Conformal Field Theory and the Alpha Space Transform

This thesis can be split into two parts. In the first, we expound the alpha space formalism [1] and extend it beyond two dimensions. By performing a Sturm-Liouville analysis of the conformal quadratic Casimir differential equation, we define an invertible integral transform which maps functions on the Lorentzian square to alpha space. We explain how poles correspond to conformal blocks and provide numerous examples of interesting densities. After lifting the crossing equation to alpha space, we present a new representation of the accompanying kernel in terms of analytic Wilson functions. We also offer some comments on Regge physics and analyticity. In the second, we investigate the perturbative renormalisation of deformed conformal field theories from the Hamiltonian perspective. We discuss the relation with conformal perturbation theory, to which we provide an explicit match up to third order in the coupling, and show how second-order anomalous dimensions in the Wilson-Fisher fixed points are straightforwardly computed in the Hamiltonian framework. We then focus on the cut-off employed in the truncated conformal space approach of Yurov and Zamolodchikov [2]. We discuss the appearance of non-covariant and non-local counterterms to second order in the cut-off, explicitly in φ4 theory, and find a smooth cut-off to tame subleading oscillations.