There is given a generalization of the theorem of Misra and Sinha [Helv. Phys. Acta 50, 99 (1977)], which shows that in the decay process the regeneration of the undecayed unstable states from the decay products cannot proceed too slowly at arbitrarily short time and, correspondingly, the deviation from exponential decay at arbitrarily short time cannot be too small either. The generalization presented here is shown to be valid in the region where the theorem of Misra and Sinha cannot be applied.
Consider an unstable quantum system that has been found undecayed at an instant s and denote by R(t,s) the rate of its regeneration into an original (undecayed) state at a later instant t+s. It is proved that the reduced evolution is a semigroup, i.e., there is no regeneration at all, provided R(t,s)1/2 can be estimated by a sufficiently regular function that is nondecreasing in s and has zero derivative with respect to t at t=0 for every s. This generalizes the theorem of Misra and Sinha [Helv. Phys. Acta 45, 619 (1972)] in a different direction than in a recent paper by Nishioka [J. Math. Phys. 29, 1860 (1988)].
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.