New Dispersive Techniques for Quantum Field Theories

Abstract

Quantum field theory has been remarkably successful in describing interactions of elementary particles, particularly through perturbative methods. However, many important regimes — including strong coupling and quantum gravity — lie beyond perturbative control, necessitating non-perturbative techniques. This thesis adopts the bootstrap approach, which constrains quantum field theories (QFTs) and conformal field theories (CFTs) by imposing physical consistency conditions such as unitarity, causality, and invariance under spacetime symmetries. A key challenge is to implement these constraints efficiently in numerical settings. For this purpose, we develop new parametric, crossing-symmetric dispersive representations for scattering amplitudes and conformal correlators. These representations encode consistency constraints compactly and enable efficient bootstrap implementation, which we apply to several physical systems.

We study non-perturbative scattering amplitudes of identical massive scalars, such as neutral pions, focusing on the high-energy behaviour of the rho-parameter — the ratio of the real to imaginary part of the forward amplitude. Contrary to expectation, we find numerical evidence for multiple sign changes in this ratio before it asymptotes at high energies. Next, we analyse tree-level, string-like scattering amplitudes of gluons. Using the bootstrap, we identify the amplitude that minimises the entanglement generated during scattering, both with and without assuming the duality hypothesis. Remarkably, we find that, in the space of duality-satisfying amplitudes, entanglement is minimised by the open superstring amplitude. We also initiate the use of machine learning methods to do numerics, providing a novel tool for exploring nonperturbative amplitudes using the bootstrap.

In the context of holographic CFTs, we use the crossing symmetric dispersion relation to derive two-sided bounds on the Wilson coefficients of the dual AdS effective field theory, establishing a precise notion of bulk locality. We then take the limit of large AdS radius 𝑅 and recover known bounds on flat-space Wilson coefficients and new bounds on their 1/𝑅 corrections. Finally, we investigate logarithmic CFTs (LCFTs) and uncover intriguing links between LCFT correlators and Ramanujan’s formulae for 𝜋. Expressing the LCFT correlators using the parametric crossing-symmetric dispersive relation leads to new CFT representations of Ramanujan’s formulae whose remarkable simplicity hints towards universal features of LCFTs.