The evolution of the phase space vector 
that we have been
examining via the classical propagator:
might appear to be of only formal significance because we cannot
evaluate the action of 
for a general Liouville operator.
The difficulty in evaluating the action of can be
seen by noting that iL is of the form
where
Thus, for any phase space function, f(x,p), it follows that
Therefore, we see that 
or
The term in parentheses is known as the commutator between 
and 
and is denoted
Thus, 
, so that the two operators do not commute with each other.
Unfortunately, this makes classical mechanics difficult. If we could
split the classical propagator according to 
,
everything would be simple because we can evaluate the action of the two operators
on the right on a phase space vector . However, because and
do not commute, the classical propagator cannot be factorized in this
simple way. This can be easily verified by Taylor expanding both sides:
However,
Fortunately, there is a theorem that we can use to factorize the classical propagator. This is the Trotter theorem, which states that
Although the proof of the theorem is somewhat beyond the scope of these lectures, it can be found in several places, in particular, see the appendix to Chapter 1 of Techniques and Applications of Path Integration by L. S. Schulman.
How does the Trotter theorem help us? Consider the approximation to
obtained by choosing M large but finite:
which can be rearranged to yield
The first of these expressions looks like approximate propagation of the
system up to a time t by M applications of the operator in brackets.
The second, therefore, looks like an approximation to the propagator for
a small time interval t/M. If we interpret this as a single
time step, 
, then we have a single-time-step approximation
to the propagator of the form
The left side is the exact propagator, 
, for a time step, .
The right side represents an approximate propagator 
.
There are several things to note about . First,
. Thus, is
unitary and, therefore, preserves the time-reversibility of the
dynamics:
Second, by Taylor expansion, it can be shown that is accurate
to order 
:
where the last line indicates equivalence up to second order in . Thus,
