The Liouville equation for a system subject to a time dependent external field is given by

(8.20)

where we have defined the time dependent f-Liouvillean, iL(t). This equation tells you that if you sit at a fixed point in phase space denoted by the dummy variable Γ, the density of phase points near Γ, changes with time in accord with (8.20). In the derivation of this equation we related the partial derivative of f(t) to various fluxes in phase space at the same value of the explicit time.

We define the distribution function propagator U^†_R(0,t) which advances the time dependence of the distribution function from 0 to t, by

(8.21)

In this equation U_R ^†(0,t) is the adjoint of U_R(0,t). It is therefore closely related to U_L(0,t) except that the Liouvilleans appearing in equation (8.3.7) are replaced by their adjoints iL(s_i). Combining equation (8.5.2) with the Liouville equation (8.5.1) we find that U_R ^†(0,t) satisfies the following equation of motion

(8.22)

The formal solution to this operator equation is

(8.23)

In distinction to the propagator for phase variables, the integration limits imply that t > s₁ > s₂ > .... > s_n , so that the f-Liouvilleans are left time ordered. The time arguments increase as we go from the right to the left. This is opposite to the time ordering in the p-propagator U_R(0,t) but the definition of U_R ^†(0,t) is consistent with the definition of U_L(0,t).

For the f-propagator U†(0,t), the usual associative law is satisfied as the time arguments are ordered right to left,

(8.24)

This equation can be verified directly using similar arguments to those used in §8.4.