When a system is subject to time dependent external fields the equations of motion for both the distribution function and phase variables, become quite complex. There are two time dependences in such a system. One is associated with the time at which you wish to know the phase position Γ(t) and the other is associated with the explicit time dependence of the field, F_e(t). In order to deal with this complexity in a convenient way we introduce a more compact notation for the propagator. Apart from some important notational differences the initial development parallels that of Holian and Evans (1985). We define the p-propagator U_R(0,t) to be the operator which advances a function of Γ only, forward in time from 0 to t (the meaning of the subscript will emerge later). That is

(8.1)

The operator U_R(0,t) operates on all functions of phase located to its right. The equations of motion for the system at time t, which are themselves a function of phase Γ, are given by

(8.2)

The notation dΓ(Γ(t),t)∕dt implies that the derivative should be calculated on the current phase Γ(t), using the current field F_e(t). On the other hand dΓ(Γ(0),t)∕dt implies that the derivative should be calculated on the initial phase Γ(0), using the current field F_e(t). The p-propagator U_R(0,t) has no effect on explicit time. Its only action is to advance the implicit time dependence of the phase, Γ.

The total time derivative of a phase function B(Γ) with no explicit time dependence (by definition a phase function cannot have an explicit time dependence) is

(8.3)

where we have introduced the time dependent p-Liouvillean, iL(t) ≡ iL(Γ,t) which acts on functions of the initial phase Γ, but contains the external field at the current time. The partial derivative of B with respect to initial phase Γ is simply another phase function, so that the propagator U_R(0,t) advances this phase function to time t (that is the partial derivative of B with respect to phase evaluated at time t). In writing the last line of (8.3) we have used the fact that the p-propagator is an explicit function of time (as well as phase), and that when written in terms of the p-propagator, dB(Γ(t))∕dt, must only involve the partial time derivative of the p-propagator. Equation (8.3) implies that the p-propagator U_R(0,t) satisfies an operator equation of the form

(8.4)

where the order of the two operators on the right-hand side is crucial. As we shall see shortly, U_R(0,t) and iL(t) do not commute since the propagator U_R(0,t) contains sums of products of iL(s_i) at different times s_i, and iL(s_i) and iL(s_j) , do not commute unless s_i= s_j. The formal solution of this operator equation is

(8.5)

Notice that the p-Liouvilleans are right ordered in time (latest time to the right). As Liouvilleans do not commute this time ordering is fixed. The integration limits imply that t > s₁ > s₂ > .... > s_n, so that the time arguments of the p-Liouvilleans in the expression for U_R(0,t) increase as we move from the left to the right. It is very important to remember that in generating B(t) from B(0) using (8.5), if we write the integrals as say, a trapezoidal approximation it is the Liouvillean at the latest time iL(t), which attacks B(0) first. The Liouvilleans attack B in an anti-causal order. We will have more to say on this issue in §8.4.

We can check that (8.5) is the solution to (8.4) by differentiating with respect to time. We see that, ₀ ∫ ^∞ ds₁ disappears and the argument iL(s₁), changes to iL(t). This term appears on the right hand side, as it must to satisfy the differential operator equation. It is easy to derive an equation for the incremental p-propagator U_R(τ,t) which advances a phase function from time τ to t,

(8.6)

Our convention for the time arguments of the U-propagators is that the first argument (in this case τ), is the lower limit of all the integrals. The second argument (in this case t), is the upper limit of the first integral.