Abstract A quantum model of the scattering of a particle via absorption and re-emission at the origin is described. The model is capable of exhibiting an arbitrary number of resonances, and the associated unstable states are discussed. The emphasis is on obtaining exact results. It is found that the exponential decay law is of limited application, and that in all cases of non-zero coupling the decay goes ultimately as t−3. Finally, it is pointed out that the treatment of unstable states in this type of model depends only on a knowledge of the phase shift as a function of radial momentum, and a possible phenomenological method is suggested.
Abstract We describe- in a didactical and detailed way- the so-called Lee model (which shares similarities with the Jaynes-Cummings and Friedrichs models) as a tool to study unstable quantum states/particles. This Lee model is based on Quantum Mechanics (QM) but possesses some of the features of Quantum Field Theory (QFT). The decay process can be studied in great detail and typical QFT quantities such as propagator, one-loop resummation, and Feynman rules can be introduced. Deviations from the exponential decay law as well as the Quantum Zeno effects can be studied within this framework. The survival probability amplitude as a Fourier transform of the energy distribution, the normalization of the latter, and the Breit-Wigner limit can be obtained in a rigorous mathematical approach.
After reviewing the description of an unstable state in the framework of Lee Hamiltonians (valid both for Quantum Mechanics (QM) and Quantum Field Theory (QFT)), we consider some theoretical aspects of non-exponential decays: the case of two decay channels, the broadening of the energy spectrum at short times, the effect of an imperfect measurement, the link to QFT, and the decay of an unstable moving particle with definite momentum. All the presented effects were not confirmed in experiments, hence are at the present stage predictions.
We consider a family of states in a model of nonrelativistic quantum electrodynamics, with positive Hamiltonian
H. For a given initial state
Ψ, the return probability amplitude
RΨ(t)=(Ψ,exp(−iHt)Ψ) may be written for positive times, as the sum of an exponentially decaying term and a correction
O(t1), for large times
t and small coupling constants (Theorem 4.1). The correction term is seen to be related both to the positivity of
H and to the existence of the virtual process of regeneration of the decaying state from the decay products, which is shown to be essentially quantum field theoretic, i.e., not present in nonrelativistic Schroedinger quantum mechanics. Some implications of this fact are analysed from the point of view of a general picture of irreversibility and the arrow of in quantum field theory. Finally, we make a first application of a time-energy uncertainty theorem to a quantum field theoretic model, in order to find a lower bound to the energy fluctuation in the state
Ψ (Theorem 5.2). In the process, it is also suggested that the time of sojourn
τH(Ψ)=∫0∞∣RΨ(t)∣2dt is the most natural quantity to consider in connection with the decay of unstable atoms or particles: it is proved to coincide with the the average lifetime of the decaying state, a standard quantity in quantum probability. No use is made of complex energies associated to analytic continuations of the resolvent operator to unphysical Riemann sheets.
We elucidate the physical nature of ghosts above the multi-particle threshold by contrasting them with unstable particles in quantum field theory. We first consider the asymptotic formulation, where ordinary positive-norm one-particle states can be unstable and decay, whereas ghosts survive asymptotically without decaying, yet admit no particle interpretation due to interference with the multi-particle component which masks the negative-norm one-particle state. This distinction originates from two different analytic structures of the dressed propagator, whose complex conjugate poles lie on the first or second Riemann sheet in the ghost or ordinary case, respectively. Ghost resonances are, in principle, phenomenologically distinguishable from ordinary ones, being narrower and exhibiting weaker interference between positive- and negative-energy peaks. We then formulate the quantum field theory in a finite interval of time and, working within a suitable approximation for the dressed propagator, find that finite-time effects amplify differences in the resonant behavior and give rise to new features, such as higher peaks in ghost resonances. Distinct temporal regimes are also identified: for times shorter than the inverse width, an approximate free-particle description is valid; at later times interactions and interference dominate, leading to decay or multi-particle masking. This study supports the absence of freely propagating ghost particles in the asymptotic limit.