On the Connection of the Scattering and Derivative Matrices with Causality

Abstract

The connection between the $S$ matrix and causality suggested by Kronig is analyzed, and it is found that the condition of causality implies that the poles of the analytical functions $S\left(k\right)\mathrm{}$ are either on the imaginary axis or in the lower half-plane. The possibility of a close connection between the properties of the derivative $R$ matrix and causality is also analyzed. Although all the properties of the $R$ matrix could not be deduced from the requirements of causality, it is considered as an encouraging preliminary result that: (1) The referred distribution of the poles of $S\left(k\right)\mathrm{}$ can be obtained from the properties of the $R$ matrix. (2) These properties of the corresponding $R$ matrix are unchanged under a transformation $S\left(k\right)\mathrm{}\to e^{\mathrm{ik}\lambda }S\left(k\right)\mathrm{}$, with $\lambda$ positive, which preserves the causal nature of the theory.