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.
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.
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.
This is the first in a series of papers. It deals with the linked-cluster summation of the perturbation series in quantum mechanics and quantum statistical physics. In particular, a formalism is developed for calculating the matrix element of the operator
exp(+sH), where
H is the quantum-mechanical Hamiltonian operator. It is found that the matrix element may be written as an ordinary
c-number integral of a
c-number function over the phase space. This function is found by doing a linked-cluster summation of the usual perturbation series. It is in the form of an exponential of an effective Hamiltonian,
Heff, which is a function of the parameter
s and the
c-number momentum and position coordinates. We have assumed that the Hamiltonian has no spin-dependent terms. The matrix element is written in such a form that it may be used for calculating the transition matrix elements in time-dependent perturbation theory as well as for calculating the canonical partition function
ZN in quantum statistical mechanics. The former is obtained by substituting $s=\ensuremath{-}\frac{\mathrm{it}}{\ensuremath{\hbar}}$ and the latter by substituting $s=\ensuremath{-}\ensuremath{\beta}$. The series for
Heff depends only on the potential and is independent of the statistics. The effects of the statistics (Bose-Einstein, Fermi-Dirac, or Maxwell-Boltzmann) are all included in a certain factor multiplying
exp(sHeff). Thus we have succeeded in separating the statistical and quantum corrections. These latter arise from the noncommutativity of the position and momentum operators. The effective Hamiltonian is an infinite series. A diagrammatic technique is given for evaluating this series. It is an expansion in powers of the coupling constant and can also be expanded in powers of Planck's constant. In general, it is not possible to give a closed-form expression for
Heff. But for most practical cases it is sufficient to keep the first few powers of Planck's constant $\ensuremath{\hbar}$, because it is very small. We therefore have an entirely new method of approximation. The statistical effect may always be included exactly. No numerical estimates are made in this paper.
In order to study the validity of the functional integral formula of the Green's function, the large time limit of the time evolution kernel exp (-iHt) of any quantum system is discussed. With the Feynman prescription, giving an infinitesimally small negative imaginary part to quadratic term of the potential in the Hamiltonian, the correct formula for this limit is obtained. It differs from the naive formula and depends on the detailed form of the potential. The usual formula holds for the perturbative expansion in terms of the potential, the proof of which is given by the path integral method. The new expression for the ground state wave function is also obtained in the course of the proof.
We analyze a modified version of the 'AgBr' Hamiltonian, solve exactly the equations of motion in terms of SU(2) coherent states, and study the weak-coupling, macroscopic limit of the model, obtaining an exponential behavior at all times. The asymptotic dominance of the exponential behavior is representative of a purely stochastic evolution and can be derived quantum mechanically in the so-called van Hove's limit (which is a weak-coupling, macroscopic limit). At the same time, a temporal behavior of the exponential type, yielding a 'probability dissipation' is closely related to dephasing ('decoherence') effects and one can expect a close connection with a dissipative and irreversible behavior. We stress the central relevance of the problem of dissipation to the quantum measurement theory and to the general topic of decoherence.