In this section we develop the operator algebra that we will need to manipulate expressions containing Liouvilleans and their associated propagators. Most of the identities which we obtain are valid for arbitrary time independent operators. Thus far we have been dealing with propagators in the time domain. For many problems it is more useful to consider their frequency dependent Laplace, or Fourier-Laplace, transforms. A useful mathematical object is the Laplace transform of the propagator. This is called the resolvent. The resolvent is an operator in the domain of the Laplace transform variable s,
(3.91)
Our first operator identity is obtained by considering two arbitrary operators A and B,
(3.92)
This identity is easily verified by operating from the right-hand side of this equation with (A+B), so
(3.93)
The operator expression (A+B)-1 is the inverse of the operator (A+B). To interpret an operator inverse of (A+B)-1, we use the series expansion
(3.94)
First we prove that the right-hand side of this expression is indeed the inverse of the operator (I+A). To do this consider
(3.95)
so that this series expansion allows us to represent the inverse of (I+A) in terms of an infinite series of products of the operator A.
Now we can investigate the Laplace transform (or resolvent) of the exponential of an operator in more detail. We use the expansion of the exponential to show that
(3.96)
This means that the resolvent of the operator, e-At, is simply (s+A)-1. We can now consider the resolvent derived from the operator (A+B), and using the first identity above, relate this resolvent to the resolvent of A. We can write
(3.97)
Substituting the Laplace integrals for the operators (s+A)-1 and (s+A+B)-1 into this equation gives
(3.98)
As the equality holds for all values of s, the integrands must be equal, so
(3.99)
This result is a very important step towards understanding the relationship between different propagators and is referred to as the Dyson decomposition when applied to propagators (Dyson, 1949). The derivation that we have used here is only valid if both of the operators A and B have no explicit time dependence. (We consider the time dependent case in Chapter 8.) If we consider the propagators exp((A+B)t) and exp(At), then a second Dyson decomposition can be obtained:
(3.100)
It is handy to use a graphical shorthand for the Dyson equation. Using this shorthand notation these two equations become,
⇐ = ← - ⇐ (◆ - o) ← (3.101)
and
⇒ = → + ⇒ (◆ - o) → (3.102)
The diamond ◆ denotes the (A+B)-Liouvillean and the circle o denotes the A-Liouvillean; the arrows ⇐ and ⇒ denote the propagators exp(-(A+B)t) and exp((A+B)t) respectively, while ← and → denote exp(-At) and exp(At) respectively. An n-1 fold convolution is implied by a chain of n arrows.
As an example of the application of this result, consider the case where B is a small perturbation to the operator A. In this case the Dyson decomposition gives the full (A+B)-propagator as the sum of the unperturbed A-propagator plus a correction term. One often faces the situation where we want to compare the operation of different propagators on either a phase variable or a distribution function. For example one might like to know the difference between the value of a phase variable A(Γ) propagated under the combined influence of the N-particle interactions and an applied external field Fe, with the value the phase variable might take at the same time in the absence of the external field. In that case (Evans and Morriss, 1984)
A(t,Fe) = ⇒ A(Γ)
= [→ + → (◆ - o) → + → (◆ - o) → (◆ - o) →
+ → (◆ - o) → (◆ - o) → (◆ - o) →
+ → (◆ - o) → (◆ - o) → (◆ - o) → (◆ - o) →
+ ................. ] A(Γ)
Therefore we can write,
(3.103)
This equation is of limited usefulness because in general, ◆ and →, do not commute. This means that the Liouvillean frequently represented by ◆, is locked inside a convolution chain of propagators with which it does not commute. A more useful expression can be derived from (3.102) by realising that ◆ commutes with its own propagator namely, ⇒. Similarly o commutes with its own propagator, →. We can 'unlock' the respective Liouvilleans from the chain in (3.102) by writing,
⇒ = → + ◆ ⇒ → - ⇒→ o (3.104)
We can recursively substitute for ⇒, yielding,
⇒ = → + ◆ → → - →→ o
+ ◆ ◆ → →→ - 2◆ → →→ o + →→→ o o
+ ........ (3.105)
Now it is easy to show that,
(3.106)
Thus (3.105) can be written as,
⇒ = {1 + t (◆ - o) + ( t2 /2!) (◆ ◆ - 2 ◆ o + o o)
+ ( t3 /3!) (◆ ◆ ◆ - 3 ◆ ◆ o + 3 ◆ o o - o o o )
+ ........}→ (3.107)
This equation was first derived by Evans and Morriss (1984). Its utility arises
from the fact that by 'unrolling' the Liouville operators to the left and the
propagator to the right, explicit formulae for the
expansion can usually be derived. A limitation of the formula is that successive
terms on the right hand side do not constitute a
power series expansion of the difference in the two propagators in powers of the
difference between the respective Liouvilleans. To be more explicit, the term, 
(◆ ◆ ◆ - 3 ◆ ◆ o + 3 ◆
o
o - o
o
o ) is not in
general of order (◆ - o)3.
If A and B are non commuting operators then the operator expression exp(A)exp(B) can be written in the form exp(C) where C is given by
(3.108)
The notation [,] is the usual Quantum Mechanical commutator. A rearrangement of this expansion, known as the Magnus expansion is well known to quantum theorists (Magnus, 1954). Any finite truncation of the Magnus expansion for the time displacement operator, gives a unitary time displacement operator approximation (Pechukas and Light, 1966). This result has not proved as useful for nonequilibrium statistical mechanics as it is for quantum theory. We give it here mainly for the sake of completeness.