A notation which is common in quantum mechanics is to refer to the phase and distribution function propagators as right and left ordered exponentials (expR and expL) respectively. To exploit this notational simplification we introduce the time ordering operators TR and TL. The operator TR simply reorders a product of operators so that the time arguments increase from left to right. In this notation we write the p-propagator UR(0,t) as
(8.25)
Using the series expansion for the exponential this becomes
(8.26)
Taking this series term by term the first two terms are trivial. We will consider the second order term in some detail.
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(8.27)
The time arguments in the first integral are time ordered from left to right so the operator will have no effect. In the second integral the order of the integrations can be interchanged to give
(8.28)
The second form is obtained by relabelling the dummy variables s1 and s2. Now both integrals have the same integration limits, and after the operation of TR both integrands are the same, so the second order term is
Using exactly the same steps we can show that each of the higher order terms are the same as those in the original representation of UR(0,t). After manipulating the integrals to obtain the same range of integration for each term of a particular order, the integrand is the sum of all permutations of the time arguments. At the nth order there are n! permutations, which after the operation of TR are all identical. This n! then cancels the (n!)-1 from the expansion of the exponential, and the result follows. Using the same arguments, the f-propagator UR †(0,t) also be written in this form
(8.29)
The use of the time ordering operator can realise considerable simplifications in many of the proofs that we have given.
Using time ordered exponentials, the composition theorem can be derived quite easily.
(8.30)
Because the exponentials are already right ordered we can write them as,
(8.31)