Macroscopic Causality Conditions and Properties of Scattering Amplitudes

Abstract

Two causality conditions that refer only to mass‐shell quantities are formulated and their consequences explored. The first condition, called weak asymptotic causality, expresses the requirement that some interaction between the initial particles must occur before the last interaction from which final particles emerge. This condition is shown to imply that if a two‐body scattering function is analytic except for singularities in the energy variable at normal thresholds, then (a) the physical scattering functions in two adjacent parts of the physical region separated by any normal threshold are parts of a single analytic function; (b) the path of continuation joining these two parts bypasses the singularity in the upper half‐plane of the energy variable; and (c) the integral over the physical function can be represented as an integral over a contour that is distorted into the upper‐half energy plane (hence not, for example, by a principal‐value integral). Singularities possessing finite derivatives of all orders with respect to real variations of the energy are not encompassed by this result. The second causality condition, called strong asymptotic causality, expresses the requirement that, apart from contributions whose effects fall off faster than any inverse power of Euclidean distance, momentum‐energy is carried over macroscopic distances only by stable physical particles. This condition implies that all n‐particle scattering functions (n ≥ 4) are analytic, apart from infinitely differentiable singularities, at physical points not lying on any positive‐α Landau surface. Moreover, the scattering functions on the two sides of any such Landau surface are analytically connected by a path that passes around the singularity surface in a well defined manner, which is the same as in perturbation theory. Thus, apart from possible infinitely differentiable singularities, the physical region singularity structure is derived from a mass‐shell causality requirement. Several properties of the set ${\mathcal{L}}^{+}$ of physical region positive‐α Landau surfaces are derived.