Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory

This article is an introduction to two currently very active research programs, the Conformal Bootstrap and Scattering Amplitudes. Rather than attempting full surveys, the emphasis is on common ideas and methods shared by these two seemingly very different programs. In both fields, mathematical and physical constraints are placed directly on the physical observables in order to explore the landscape of possible consistent quantum field theories. We give explicit examples from both programs: the reader can expect to encounter boiling water, ferromagnets, pion scattering, and emergent symmetries on this journey into the landscape of local relativistic quantum field theories. The first part is written for a general physics audience. The second part includes further details, including a new on-shell bottom-up reconstruction of the $\mathbb{CP}^1$ model with the Fubini-Study metric arising from re-summation of the $n$-point interaction terms derived from amplitudes.