Bootstrapping amplitudes of scalar particles

Quantum Field Theories are a central object of interest of modern physics, describing fundamental interactions of matter. However, current methods give limited insight into strongly coupling theories. S-matrix bootstrap program, described in this thesis, aims to provide insight into these strongly coupled theories, by mapping the space of consistent scattering amplitudes that obey properties imposed by unitarity, Lorentz invariance and causality. This thesis aims to outline the basic principles of the program, along with steps to construct numerical S-matrix bootstrap experiment, presents the \texttt{Wolfram Mathematica} library developed by author to automatize parts of the procedure that are shared among experiments, and finally discusses the research on 4d conformal field theories, which, by relevant deformation, can be turned into interacting QFTs, in which bootstrapping the scattering amplitudes allowed finding lower bounds on $a$-anomaly coefficient in the investigated cases.