Analytic Properties of Perturbation Expansions

A method is introduced for locating and interpreting the singularities in finite-order perturbation expansions of Green's functions, in cases where the external momenta are complex vectors with real scalar products. In particular, when the external momenta form a real Euclidean set, there is a simple graphical construction, applicable to all finite orders. As an example of a case in which the external momenta form a real set in a space of signature (+, +; -, -), a feature of Mandelstam's representation for scattering amplitudes is interpreted in perturbation theory. Finally, the ranges of values of momentum-transfer for which (in perturbation theory) dispersion relations certainly hold are enlarged, in particular by using the fact that the pion is pseudoscalar.