Explorations in the Space of S-Matrices
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S-matrix is one of the fundamental observables of the quantum theory of relativistic particles. The quantum dynamics of relativistic particles can be abstractly understood 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. In this talk, I will discuss 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). The discussion will be centred on a crossing symmetric dispersive representation of scattering amplitudes due to Auberson and Khuri (1972), which enables us to carry out this exploration. In particular, the dispersion kernel turns out to be a univalent function in a suitable complex variable. Univalent functions are known to satisfy various bounding relations. The most famous of them is the de Branges’ theorem, previously known as the Bieberbach conjecture, which bounds the Taylor coefficient of univalent functions. Using this theorem, we put double-sided bounds on Wilson coefficients of EFT amplitudes. Using another theorem, the Koebe growth theorem, we were able to put double-sided bounds on the amplitude itself. We also explore the connection with another kind of function from GFT, the typically real function, which is also known to satisfy various bounding relations. 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 turn our attention to holographic S-matrices. The conjectural $AdS/CFT$ holography provides a way to construct flat space scattering amplitudes from the Mellin amplitudes of a conformal field theory (CFT) by taking a large radius limit of the dual $AdS$ space. Various analytic properties of flat space scattering amplitudes are encoded in corresponding properties of the CFT Mellin amplitude. Flat space $2-2$ scattering amplitudes are known to satisfy high energy bounds called the Froissart-Martin bound which follows from axiomatic analyticity and unitarity properties of the S-matrix. Froissart-Martin bound is one of the robust consistency tests for a flat space scattering amplitude. Therefore if a holographic construction of the S-matrix is to work, one should be able to obtain a systematic derivation of the Froissart-Martin bound starting with $4-$point Mellin amplitude for a holographic CFT. We provide such a derivation in the second part of the thesis. We find that our holographic derivation gives the exact Froissart-Martin bound in $4$ spacetime dimensions, while in greater spacetime dimensions, we get weaker bounds. We attempt to argue the possible reason for this behaviour.