Dyson time operator

Here, P is the Dyson time-ordering operator and Q(l)IP is given by the Heisenberg transformation Eq. (G.6), that is,... 

The different situations illustrated in Fig. 6 correspond more and more to approximate models when passing up to down. The top of the figure given the case described by Eq. (126) and corresponds to the reference quantum indirect damping. Below, is depicted the situation described by Eq. (146), where the Dyson time ordering operator is ignored in the quantum model. Further below is given the behavior that corresponds to both Eqs. (152) and (174), that is, to weak approximations on the classical limit of the quantum model [Eq. (152)] and to the semiclassical model [Eq. (174)]. At last, at the bottom we visualize the semiclassical model of Robertson and Yarwood [described by Eq. (185)]. 

Here, P is the Dyson time-ordering operator [57], Q (t)IP is the coordinate in the interaction picture with respect to the thermal bath and to the diagonal part of the Hamiltonian of the H-bond bridge, and the notation (( )e)siow has the meaning of a partial trace on the thermal bath and on the H-bond bridge coordinates. 

As a matter of fact, the Boson normal-ordering procedure allows us to get the formal solution of the IP time-evolution operator involving the Dyson timeordering operator P. Also, observe that within the Bosons representation, the IP coordinate is... 

Besides P, which showed not be confused with the momentum operator P of the H-bond bridge, is the Dyson time-ordering operator [57] acting on the Taylor expansion terms of the exponential operator in such a way so that the time arguments involved in the different integrals will be t > t > t". 

This is the reason for the use of the Dyson time-ordering operator. 

Because the time order of the incremental evolution operators is destroyed by the integral in the exponent, the so-called Dyson time ordering operator is introduced to reestablish the lost time order. 

The use of the Dyson operator is required because the time dependant IP Q coordinates at different times does not commute as it will appear below. 

In any time segment tj — tj i), using the decomposition of the quantum-classical Liouville operator into diagonal and off-diagonal J parts in (43), the quantum-classical propagator may be written in Dyson form as... 

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