Construction of the Perturbation Series for Transition Amplitudes from their Analyticity and Unitarity Properties

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Abstract
The analyticity properties of transition amplitudes are used in conjunction with the unitarity requirements to generate successive terms in the perturbation series, without referring to a specific Lagrangian. In the sixth and higher orders, production is neglected in the unitarity condition; subject to this approximation, it is found that the series can be so constructed. No analyticity properties which have not been rigorously proved need be employed, and the terms are found to satisfy the double dispersion representation. By examining the connection between this method and the conventional calculation of the perturbation series, the types of spectral function corresponding to different Feynman diagrams can be found. Formulas are given for the regions in which the spectral functions are nonzero.