Toledo, Jonathan2016-09-142016-09-142016-09-142016-07-15http://hdl.handle.net/10012/10841This thesis is devoted to a two-pronged study of non-perturbative quantum field theory. In Part I we focus on the four-dimensional super conformal $\mathcal{N}=4$ Yang Mills theory. We compute smooth Wilson loops and correlation functions in the strong-coupling regime of the theory using the classical integrability of the dual string theory as our main tool. In both cases the solution is given as a set of integral equations of thermodynamic bethe ansatz type. The correlation function and Wilson loop are then written in terms of the corresponding TBA free energy. The equations for the Wilson loop expectation value can be used for generic smooth contours embeddable in an $\mathbb{R}^{1,1}$ subspace of $\mathbb{R}^{1,3}$. In Part II we ask general questions about the allowed space of massive quantum field theories based only on crossing symmetry and unitarity. We approach this question in two ways. First we consider putting massive QFT into an AdS box and study the conformal boundary theory using standard conformal bootstrap tools. We call this procedure the {\it boundary bootstrap}. It is applicable in any dimension but takes its simplest form for a $1+1$ dimensional bulk QFT where we use it to obtain rigorous bounds on allowed QFT couplings. For $1+1$ dimensional QFT we are also able to obtain rigorous bounds directly in flat space using unitarity, crossing symmetry and analyticity of the S-matrix. The bounds obtained in this way agree perfectly with those obtained from the boundary bootstrap.enquantum field theorynon-perturbative quantum field theoryN=4 super yang millsS-matrix bootstrapDoctoral Thesis