Mandelstam analyticity: single-function description across the entire Mandelstam plane

Establish rigorously that there exists a single analytic function T(s,t) that simultaneously describes a given 2→2 scattering process and all of its crossed channels throughout the entire Mandelstam plane, including all three physical regions, under the assumptions of analyticity and crossing symmetry.

Background

In discussing 2→2 scattering with Mandelstam variables s, t, and u, the chapter emphasizes that analyticity and crossing symmetry motivate viewing the amplitude as a single analytic function T(s,t) defined over the full Mandelstam plane, covering all physical regions and crossed channels. This global analyticity assumption is historically associated with the Mandelstam analyticity program in S-matrix theory.

While the text notes that this viewpoint can be justified from non-relativistic scattering theory, axiomatic quantum field theory, and analyses of perturbative Feynman diagrams, it explicitly refers to the statement as a conjecture rather than a theorem. A rigorous, fully general proof remains a central conceptual challenge in the analytic S-matrix framework.