Causality and the Dispersion Relation: Logical Foundations
Abstract
"Strict causality" is the assumption that no signal whatsoever can be transmitted over a spacelike interval in spacetime, or that no signal can travel faster than the velocity of light in vacuo. In this paper a rigorous proof is given of the logical equivalence of strict causality ("no output before the input") and the validity of a dispersion relation, e.g., the relation expressing the real part of a generalized scattering amplitude as an integral involving the imaginary part. This proof applies to a general linear system with a timeindependent connection between the output and a freely variable input and has the advantage over previous work that no tacit assumptions are made about the analytic behavior or singlevaluedness of the amplitude, but, on the contrary, strict causality is shown to imply that the generalized scattering amplitude is analytic in the upper half of the complex frequency plane. The dispersion relations are given first as a relation between the real and imaginary parts of the generalized scattering amplitude and then in terms of the complex phase shift.
 Publication:

Physical Review
 Pub Date:
 December 1956
 DOI:
 10.1103/PhysRev.104.1760
 Bibcode:
 1956PhRv..104.1760T