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.
Highlights
Speaking, the S-matrix in quantum field theory is defined only by using wave packets; see any textbook, e.g., Refs. [1,2]
Stueckelberg correctly pointed out in 1951 that the plane-wave S-matrix with finite time interval exhibits an extra ultraviolet (UV) divergence coming from the interaction point at the boundary in time [12]
In order to remove it within the plane-wave formalism, a phenomenological factor has been introduced so that the uncertainty of the initial and final times of the process can be taken into account
Summary
The S-matrix in quantum field theory is defined only by using wave packets; see any textbook, e.g., Refs. [1,2]. In order to remove it within the plane-wave formalism, a phenomenological factor has been introduced so that the uncertainty of the initial and final times of the process can be taken into account. This has led to the violation of unitarity, and the necessary modification of the S-matrix to cure the pathology has become complicated and rather intractable. In this paper we revisit the Gaussian wave-packet formalism to derive Fermi’s golden rule. 3, we reformulate the Gaussian S-matrix and present a systematic procedure to separate the bulk contribution from the boundary ones. In Appendix A we review the Gaussian wave-packet formalism for the scalar, spinor, and vector. In Appendix D we present possible expressions for the boundary limit
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