Extension of Levinson's Theorem to the Relativistic Case

Abstract

A generalization of Levinson's theorem is proved. The proof requires that the elastic partial-wave scattering amplitude satisfy a dispersion relation, and that the $\frac{N}{D}$ integral equation be of Fredholm type with nonzero determinant. Inelastic processes are taken into account fully by means of a complex phase shift. The high-energy behavior of the imaginary part of the phase shift is subject to mild restrictions. For spinless particles the theorem states that ${\delta }_l\left(\infty \right)=(-n_b+n_c)\pi$. The real part ${\delta }_l$ of the phase shift is normalized to zero at threshold. $n_b$ is the number of "particle poles"; i.e., elementary particle poles or bound state poles of the amplitude. $n_c$ is the number of Castillejo-Dalitz-Dyson (CDD) poles of the $D$ function. An unfamiliar aspect of the CDD ambiguity is discussed. For complete generality in computing particle poles from a given left cut discontinuity, a new sort of CDD pole must be admitted at real energies below threshold. This type of pole is to be associated with a stable particle with energy below threshold, whereas an ordinary CDD pole corresponds to an unstable particle above threshold.