An exact general formula for the matrix elements of the evolution operator in quantum theory is established. The formul ("ABC-formula") has the form =exp(At+B+C(t)). The constants A and B and the decreasing function C(t) are computed in perturbation theory. The ABC-formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory. The formula includes the higher order corrections to the well known Weisskopf-Wigner approximation and to the stochastic (van Hove) limit which are widely used in considerations of problems of radiation, decay, quantum decoherence, derivation of master and kinetic equations etc. The function C(t) admits an interpretation as an analogue of the autocorrelation function describing quantum chaos for the quantum baker's map.
There exists the well-known approximate expression describing the large time behavior of matrix elements of the evolution operator in quantum theory: < U(t)>≃ exp (at). This expression plays the crucial role in considerations of problems of quantum decoherence, radiation, decay, scattering theory, stochastic limit, derivation of master and kinetic equations etc. This expression was obtained in the Weisskopf–Wigner approximation and in the van Hove (stochastic) limit. We derive the exact general formula which includes the higher order corrections to the above approximate expression: <U(t)>= exp (At+B+C(t)). The constants A and B and the oscillating function C(t) are computed in perturbation theory. The method of perturbation of spectra and renormalized wave operators is used. The formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory.
There exists the well known approximate expression describing the large time behaviour of matrix elements of the evolution operator in quantum theory: =exp(at)+... This expression plays the crucial role in considerations of problems of quantum decoherence, radiation, decay, scattering theory, stochastic limit, derivation of master and kinetic equations etc. This expression was obtained in the Weisskopf-Wigner approximation and in the van Hove (stochastic) limit. We derive the exact general formula which includes the higher order corrections to the above approximate expression: =exp(At+B+C(t)). The constants A and B and the oscillating function C(t) are computed in perturbation theory. The method of perturbation of spectra and renormalized wave operators is used. The formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory.
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.
The time evolution of quantum many body systems in the relativistic quantum field theory is studied. We use Shibata-Hashitsume’s projection operator method which gives a systematic expansion of the Heisenberg equation of motion in terms of the interaction Hamiltonian. We calculate the time evolution of expectation values of creation and annihilation operators in a renormalizable model with unstable particles in a vacuum and in a medium. It is shown how to renormalize the ultraviolet divergences appearing in the equation of motion and calculate the medium effects on the time evolution. The mass and the wavefunction renormalizations are carried out in the lowest order. We find that the results coincide with the on-mass-shell values of the self-energy and its derivative which are calculated by the usual Feynman rule. We also discuss the initial divergence which remains at the initial time after mass renormalization. It is suggested that the projection operator method can be applied to the relativistic quantum field theory.
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.