

Progress in Conformal Field Theories

In contrast, for conformal field theories (CFTs) there has been remarkable progress in recent years. The key ingredient here is the operator product expansion, which both significantly constrains correlation functions and whose associativity conditions can be analyzed numerically.

Limitations of Current Techniques

These techniques are, however, only suitable for theories with a conformal symmetry, which normally only emerges at the endpoints of finely tuned RG flows.

Need for New Methods

To advance the non-perturbative understanding of strongly coupled non-conformal QFTs, new methods are needed. Among other things, it is imperative to grasp the non-perturbative structure of scattering amplitudes.

Proposed Approaches

I propose to follow two interlinked paths based on recent results:

1. The first approach concerns the flat-space limit of a QFT in a fixed hyperbolic background known as Anti de-Sitter (AdS) space.
2. The second approach can be called the numerical S-matrix bootstrap.

Both approaches hold great promise, but by pursuing them jointly, I will be able to optimally leverage insights from one approach into the other.