Volume IV - Complex Angular Momentum in Quantum Field Theory

Rather soon after Regge's pioneering work of 1959 the main interest for the concept of complex angular momentum left the original framework of potential scattering and moved toward the phenomenological implications of Regge theory and its connection with Particle Physics. Particles were found to be organized in families which can be represented by trajectories in the complex angular momentum plane parametrized by the squared center of mass energy. Therefore, the idea that the scattering amplitudes are analytic functions of the angular momentum and that the associated (polar) singularities can be used to describe elementary particles became very popular. However, the rigorous connection with Quantum Field Theory, i.e., the reference theoretical framework for elementary particles, was missing. In particular, the analyticity with respect to the complex angular momentum variables was not in the list of analyticity properties rigorously derived from the axioms of Quantum Field Theory. The program developed in this series of papers concerns primarily with showing that Regge's philosophy regarding analyticity and singularities is conceptually implied by the axiomatic framework of Quantum Field Theory. The mathematical methods employed in these studies are rather sophisticated and concern essentially with the harmonic analysis on complex hyperboloids and its relation with the harmonic analysis developed on the complex sphere.