Toward a QFT treatment of nonexponential decay

TL;DR

This paper explores the properties of the survival probability of unstable quantum states using a Lee Hamiltonian framework, drawing parallels with Quantum Field Theory to better understand nonexponential decay behaviors.

Contribution

It introduces a QFT-inspired approach to analyze the survival probability of unstable states, providing a detailed derivation and highlighting differences with genuine relativistic QFT.

Findings

Survival probability amplitude is the Fourier transform of the energy distribution.

The approach closely resembles QFT with propagators and Feynman rules.

Identifies differences between Lee Hamiltonian and relativistic QFT.

Abstract

We study the properties of the survival probability of an unstable quantum state described by a Lee Hamiltonian. This theoretical approach resembles closely Quantum Field Theory (QFT): one can introduce in a rather simple framework the concept of propagator and Feynman rules, Within this context, we re-derive (in a detailed and didactical way) the well-known result according to which the amplitude of the survival probability is the Fourier transform of the energy distribution (or spectral function) of the unstable state (in turn, the energy distribution is proportional to the imaginary part of the propagator of the unstable state). Typically, the survival probability amplitude is the starting point of many studies of non-exponential decays. This work represents a further step toward the evaluation of the survival probability amplitude in genuine relativistic QFT. However, although many similarities exist, QFT presents some differences w.r.t. the Lee Hamiltonian which should be studied in the future.