We will assume that t > 0. Intuitively it is obvious that the inverse of U_R(0,t), which we write as U_R(0,t)^-1, should be the propagator that propagates backwards in time from t to 0. From (8.6) we can write down

(8.7)

Before proceeding further we will introduce an identity which is useful for manipulating these types of integrals. Often we will have a pair of integrals which we want to exchange. The limits of the inner most integral depend on the integration variable for the outer integral. The result we shall use is the following, that

(8.8)

As can be seen from Figure 8.1, the range of integration for both integrals is the same. If we approximate the integral as a sum we see that the difference is in the order in which the contributions are summed. As long as the original integral is absolutely convergent the result is true. We will assume that all integrals are absolutely convergent.

It is illustrative to develop other representations of U_R(0,t)^-1 so we consider the expression (8.7) term by term,

+ ...... (8.9)

Interchanging the integration limits in every integral gives a factor of minus one for each interchange.

+ ...... (8.10)

We can use the integral interchange result (8.8) on the third term on the RHS (note that the integrand is unchanged by this operation). In the fourth term we can use the interchange result three times to completely reverse the order of the integrations giving,

+ ...... (8.11)

The final step is to relabel the dummy integration variables to give

(8.12)

As t > 0, an examination of the integration limits reveals that the Liouvilleans in this expression are left-ordered. Comparing this expression with the definition of U_R(0,t) there are two differences, the time ordering and the factor of (-)ⁿ. We now define the operator U_L(0,t) to be equal to the RHS of (8.12), so we have

(8.13)

and

(8.14)

From this definition of U_L(0,t), it can be shown that U_L(0,t) satisfies the operator equation

(8.15)

This result can be obtained by differentiating the definition of U_L(0,t), (8.13), or by differentiating U_R(0,t)^-1, (8.7), directly. Equation (8.3.9) allows us to verify that U_L(0,t) is the inverse of U_R(0,t) in a new way. First we note that U_L(0,t) U_R(0,t) = 1 is true for t=0. Then differentiating with respect to time we find that,

= 0, ∀t. (8.16)

As the result is true at t=0, and the time derivative of each side of the equation is true for all time, the result is true for all time.