Particle decay in Gaussian wave-packet formalism revisited

TL;DR

This paper derives Fermi's golden rule within the Gaussian wave-packet formalism of quantum field theory, addressing boundary effects and maintaining unitarity, thus refining the understanding of particle decay processes.

Contribution

It provides a systematic method to separate bulk and boundary contributions in particle decay calculations within a finite time, preserving unitarity and clarifying boundary effects.

Findings

Derived Fermi's golden rule in Gaussian wave-packet formalism.

Separated bulk and boundary contributions systematically.

Clarified the physical significance of boundary deviations.

Abstract

We derive the Fermi's golden rule in the Gaussian wave-packet formalism of quantum field theory, proposed by Ishikawa, Shimomura, and Tobita, for the particle decay within a finite time interval. We present a systematic procedure to separate the bulk contribution from those of time boundaries, while manifestly maintaining the unitarity of the $S$-matrix unlike the proposal by Stueckelberg in 1951. We also revisit the suggested deviation from the golden rule and clarify that it indeed corresponds to the boundary contributions, though their physical significance is yet to be confirmed.