Conformal field theories play a central role in theoretical physics with many applications ranging from condensed matter to string theory. The conformal bootstrap studies conformal field theories using mathematical consistency conditions and has seen great progress over the last decade. In this thesis we present an implementation of analytic bootstrap methods for perturbative conformal field theories in dimensions greater than two, which we achieve by combining large spin perturbation theory with the Lorentzian inversion formula. In the presence of a small expansion parameter, not necessarily the coupling constant, we develop this into a systematic framework, applicable to a wide range of theories. The first two chapters provide the necessary background and a review of the analytic bootstrap. This is followed by a chapter which describes the method in detail, taking the form of a practical guide to large spin perturbation theory by means of a step-by-step implementation. The second part of the thesis presents several explicit implementations of the framework, taking examples from a number of well-studied conformal field theories. We show how many literature results can be reproduced from a purely bootstrap perspective and how a variety of new results can be derived.
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