Dual S-matrix Bootstrap I: 2D Theory

TL;DR

This paper introduces a dual optimization approach to the 2D S-matrix bootstrap, providing rigorous bounds on quantum field theory observables using unitarity, crossing symmetry, and analyticity, and demonstrates its effectiveness on a two-particle system.

Contribution

It develops a dual formulation of the 2D S-matrix bootstrap that matches traditional bounds and applies it to a two-particle gapped system as a toy model.

Findings

Dual approach yields the same bounds as the primal formulation.

Numerical optimization of bounds is feasible and converges.

Application to a two-particle system demonstrates the method's utility.

Abstract

Using duality in optimization theory we formulate a dual approach to the S-matrix bootstrap that provides rigorous bounds to 2D QFT observables as a consequence of unitarity, crossing symmetry and analyticity of the scattering matrix. We then explain how to optimize such bounds numerically, and prove that they provide the same bounds obtained from the usual primal formulation of the S-matrix Bootstrap, at least once convergence is attained from both perspectives. These techniques are then applied to the study of a gapped system with two stable particles of different masses, which serves as a toy model for bootstrapping popular physical systems.