Nonperturbative aspects of scattering amplitudes

In this thesis we study how physical principles imposed on the S-matrix, such as Lorentz invariance, unitarity, crossing symmetry and analyticity constrain quantum field theories at the nonperturbative level. We start with a pedagogical introduction to the S-matrix bootstrap and showcase some basic consequences, such as the Froissart bound on the total cross-section. We then more carefully revisit and extend old results from 1960s including an interesting relation between large spin, low energy data and Landau singularities. We revisit the Aks theorem asserting the necessity of particle production in relativistic scattering in $d > 2$. We establish the nonperturbative support for the spectral density, which is needed for a double dispersive representation of the scattering amplitude (or Mandelstam representation). This is a result of an extensive Landau and graph-theoretical analysis in which the leading inelastic Landau curves are determined and in particular an infinite subclass of these curves are found to accumulate at finite energies. We also consider the scattering of heavier particles where so-called anomalous thresholds are present. We show that anomalous thresholds are a consequence of the aforementioned principles and a key extra assumption: analyticity in the mass. We find a nonperturbative formula for the imaginary part across the anomalous threshold which, in particular, is shown \emph{not} to be positive definite. Finally, we add form factors and spectral densities of local operators to the bootstrap setup, such as the stress-energy tensor. This let us include information about the UV conformal field theory and allows to better target certain quantum field theories such as Ising field theory.