Causality and the Dispersion Relation: Logical Foundations

John S. Toll
Phys. Rev. 104, 1760 – Published 15 December 1956
PDFExport Citation

Abstract

"Strict causality" is the assumption that no signal whatsoever can be transmitted over a space-like interval in space-time, 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 time-independent 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 single-valuedness 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.

  • Received 31 January 1956

DOI:https://doi.org/10.1103/PhysRev.104.1760

©1956 American Physical Society

Authors & Affiliations

John S. Toll

  • Princeton University, Princeton, New Jersey and University of Maryland, College Park, Maryland

References (Subscription Required)

Click to Expand
Issue

Vol. 104, Iss. 6 — December 1956

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review Journals Archive

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×