Exclusion Plots from Pure Thought: The S-Matrix Bootstrap

In physics, exclusion plots are typically the domain of experimentalists, who rely on collider data or astrophysical observations to rule out regions of parameter space. In this talk, I will review a research program, called the S-Matrix Bootstrap, that generates theoretical exclusion plots using nothing but the fundamental principles of quantum field theory.

The S-Matrix Bootstrap revisits the idea that the laws of physics are often dictated by internal consistency rather than microscopic details. While the traditional approach starts with a Hamiltonian or a Lagrangian and computes observables, the bootstrap philosophy asks a converse question: What is the space of all mathematical theories consistent with fundamental principles such as causality, locality, and unitarity? In recent years, this abstract question has been transformed into a precision tool through the use of convex optimization and semidefinite programming.

I will describe how these numerical techniques allow us to rigorously map the geometry of the "theory space." We will explore how the bootstrap distinguishes between the "landscape" of consistent physical theories and the "swampland" of theories that appear valid at low energies but cannot be completed in the ultraviolet. I will discuss general applications of this framework, ranging from constraining Wilson coefficients of effective field theories to bounding coupling constants of strongly interacting systems in various dimensions.

Finally, I will highlight a remarkable empirical observation I call the "Optimization Hypothesis": physical theories describing nature often inhabit the boundaries of the allowed regions identified by the bootstrap. This suggests that real-world physics—whether in the context of high-energy particles or critical phenomena in statistical mechanics—is, in a specific geometric sense, optimized.