Replacing the Breit-Wigner amplitude by the complex delta function to describe resonances

TL;DR

This paper proposes replacing the Breit-Wigner amplitude with the complex delta function in resonance descriptions, enabling causal results without extending energy integrations beyond the physical spectrum.

Contribution

It introduces a novel approach to resonance modeling by substituting the Breit-Wigner amplitude with the complex delta function, maintaining physical energy bounds.

Findings

Achieves causal results without spectrum extension

Simplifies resonance calculations within physical energy range

Provides a new mathematical framework for resonance analysis

Abstract

Whenever the Breit-Wigner amplitude appears in a calculation,there are many instances (e.g., Fermi's two-level system and the Weisskopf-Wigner approximation) where energy integrations are extended from the scattering spectrum of the Hamiltonian to the whole real line. Such extensions are performed in order to obtain a desirable, causal result. In this paper, we recall several of those instances and show that substituting the Breit-Wigner amplitude by the complex delta function allows us to recover such desirable results without having to extend energy integrations outside of the scattering spectrum.