Dispersion relations, ‘causal’ inequalities and the completeness of the set of eigenstates in non-relativistic potential scattering

Abstract

The connection between the causality condition (as formulated by Van Kampen) and the dispersion relations satisfied by the scattering matrix in the non-relativistic problem is considered and it is shown that in a certain limit, which is attained by explicitly abandoning the mathematical framework of quantum mechanics, the requirement of causality reduces to a restriction on the system which is in general not sufficient to guarantee the dispersion relations, namely, the ‘causal’ inequality (for a given partial wave) first derived by Wigner. It is then shown that when the problem is reconsidered under the usual mathematical framework, or, more precisely, under the general assumption that the wave functions describing the system form a complete set of states, not only the dispersion relations, but also Van Kampen's causality condition and Wigner's inequalities can be rigorously derived from it.