  S-matrix is one of the fundamental observables of the quantum theory of
relativistic particles. There have been attempts to understand the quantum
dynamics of relativistic particles abstractly in terms of S-matrix bypassing a
Lagrangian formulation of quantum field theory. Equivalently, the space of
possible S-matrices defines an abstract theory space. This thesis examines how
to constrain the spectrum of physical theories in the theory space using the
basic physical requirements of Poincare invariance, quantum unitarity, and
causality. The thesis discusses two distinct but related ways of such
exploration. The first part of the thesis explores a novel mathematical way of
cruising the space of S-matrices using the techniques from geometric function
theory (GFT). This analysis leads to rigorous two-sided bounds on Wilson
coefficients in the effective field theories (EFT). Using these GFT techniques,
we study elastic scattering amplitudes of identical massive scalar Bosons, EFT
amplitudes of elastic 2-2 photon, and graviton scattering and try to constrain
the space of low energy effective field theories. In the second part of the
thesis, we work out a systematic derivation of the Froissart-Martin bound
starting with $4-$point Mellin amplitude for a holographic CFT.
