From the S-matrix to the lattice: bootstrapping QFTs

Abstract

In this thesis, I present our attempts to formulate new bootstrap approaches to Quantum Field Theories. The main goal of these approaches is to offer complementary techniques to the Effective Field Theory paradigm. Namely, techniques that allow one to constrain IR observables using UV knowledge. For example, we may assume the existence of a local UV completion or detailed information about a UV lattice description. The first approach we present utilizes $S$-matrix bootstrap to turn ``experimental" data, the low energy scattering phase, into rigorous information about the existence or nonexistence of resonances in a two-dimensional theory of massive particles. As a zeroth step towards generalization to higher dimensions, we also report on progress in the study of scattering in Chern-Simons theory coupled to fundamental matter, which we hope will enable formulation of its $S$-matrix bootstrap problem in the near future. It will be the first $S$-matrix bootstrap problem studied in which Anyonic properties are directly involved. In order to set up the second approach, we have studied the line operators that can appear in the conformal Chern-Simons theory coupled to fundamental matter. Their endpoints are the Anyons of the conformal theory. The lines obey constraints from conformal symmetry and a set of loop equations which allow setting up a bootstrap for correlators of straight lines. In the third approach, we show that equations of motion together with reflection positivity severely constrain correlators of the Ising statistical model on the lattice. Namely, we derive rigorous two-sided bounds on spin correlators through semi-definite programming.